Applied Analysis Seminar
Questions or comments?

Posted October 14, 2003

3:30 pm - 4:30 pm Tuesday, September 23, 2003 Lockett 240
Stephen Shipman, Mathematics Department, LSU

Boundary projections and Helmoltz resonances 1

Applied Analysis Seminar
Questions or comments?

Posted October 14, 2003

3:30 pm - 4:30 pm Tuesday, September 30, 2003 Lockett 240
Stephen Shipman, Mathematics Department, LSU

Boundary projections and Helmoltz resonances 2

Applied Analysis Seminar
Questions or comments?

Posted October 23, 2003

3:30 pm Lockett 240
Harris Wong, Department of Mechanical Engineering

A d-function model of facets and its applications

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett 240
Stephanos Venakides, Department of Mathematics, Duke University

The Semiclassical Limit of the Focusing Nonlinear Schroedinger Equation

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett 240
Robert Lipton, Mathematics Department, LSU

Field Fluctuations, Spectral Measures, and Moment Problems

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

Last modified November 6, 2003

Christo Christov, University of Louisiana at Lafayette

Nonlinear Waves and Quasi-Particles: The Emerging of a New Paradigm

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

Last modified November 6, 2003

Karsten Thompson, Department of Chemical Engineering, Louisiana State University

Modeling Multiple-scale Phenomena in Porous Materials

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett 240
Endel Iarve, Materials Directorate Wright Patterson Air Force Base and the University of Dayton Research Institute, Dayton Ohio

Mesh-independent modeling of cracks by using higher-order shape functions

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm The Deans conference room 3225, CEBA
Endel Iarve, Materials Directorate Wright Patterson Air Force Base and the University of Dayton Research Institute, Dayton Ohio

Effect of splitting on tensile strength distribution of unidirectional carbon fiber composites

Special Civil Engineering and Applied Analysis Seminar

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

2:00 pm Lockett 285
Boris Baeumer, University of Otago, New Zealand

Fractal Transport and Dispersion: Limits of Continuous Time Random Walks

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett 240
Ricardo Estrada, Mathematics Department, LSU

Distributional Solutions of Singular Integral Equations

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett 240
Wilfrid Gangbo, Department of Mathematics, Georgia Institute of Technology

Inequalities for generalized entropy and optimal transportation

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett 240
Yitshak Ram, Department of Mechanical Engineering, Louisiana State University

Inverse Problems and Eigenvalue Assignment in Vibration and Control

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

Lockett 240
Yuri Antipov, Mathematics Department, LSU

Functional-difference equations and applications

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett 240
Jay Walton, Department of Mathematics, Texas A & M University

Dynamic Fracture Models in Viscoelasticity

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett 240
Manuel Tiglio, Department of Physics, Louisiana State University

Summation by parts and dissipation for black hole excision

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett 240
Stephen McDowall, Department of Mathematics, Western Washington University Priklonsky

Total boundary determination of electromagnetic material parameters from boundary data

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett Hall
Oscar Bruno, Department of Applied and Computational Mathematics, California Institute of Technology

New high-order, high-frequency methods in computational electromagnetism

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett Hall 240
Mayank Tyagi, Mechanical Engineering Department, Louisiana State University.

Issues in Large Eddy Simulations of Complex Turbulent Flows

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett Hall 240
Vladimir Priklonsky, Moscow State University

Tidal Flow and Transport Model

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett Hall 240
Paul Martin, Department of Mathematical and Computer Science, Colorado School of Mines, Golden

Fundamental solutions and functionally graded materials

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:00 pm Lockett Hall 240
Boris Belinskiy, Department of Mathematics, University of Tennessee at Chattanooga

Boundary Value Contact Problems

Applied Analysis Seminar
Questions or comments?

Posted October 24, 2003

3:30 pm Lockett Hall 240
Jannette Frandsen, Department of Civil & Environmental Engineering, LSU

A Tuned Liquid Damper Model for Frequency Response Predictions of a Coupled System

Applied Analysis Seminar
Questions or comments?

Posted August 20, 2003

Last modified September 17, 2003

Gunter Lumer, University of Mons-Hainaut and Solvay Institute for Physics and Chemistry, Brussels

Multiparameter dynamics in macrophysics of clouds on flat and general surfaces, or in certain supply-management aspects

Applied Analysis Seminar
Questions or comments?

Posted September 4, 2003

Last modified September 17, 2003

Jung-Han Kimn, Mathematics Department, LSU

Overlapping Domain Decomposition Methods

Applied Analysis Seminar
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Posted September 4, 2003

Last modified September 23, 2003

Blaise Bourdin, Department of Mathematics and Center for Computation & Technology, LSU

Brittle fracture seen as a free discontinuities problem

Applied Analysis Seminar
Questions or comments?

Posted September 17, 2003

4:00 pm - 5:00 pm 277, Lockett Hall
Horst Beyer, Max Planck Institute for Gravitational Physics, Golm, Germany, and Dept. of Mathematics, LSU

On the Stability of the Kerr Black Hole

Applied Analysis Seminar
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Posted September 23, 2003

4:00 pm - 5:00 pm 277, Lockett Hall
Olivier Sarbach, Dept. of Mathematics and Dept. of Physics & Astronomy, LSU

The initial-boundary value formulation of Einstein's equations

Applied Analysis Seminar
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Posted August 14, 2003

Last modified October 23, 2003

Gilles Francfort, Université Paris Nord, France

Brittle fracture evolution: a variational standpoint.

Applied Analysis Seminar
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Posted August 14, 2003

Last modified November 4, 2003

Yonggang Huang, Dept. of Mechanical Engineering, University of Illinois at Urbana-Champaign

The fundamental solution of intersonic crack propagation

Applied Analysis Seminar
Questions or comments?

Posted October 28, 2003

4:00 pm - 5:00 pm 235, Lockett Hall
Andrej Cherkaev, University of Utah

TBA

To be followed by a "Special Fluid Dynamic" seminar at the Chimes.

Applied Analysis Seminar
Questions or comments?

Posted August 27, 2003

Last modified September 18, 2003

Andri Gretarsson, California Institute of Technology and LIGO livingston observatory

Detecting Gravitational Waves

Applied Analysis Seminar
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Posted October 23, 2003

Last modified October 27, 2003

Peter Y Huang, LSU, Department of Mechanical Engineering

Direct Numerical Simulation of Multiphase Flows in Newtonian and Non-Newtonian Fluids

Applied Analysis Seminar
Questions or comments?

Posted November 17, 2003

4:00 pm - 5:00 pm 277, Lockett Hall
Darko Volkov, Department of Mathematical Sciences, New Jersey Institute of Technology

Integral equation methods for the statics and the dynamics of an
electrified fluid bridge

Applied Analysis Seminar
Questions or comments?

Posted January 8, 2004

Last modified January 22, 2004

Petr Kloucek, Computational and Applied Mathematics department, Rice University

Stochastic Modeling of the Functional Crystalline Materials

Applied Analysis Seminar
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Posted January 22, 2004

Last modified February 11, 2004

Vladimir Mityushev, Institute de Physique du Globe de Paris (France), and Pedagogical University in Slupsk (Poland)

Effective properties of composites with unidirectional cylindrical fibers

Applied Analysis Seminar
Questions or comments?

Posted January 9, 2004

Last modified February 16, 2004

Marcus Sarkis, Institito de Matematica Pura e Aplicada (IMPA, Brazil) and Worcester Polytechnic Institute

Schwarz Methods for Partial Differential Equations

Applied Analysis Seminar
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Posted January 14, 2004

Last modified March 1, 2004

Ricardo Estrada, Mathematics Department, LSU

Distributional Radius of Curvature

We show that any continuous plane path that turns to the left has a

well-defined distribution, that corresponds to the radius of curvature of

smooth paths. As a byproduct, we will learn to divide by 0! These ideas were

inspired by a talk by Professor H. Wong in the Applied Analysis Seminar some

months ago, where he showed how to use Dirac delta functions to model facets

in crystals.

Applied Analysis Seminar
Questions or comments?

Posted February 3, 2004

Last modified March 10, 2004

Gregory Kriegsmann, New Jersey Institute of Technology

Complete Transmission Through a Two-Dimensional Diffraction Grating

Abstract: The propagation of a normally incident plane electromagnetic wave through

a two-dimensional metallic grating, is modeled and analyzed using S-Matrix

theory. The period of the structure $A$ is on the order of the incident wave

length $lambda$, but the height of the channel $H$ separating the grating

elements is very small in comparison. Exploiting the small parameter $H/A$

an approximate transmission coefficient is obtained for the grating. For a

fixed frequency this coefficient is $O(H/A)$ due to the thinness of the channel.

However, near resonant lengths it is $O(1)$. That is, for certain widths

the structure is transparent. Similarly, for a fixed length the transmission

coefficient has the same resonant features as a function of frequency. This latter

feature makes this grating potentially useful as a selective filter.

Applied Analysis Seminar
Questions or comments?

Posted February 11, 2004

Last modified March 15, 2004

John Strain, University of California Berkeley

High-order fractional step methods for constrained differential equations

Applied Analysis Seminar
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Posted April 12, 2004

3:40 pm - 4:30 pm 235 Lockett Hall
Horst Beyer, Max Planck Institute for Gravitational Physics, Golm, Germany, and Dept. of Mathematics, LSU

On some vector analogues of Sturm-Liouville operators

Abstract: The talk considers a general class of densely defined, linear symmmetric operators in Hilbert space, which originate from the separation of vector partial differential operators (PDO) in three dimensions, which are invariant under the rotation group. Those PDO describe spheroidal Lagrangian adiabatic oscillations of spherically symmetric newtonian stars (treated as ideal fluids) in the so called `Cowling approximation' in stellar pulsation theory. Their extension properties turn out to be very similar to that of minimal Sturm-Liouville operators. In particular close analogues of Weyl's famous theorems hold. On the other hand the spectral properties of their self-adjoint extensions are quite different. In particular every extension has a non-trivial essential spectrum. Finally, a result is given which allows to determine the resolvent of the self-adjoint extensions, which are perturbed by a `matrix' of integral operators of a specific general type. Those perturbed operators are generalizations of operators governing spheroidal adiabatic oscillations of spherically symmetric stars.

Applied Analysis Seminar
Questions or comments?

Posted January 9, 2004

Last modified April 13, 2004

Guillermo Goldsztein , School of Mathematics, Georgia Institute of Technology

Perfectly plastic heterogeneous materials

Applied Analysis Seminar
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Posted March 1, 2004

Last modified April 16, 2004

Daniel Sage, Mathematics Department, LSU

Racah coefficients, subrepresentation semirings, and composite materials--An application of representation theory to material science

Applied Analysis Seminar
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Posted April 26, 2004

Last modified May 3, 2004

Helena Nussenzveig Lopes, Universidade Estadual de Campinas (Brasil) and Penn State University

On vortex sheet evolution

Visit supported by Visiting Experts Program in Mathematics, Louisiana

Board of Regents. LEQSF(2002-04)-ENH-TR-13

Abstract

Applied Analysis Seminar
Questions or comments?

Posted September 10, 2004

Last modified October 13, 2004

Enrique Reyes, University of New Orleans

Practical finite difference modeling approaches to environmental problems: Louisiana coastal land loss.

Applied Analysis Seminar
Questions or comments?

Posted October 13, 2004

3:30 pm 239, Lockett Hall
Stephen Shipman, Mathematics Department, LSU

Anomalous electromagnetic transmission mediated by guided modes

Applied Analysis Seminar
Questions or comments?

Posted October 26, 2004

3:30 pm 239, Lockett Hall
Jonathan Dowling, Louisiana State University, Department of Physics
Hearne Professor of Theoretical Physics at LSU,
Quantum Sciences and Technologies Group

Effective densities of state

Applied Analysis Seminar
Questions or comments?

Posted October 26, 2004

3:30 pm - 4:30 pm 239, Lockett Hall
Petr Plechak, Mathematics Institute, University of Warwick
Candidate for Associate Professor Position in Scientific Computation

TBA

Applied Analysis Seminar
Questions or comments?

Posted October 3, 2004

Last modified October 26, 2004

Michel Jabbour, University of Kentucky

TBA

Applied Analysis Seminar
Questions or comments?

Posted October 27, 2004

Last modified January 27, 2005

Asher Rubinstein, Department of Mechanical Engineering, Tulane University

Failure Analysis of Thermal Barrier Coatings

Applied Analysis Seminar
Questions or comments?

Posted October 26, 2004

Last modified January 27, 2005

Susanne Brenner, Department of Mathematics, University of South Carolina

Additive Multigrid Theory

Applied Analysis Seminar
Questions or comments?

Posted October 26, 2004

Last modified February 14, 2005

Béatrice Rivière, Department of Mathematics, University of Pittsburgh

Discontinuous Galerkin methods for incompressible flows

Applied Analysis Seminar
Questions or comments?

Posted February 9, 2005

Last modified March 31, 2005

Alexander Figotin, University of Californina at Irvine

Conservative extensions of dispersive dissipative systems

Applied Analysis Seminar
Questions or comments?

Posted March 31, 2005

3:30 pm - 4:30 pm Lockett 285
John Willis, Cambridge University
Fellow, Royal Society of London (FRS)

Radon transforms in Solid Mechanics

Applied Analysis Seminar
Questions or comments?

Posted April 26, 2005

Last modified April 27, 2005

Blaise Bourdin, Department of Mathematics and Center for Computation & Technology, LSU

From Geman and Geman to Mumford-Shah

This talk focuses on the issues raised by an apparently simple problem: extending Geman and Geman's weak-membrane model for the segmentation of signals to that of images. I will briefly describe the problems of image and signal segmentation, then present Geman and Geman's approach. I will illustrate the issue with its intuitive multi-dimensional extension. Then, I will present how one can derive the Mumford-Shah functional as the Gamma limit of a weak-membrane energy, and then extend it to the 2D and 3D cases. Time permitting, I will then

present numerical schemes based on the Mumford-Shah problem.

Applied Analysis Seminar
Questions or comments?

Posted September 29, 2005

Last modified October 6, 2005

Robert Lipton, Mathematics Department, LSU

Multi-scale Stress Analysis

Many structures are hierarchical in nature and are made up of substructures distributed across several length scales. Examples include aircraft wings made from fiber reinforced laminates and naturally occurring structures like bone. From the perspective of failure initiation it is crucial to quantify the load transfer between length scales. The presence of geometrically induced stress or strain singularities at either the structural or substructural scale can have influence across length scales and initiate nonlinear phenomena that result in overall structural failure. In this presentation we examine load transfer between length scales for hierarchical structures when the substructure is known exactly or only in a statistical sense. New mathematical objects dubbed macrostress modulation functions are presented that facilitate a quantitative description of the load transfer in hierarchical structures. Several concrete physical examples are provided illustrating how these quantities can be used to quantify the stress and strain distribution inside multi-scale structures. It is then shown how to turn the problem around and use the macrostress modulation functions to design graded microstructures for control of local stress.

Applied Analysis Seminar
Questions or comments?

Posted October 11, 2005

3:30 pm 241 Lockett Hall
Robert Lipton, Mathematics Department, LSU

Differentiation of G-limits and weak L-P estimates for sequences

Applied Analysis Seminar
Questions or comments?

Posted October 6, 2005

Last modified October 18, 2005

Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

On the distance between homotopy classes of $S^1$-valued maps

Abstract: Certain Sobolev spaces of $S^1$-valued functions can be written as a disjoint union of homotopy classes. The problem of finding the distance between different homotopy classes in such spaces is considered. In particular several types of one-dimensional and two-dimensional domains are studied. Lower bounds are derived for these distances. Furthermore, in many cases it is shown that the lower bounds are sharp but are not achieved.

Applied Analysis Seminar
Questions or comments?

Posted November 11, 2005

Last modified November 23, 2005

Corey Redd, Department of Mathematics, LSU

Capturing Deviation from Ergodicity at Different Scales

Many researchers are interested in the topics of ergodicity and mixing, and

more importantly in methods by which these quantities can be measured. As

these properties may register differently based upon the space under

observation, it is also important that any measure be able to be applied at

different scales. Up to now, an energy based measure (L-2 norm) has

traditionally been used to assess the ergodicity and/or mixing of a system.

This method is less than ideal in part due to its non-uniqueness and

difficulty with assessment on varying scales. I will present a Lagrangian

based, multiscale method for measuring ergodicity that will attempt to address

these issues.

This talk will begin with background information on ergodicity and mixing and

the relationship between the two. From the abstract definitions, I will

derive an equation that will measure ergodicity on multiple scales. Following

that, results will be presented from some initial computations of the metric

on several test maps. Finally, computational issues will be discussed that

are specific to measuring ergodicity, as well as in comparison to a mixing

measure.

Applied Analysis Seminar
Questions or comments?

Posted January 23, 2006

Last modified February 7, 2006

Robert B. Haber, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign

Space-time Discontinuous Galerkin Methods for Multi-scale Solid Mechanics

Applied Analysis Seminar
Questions or comments?

Posted March 8, 2006

Last modified March 21, 2006

Ken Mattsson, Center for Integrated Turbulence Simulations, Stanford University

Towards time stable and high order accurate schemes for realistic applications

For wave propagation problems, the computational domain is often large compared to the wavelengths, which means that waves have to travel long distances during long times. As a result, high order accurate time marching methods, as well as efficient high order spatially accurate schemes (at least 3rd order) are required. Such schemes, although they might be G-K-S stable (convergence to the true solution as delta x -> 0), may exhibit a non-physical growth in time, for realistic mesh sizes. It is therefore important to device schemes, which do not allow a growth in time that is not called for by the differential equation. Such schemes are called strictly (or time) stable. We are particularly interested in efficient methods with a simple data structure that parallelize easily on structured grids. High order accurate finite difference methods fulfill these requirements. Traditionally, a successful marriage of high order accurate finite difference and strict stability was a complicated and highly problem dependent task, especially for realistic applications. The major breakthrough came with the construction (Kreiss et al., in 1974) of non-dissipative operators that satisfy a summation by parts (SBP) formulation, and later with the introduction of a specific procedure (Carpenter et al., in 1994) to impose boundary conditions as a penalty term, referred to as the Simultaneous Approximation Term (SAT) method. The combination of SBP and SAT naturally leads to strictly stable and high order accurate schemes for well-posed linear problems, on rectangular domains. During the last 10 years, the methodology has been extended to handle complex geometries and non-linear problems. In this talk I will introduce the original SBP and SAT concepts, and further discuss the status today and the focus on future applications. In particular I will discuss some recent developments towards time stable and accurate hybrid combinations of structured and unstructured SBP schemes, making use of the SAT method.

Applied Analysis Seminar
Questions or comments?

Posted January 26, 2006

Last modified March 11, 2006

Yaniv Almog, Department of Mathematics, LSU

Boundary layers in superconductivity and smectic liquid crystals

Applied Analysis Seminar
Questions or comments?

Posted April 13, 2006

3:30 pm - 4:30 pm Lockett, 235
Anna Zemlyanova, Department of Mathematics, LSU

The problem on reinforcement and repair of a hole in a plate with a patch

It is known that holes in a thin plate create undesirable stress concentration and can lead to the formation of cracks from the edge of the hole. I will consider the mechanical problem of repair of the hole by a two-dimensional patch. This problem will be reduced to the system of three singular integral equations. Uniqueness of the solution of the system will be proved. Numerical results will be given for some particular cases.

Applied Analysis Seminar
Questions or comments?

Posted May 2, 2006

Last modified May 20, 2006

Gnana Bhaskar Tenali, Mathematics, Florida Institute of Technology

FIXED POINT THEOREMS IN PARTIALLY ORDERED METRIC SPACES AND APPLICATIONS

Abstract. I?ll talk about some recent progress made on fixed point theorems in partially ordered metric spaces. In particular, I will discuss

a fixed point theorem for a mixed monotone mapping in a metric space endowed with a partial order, using a weak contractivity

type of assumption. Besides including several recent developments, such a theorem can be used to investigate a large class of problems.

As an application we discuss the existence and uniqueness of solution for a periodic boundary value problem.

Applied Analysis Seminar
Questions or comments?

Posted June 20, 2006

Last modified July 10, 2006

Mathias Stolpe, Institut for Mathematik, Danmarks Tekniske Universitet

A method for global optimization of the stacking sequence in laminated composite shell structures

Applied Analysis Seminar
Questions or comments?

Posted July 20, 2006

Last modified July 21, 2006

Fernando Fraternali, California Institute of Technology and Università di Salerno

Free Discontinuity Approaches to Fracture and Folding

Applied Analysis Seminar
Questions or comments?

Posted November 1, 2006

Last modified November 10, 2006

Blaise Bourdin, Department of Mathematics and Center for Computation & Technology, LSU

Numerical implementation of a variational model of brittle fracture

Abstract: Fracture mechanics is a very active area of research, with vital applications. In recent years, the unexpected collapse of terminal 2F at Charles de Gaulle airport in France or the Columbia space shuttle disintegration upon re-entry illustrate the importance of a better understanding and numerical simulation of the mechanism of fracture. In the area of brittle fracture, the most widely accepted theories are based on Griffith’s criterion and limited to the propagation of an isolated, pre-existing crack along a given path. Extending Griffith’s theory into a global minimization principle, while preserving its essence, the concept of energy restitution in between surface and bulk terms, G. Francfort and J.-J. Marigo proposed a new formulation for the brittle fracture problem. The basis of their model is the minimization of a total energy with respect to any admissible displacement and crack field. The main advantage of this approach is to be capable of predicting the initiation of new cracks, computing their path, and accounting the interactions betwenn several cracks, in two and three space dimensions. Of course, this has a price both theoretically and numerically. In particular, in order to acheive global minimization with respect to any crack set, one has to devise special numerical methods. After briefly reviewing the issues of brittle fracture mechanics, I will present the Francfort-Marigo model. I will rapidly describe some elements of its analysis, and present a numerical approximation based on the properties of Gamma-convergence. I will derive necessary optimality condiions with respect to the global time evolution, and show how to use them in a minimization algorithm. Then, I will present some extensions of the original model, accounting for body forces (under some restrictions) or thermal loads, and describe how to adapt the numerical implementation. I will illustrate my talk with several large scale two and three dimensional experiments.

Applied Analysis Seminar
Questions or comments?

Posted November 1, 2006

Last modified November 19, 2006

Jung-Han Kimn, Mathematics Department, LSU

Parallel Implementation of Domain Decomposition Methods

Abstract: Many important problems from current industrial and academic research, including the numerical solution of partial differential equations, generate extremely large data sets beyond the capacity of single-processor computers. Parallel computation on multiple-processor super computers is therefore the key to increasing performance but efficient parallel algorithms for multiple-processor super computers with huge number of processors are still needed. Domain Decomposition methods comprise an important class of parallel algorithms that are naturally parallel and flexible in their application to a sweeping range of scientific and engineering problems. This talk gives a brief discussion of some issues when we implement parallel domain decomposition methods. We will present some of our recent theoretical and numerical results for parallel domain decomposition methods for elliptic and hyperbolic partial differential equations.

Applied Analysis Seminar
Questions or comments?

Posted October 11, 2006

Last modified November 27, 2006

Michael Mascagni, Department of Computer Science, Florida State University

Using Simple SDEs (Stochastic Differential Equations) to Solve Complicated PDEs (Partial Differential Equations)

This talk begins with an overview of methods to solve PDEs based on the representation of point solutions of the PDEs as expected values of functionals of stochastic processes defined by the Feynman-Kac formula. The particular stochastic processes that arise in the Feynman-Kac formula are solutions to specific SDEs defined by the characteristics of the differential operator in the PDE. The Feynman-Kac formula is applicable to wide class of linear initial and initial-boundary value problems for elliptic and parabolic PDEs. We then concentrate our attention on elliptic boundary value problems that arise in applications in materials science and biochemistry. These problems are similar in that the PDEs to be solved are rather simple, and hence the associated SDEs that arise in the Feynman-Kac formula are likewise simple. However, the geometry of the problem is often complicated and amenable to several acceleration approaches particular to these simple SDEs. We will specifically describe the walk on spheres, Greens function first passage, last passage, walk on the boundary, and walk on subdomains methods in this context. These methods will be presented in the setting of several applications studied by the author and his research collaborators.

Applied Analysis Seminar
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Posted November 1, 2006

Last modified December 3, 2006

Robert Lipton, Mathematics Department, LSU

Homogenization and Field Concentrations in Heterogeneous Media

Applied Analysis Seminar
Questions or comments?

Posted November 1, 2006

Last modified December 11, 2006

Michael Stuebner, Louisiana State University

An inverse homogenization approach to stress constrained structural design

The presentation addresses the problem of optimal design of microstructure in composite materials. A computational method for grading the microstructure for the control of local stress in the vicinity of stress concentrations is developed. The method is based upon new rigorous multiscale stress criteria connecting the macroscopic or homogenized stress to local stress fluctuations at the scale of the microstructure. The approach is applied to different type of design problems.

Applied Analysis Seminar
Questions or comments?

Posted January 29, 2007

Last modified February 1, 2007

Paul Saylor, University of Illinois

Stanford's Foresight and Forsythe's Stanford

Abstract:

What Stanford Was Like

What the Time Was Like

Over A Four Year Period

Starting with the Arrival of This New Man

Professor George Forsythe, In 1957

Plus A Bonus Look-Ahead to the Future

Applied Analysis Seminar
Questions or comments?

Posted January 29, 2007

11:00 am - 12:00 pm Johnston Hall Room 338
Fengyan Li, Rensselaer Polytechnic Institute

Recent development in nonconforming methods for Maxwell equations

In this talk, I will discuss some recent developments in computational electromagnetism. Two schemes are formulated for the reduced time-harmonic Maxwell equations. One is using the classical nonconforming finite elements, the other is based on the interior penalty type discontinuous Galerkin methods. The operators in these schemes naturally define two Maxwell eigensolvers which are spurious free. Theoretical and numerical results will be presented to demonstrate the performance of these methods. This is joint work with Susanne Brenner and Li-yeng Sung (LSU).

Applied Analysis Seminar
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Posted January 17, 2007

Last modified February 15, 2007

Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

On a minimization problem with a mass constraint involving a potential vanishing on two curves

We study a singular perturbation type minimization problem with a mass constraint over a domain or a manifold, involving a potential vanishing on two curves in the plane. We describe the asymptotic behavior of the energy as the parameter epsilon goes to zero, and in particular, how it depends on the geometry of the domain. In the case of the problem on the sphere we give a precise description of the limiting behavior of both the minimizers and their energies. This is a joint work with Nelly Andre (Tours).

Applied Analysis Seminar
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Posted January 29, 2007

3:40 pm - 4:30 pm 284 Lockett Hall
Alexander Pankov, College of William and Mary

Gap solitons, periodic NLS, and critical point theory

Abstract: Here a gap soliton means a spatially exponentially localized standing wave solution of periodic nonlinear Maxwell equations, having a carrier frequency in a spectral gap. There is an enormous literature devoted to study of what should be gap solitons by means of approximate methods, e.g., envelope function approach, and numerical simulations (basically, in one dimension). These results provide a lot of information about such solutions, say, their shape. However, the existence of gap solitons is not a clear issue. In this talk we discuss the existence problem in the case of periodic Akhmediev-Kerr medium. We consider two-dimensional case and look for (TM) polarized solutions. Then the problem reduces to a (two-dimensional) periodic stationary NLS with cubic nonlinearity. To study this equation we employ critical point theory (specifically, the linking theorem) together with the so-called periodic approximations. This leads to the existence of TM gap solitons and provides an estimate for the rate of exponential decay. Finally, we discuss certain open mathematical problems.

Applied Analysis Seminar
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Posted March 8, 2007

Last modified March 13, 2007

Hong Zhang, Dept. of Computer Science, Illinois Institute of Technology and Mathematics and Computer Science Division, Argonne National Laboratory

Eigenvalue Problems in Nanoscale Material Modeling

Together with a group of material scientist, we intend to calculate the atomic and electronic structure of nanoparticles on a quantum-mechanical level. The mathematical core of this modeling is a sequence of large and sparse eigenvalue problems. In this talk, I will present the special requirements of the solutions, the challenges on the computational method, our algorithmic approach and software development. Numerical implementation on the advanced distributed computers will be demonstrated.

This work also demonstrates how to efficiently develop special-purpose application code on the top of available parallel software packages. By the end of the talk, as a PETSc developer, I will give a demo on using PETSc (Portable, Extensible Toolkit for Scientific Computation) as a tool for large scale numerical simulation.

Applied Analysis Seminar
Questions or comments?

Posted February 12, 2007

Last modified April 23, 2007

Lia Bronsard, Mc Master University,

Ginzburg-Landau vortices concentrating on curves.

We study the two-dimensional Ginzburg-Landau functional for superconductivity and the related Gross-Pitaevskii functional for Bose-Einstein Condensate. In a convex simply-connected domain, Serfaty has shown that the vortices accumulate around a single point in the domain as the Ginzburg--Landau parameter $kappatoinfty$. Our previous papers (with Aftalion and Alama) on multiply connected domains show that vortices may instead accumulate on an appropriate curve as $kappatoinfty$. In our recent result with S. Alama and V. Millot, we study the number and distribution of these vortices along the curve of concentration. Their distribution is determined by a classical problem from potential theory.

Applied Analysis Seminar
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Posted August 16, 2007

Last modified August 22, 2007

Shinnosuke Oharu, Chuo University, Japan

Ecological models of red tide plankton in the coastal ocean.

Abstract: This talk will be concerned with a mathematical model consisting of an ecological model for a specific species of plankton and an ocean model, numerical models consistent with the PDE models, and computer simulations by means of new CFD methods.

Applied Analysis Seminar
Questions or comments?

Posted September 14, 2007

Last modified September 21, 2007

Édouard Oudet, Laboratoire de Mathématiques, Université de Savoie, France

Constant width bodies in dimension 3

A body (that is, a compact connected subset *K* of **R**^{n}) is said to be of *constant width α* if its projection on any straight line is a segment of length *α>0*, the same value for all lines.

We present in this talk a complete analytic parametrization of constant width bodies in dimension 3 based on the median surface: more precisely, we define a bijection between some space of functions and constant width bodies. We compute simple geometrical quantities like the volume and the surface area in terms of those functions. As a corollary we give a new algebraic proof of Blaschke's formula. Finally, we present some numerical computations based on the preceding parametrization.

É. Oudet will be visiting the department this week (9/24 - 9/28). If you want to schedule a meeting with him, contact B. Bourdin.

Applied Analysis Seminar
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Posted September 28, 2007

3:40 pm - 4:30 pm Lockett Hall 233
Burak Aksoylu, Department of Mathematics and CCT

Physics-based preconditioners for solving PDEs on highly heterogeneous media

Eigenvalues of smallest magnitude become a major bottleneck for iterative solvers especially when the underlying physical properties have severe contrasts. These contrasts are commonly found in many applications such as composite materials, geological rock properties and thermal and electrical conductivity.

The main objective of this work is to construct a method as algebraic as possible that could efficiently exploit the connectivity of highly heterogeneous media in the solution of diffusion operators. We propose an algebraic way of separating binary-like systems according to a given threshold into high- and low-conductivity regimes of coefficient size O(m) and O(1), respectively where m >> 1. The condition number of the linear system depends both on the mesh size Δ x and the coefficient size $m$. For our purposes, we address only the m dependence since the condition number of the linear system is mainly governed by the high-conductivity subblock. Thus, the proposed strategy is inspired by capturing the relevant physics governing the problem. Based on the algebraic construction, a two-stage preconditioning strategy is developed as follows: (1) a first stage that comprises approximation to the components of the solution associated to small eigenvalues and, (2) a second stage that deals with the remaining solution components with a deflation strategy (if ever needed). The deflation strategies are based on computing near invariant subspaces corresponding to smallest and deflating them by the use of recycled the Krylov subspaces.

Due to its algebraic nature, the proposed approach can support a wide range of realistic geometries (e.g., layered and channelized media). Numerical examples show that the proposed class of physics-based preconditioners are more effective and robust compared to a class of Krylov-based deflation methods on highly heterogeneous media. We also report on singular perturbation analysis of the stiffness matrix and the impact of the number of high-conductive regions on various matrices.

Applied Analysis Seminar
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Posted October 9, 2007

Last modified October 22, 2007

Michael Zabarankin, Stevens Institute of Technology

Generalized Analytic Functions in 3D Axially Symmetric Stokes Flows

Abstract

A class of generalized analytic functions, defined by a special case of

the Carleman system that arises from related potentials encountered in

various areas of applied mathematics has been considered. Hilbert

formulas, establishing relationships between the real and imaginary

parts of a generalized analytic function from this class, have been

derived for the domains exterior to the contour of spindle, lens,

bi-spheres and torus in the meridional cross-section plane. In

bi-spherical and toroidal coordinates, this special case of the Carleman

system has been reduced to a second-order difference equation with

respect to either the coefficients in series or densities in integral

representations of the real and imaginary parts. For spindle and lens,

the equation has been solved in the framework of Riemann boundary-value

problems in the class of meromorphic functions. For torus, the equation

has been solved by means of the Fourier transform, while for bi-spheres,

it has been solved by an algebraic method. As examples, analytical

expressions for the pressure in the problems of the 3D axially symmetric

Stokes flows about rigid spindle, biconvex lens, bi-spheres and torus

have been derived based on the corresponding Hilbert formulas.

Applied Analysis Seminar
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Posted October 4, 2007

3:40 pm - 4:30 pm Lockett 233
Thirupathi Gudi, CCT, LSU

Local Discontinuous Galerkin Methods for Elliptic Problems

Applied Analysis Seminar
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Posted October 17, 2007

Last modified November 27, 2007

Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

Global minimizers for a p-Ginzburg-Landau energy.

We study the problem of existence of global minimizers for a p-Ginzburg-Landau type energy on the plane and on the half-plane, for p>2, under a degree condition at infinity. We prove existence of a minimizer when the degree equals 1. This is joint work with Yaniv Almog, Leonid Berlyand and Dmitry Golovaty.

Applied Analysis Seminar
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Posted December 28, 2007

Last modified January 29, 2008

Peter Sternberg, Indiana University

Bifurcating solutions in a model for a superconducting wire
subjected to an applied current

Abstract: We study formally and rigorously the bifurcation to steady and time-periodic states in a model for a thin superconducting wire in the presence of an imposed current. Exploiting the PT-symmetry of the equations at both the linearized and nonlinear levels, and taking advantage of the collision of real eigenvalues leading to complex spectrum, we obtain explicit asymptotic formulas for the stationary solutions, for the amplitude and period of the bifurcating periodic solutions and for the location of their zeros or ``phase slip centers" as they are known in the physics literature. In so doing, we construct a center manifold for the flow and give a complete description of the associated finite-dimensional dynamics. This is joint work with Jacob Rubinstein and Kevin Zumbrun.

Applied Analysis Seminar
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Posted February 26, 2008

Last modified March 3, 2008

Razvan Teodorescu, Los Alamos National Laboratory

Harmonic Growth in 2D via Biorthogonal Polynomials

Evolution of planar domains (representing physical clusters) under harmonic forces is representative for many problems in mathematical physics. In certain situations, the evolution leads to finite-time singularities. I will discuss a regularization of this evolution inspired by the equilibrium distribution of eigenvalues of large random normal matrices. Connections to operator theory will also be discussed.

Applied Analysis Seminar
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Posted March 24, 2008

3:40 pm - 4:30 pm 233 Lockett Hall
Yuliya Gorb, Department of Mathematics Texas A&M University

Fictitious Fluid Approach for Justification of Asymptotics of Effective Properties of Highly Concentrated Suspensions

Abstract: The method of the discrete network approximation has been used for determining effective properties of high contrast disordered composites with particles close to touching. It is illustrated by considering a highly packed suspension of rigid particles in a Newtonian fluid. The effective viscous dissipation rate of such a suspension exhibits a singular behavior, and the goal is to derive and justify its asymptotic formula as a characteristic interparticle distance tends to zero. The main idea of the presented approach is a reduction of the original continuum problem described by partial differential equations with rough coefficients to a discrete network. This reduction is done in two steps which constitute the "fictitious fluid" approach. While previously developed techniques based on a direct discretization allowed to obtain only the leading singular term of asymptotics for special symmetric boundary conditions, we are able to capture all singular terms in the asymptotic formula of the dissipation rate for generic boundary conditions. The fictitious fluid approach also allows for a complete qualitative description of microflow in a thin gap between neighboring particles in the suspension.

Applied Analysis Seminar
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Posted February 25, 2008

Last modified April 21, 2008

Mikhail Stepanov, Department of Mathematics, The University of Arizona

Instantons in hydrodynamics

We consider the hydrodynamic type system (Navier-Stokes or Burgers

equation) with random forcing. The untypical events of high vorticity or

large velocity gradients are due to extreme realizations of the forcing.

To generate such an event one can increase the forcing amplitude or to

optimize its shape (without sacrificing the probability of such forcing to

happen). The tails of the velocity field probability distribution function

can be obtained by finding an optimal shape of forcing, which corresponds

to saddle point (instanton) approximation in the path integral describing

the velocity statistics. It will be shown how to find the instantons in

hydrodynamic systems numerically.

Applied Analysis Seminar
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Posted March 31, 2008

Last modified April 16, 2008

John W. Cain, Virginia Commonwealth University

A Kinematic Model for Propagation of Cardiac Action Potentials

Propagation of cardiac action potentials is usually modeled

with a reaction-diffusion equation known as the cable equation. However,

when studying the initiation of arrhythmias, one is primarily interested

in the progress of action potential wavefronts without regard to the

complete wave profile. In this talk, I will explain how to derive a

purely kinematic model of action potential propagation in cardiac tissue.

I will reduce a standard PDE model (the cable equation) to an infinite

sequence of ODEs which govern the progress of wave fronts in a repeatedly

stimulated fiber of cardiac tissue. The linearization of the sequence of

ODEs admits an exact solution, expressible in terms of generalized

Laguerre polynomials. Analyzing the solutions yields valuable insight

regarding nonlinear wave propagation in an excitable medium, providing

interesting physiological implications.

Applied Analysis Seminar
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Posted March 26, 2008

3:40 pm - 4:30 pm 233 Lockett Hall
Bogdan Vernescu, Worcester Polytechnic Institute

TBA

Applied Analysis Seminar
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Posted September 9, 2008

3:30 pm - 4:30 pm Monday, October 27, 2008 TBA
Dmitry Golovaty, University of Akron

TBA

Applied Analysis Seminar
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Posted September 13, 2008

Last modified October 15, 2008

Dmitry Golovaty, University of Akron

An effective model for ferronematic liquid crystals

I will discuss a nonlinear homogenization problem for ferronematics---colloidal suspensions of small ferromagnetic particles in a nematic liquid crystalline medium---in a regime when the volume fraction of weakly interacting particles is small. The energy of the suspension is given by a Ginzburg-Landau term supplemented by a Rapini-Papoular surface anchoring energy term and terms describing interaction between the suspension and the magnetic field. For a pure nematic, the energy density of interaction between the magnetic field and the nematic director is given by a quadratic term that is minimized when the director is parallel to the field. For a ferronematic, the additional, indirect coupling between the nematic and the field is introduced into the energy via anchoring of nematic molecules on the surfaces of the particles. Assuming that the particles are identical prolate spheroids with fixed positions but variable orientations, we use the method of quasisolutions to show that the influence of particles on the suspension can be accounted for by an effective nonlinear potential. For needle-like particles of large eccentricity, the model reduces to a known expression of Brochard and de Gennes. This is a joint work with C. Calderer, A. Desimone, and A. Panchenko.

Applied Analysis Seminar
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Posted October 12, 2008

Last modified October 27, 2008

Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

On a minimization problem involving a potential vanishing on two curves

This talk is concerned with a vector-valued singular perturbation problem involving a potential vanishing on two curves. We study the limiting behaviour of the minimizers, and demonstrate how it depends on the geometry of the domain. This is a joint work with Nelly Andre (University of Tours).

Applied Analysis Seminar
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Posted November 3, 2008

Last modified November 7, 2008

Phuc Nguyen, Department of Mathematics, Louisiana State University

Quasilinear and Hessian equations with super-critical exponents and singular data

Applied Analysis Seminar
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Posted November 19, 2008

Last modified December 5, 2008

Stephen Shipman, Mathematics Department, LSU

Field sensitivity to L^p perturbations of a scatterer

This will be an informal presentation as part of the weekly material science discussion group. I will discuss the title problem and some related problems I would like to solve.

Applied Analysis Seminar
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Posted February 11, 2009

3:40 pm - 4:30 pm
Monica Torres, Department of Mathematics, Purdue University

The structure of solutions of systems of hyperbolic conservation laws

Hyperbolic systems of conservations laws model many areas of physics, including fluid mechanics, acoustics, etc. One of the main challenges in the analysis of these equations is that solutions develop singularities even if the initial data is smooth. These singularities are known as shock waves. Existence theorems only show that entropy solutions belong to some $L^p$ space and satisfy an entropy inequality in the distributional sense. Therefore, an open problem is to study the structure of solutions and regularity of the shock waves. In this talk we present results in this direction, which include some Liouville-type results for systems of conservation laws.

Applied Analysis Seminar
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Posted January 20, 2009

Last modified April 17, 2009

David Dobson, Department of Mathematics, University of Utah

Electromagnetic transmission resonances in periodic hole arrays

Recently there has been increasing interest in terahertz-frequency electromagnetic radiation in the engineering community. Improved methods of generating such radiation has led to hopes of applications in communications, imaging, and spectroscopy. Unfortunately almost all materials are highly absorptive in the terahertz range, making device design difficult. One method of manipulating terahertz radiation is by filtering through thin, perforated metal plates. Such plates exhibit interesting, and sometimes unexpected transmission properties. The transmission spectrum depends strongly on both the hole pattern and the aperture shape. This talk will describe some work on developing a model for transmission through periodic hole arrays, including analysis and numerical methods. We conclude with some preliminary work on the problem of optimal design of aperture shape to produce a desired transmission spectrum.

Host: Stephen Shipman

Applied Analysis Seminar
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Posted March 13, 2009

Last modified March 15, 2009

Tadele Mengesha, Louisiana State University

TBA

Applied Analysis Seminar
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Posted April 13, 2009

3:40 pm - 4:30 pm Lockett 285
Tadele Mengesha, Louisiana State University

Sufficient conditions for smooth strong local minima

The talk addresses the conjecture that uniform quasiconvexity and uniform positivity of the second variation are sufficient for a smooth extremal to be a strong local minimizer. Our result holds for a class of variational integrals with integrands of polynomial growth at infinity. The proof is based on the decomposition of an arbitrary variation into its purely strong and weak parts. We show that these two parts act independently on the functional. The action of the weak part can be described in terms of the second variation. While the uniform positivity of the second variation prevents the weak part from decreasing the functional, the uniform quasiconvexity conditions prevent the strong part from doing the same. This is a joint work with Yury Grabovsky.

Applied Analysis Seminar
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Posted August 24, 2009

3:40 pm - 4:30 pm Room 233 Lockett Hall
Robert Lipton, Mathematics Department, LSU

Strength of Elastic - Plastic Composites Made From Random Configurations of Two Materials

Applied Analysis Seminar
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Posted September 6, 2009

3:30 pm - 4:30 pm Room 233 Lockett Hall
Yaniv Almog, Department of Mathematics, LSU

Superconductivity Near the Normal State in the presence of electric current.

We consider the linearization of the time-dependent Ginzburg-Landau near the normal state. We assume that an electric current is applied through the sample, which captures the whole plane, inducing thereby, a magnetic field. We show that independently of the current, the normal state is always stable. Using Fourier analysis the detailed behaviour of solutions is obtained as well. Relying on semi-group theory we then obtain the spectral properties of the steady-state elliptic operator. We shall also consider the spectral properties of the same elliptic operator near a flat wall, and obtain the critical current in the limit of small and large normal conductivity

Applied Analysis Seminar
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Posted September 14, 2009

3:40 pm - 4:30 pm Room 233 Lockett Hall
Scott Armstrong, Department of Mathematics, Louisiana State University

Self-similar solution and long-time asymptotics for fully nonlinear parabolic equations

I will present results on the existence and uniqueness of a self-similar solution of a fully nonlinear, parabolic equation (an example of which include the Bellman-Isaacs equation arising in the theory of stochastic optimal control and stochastic differential game theory). As an application, we are able to describe the long-time behavior of solutions to the Cauchy problem, and derive a conservation law which generalizes the conservation of mass in the case of the heat equation. The scaling invariance property of the self-similar solution depends on the nonlinear operator, and is in general different from that of the heat kernel. We will see that this difference has an interesting interpretation in terms of controlled diffusion processes. This work is joint with M. Trokhimtchouk.

Applied Analysis Seminar
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Posted August 27, 2009

Last modified September 23, 2009

Matthew Knepley, Computation Institute, University of Chicago

Tree-based methods on GPUs

Abstract: We examine the performance of the Fast Multipole Method on heterogeneous computing devices, such as a CPU attached to an Nvidia Tesla 1060C card. The inherent bottleneck imposed by the tree structure is ameliorated by a refactoring of the algorithm which exposes the fine-grained dependency structure. Also, common reduction operations are refactored in order to maximize throughput. These optimizations are enabled by our concise yet powerful interface for tree-structured algorithms. Examples of performance are shown for problems arising from vortex methods for fluids

Applied Analysis Seminar
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Posted September 14, 2009

3:40 pm - 4:30 pm Room 233 Lockett Hall
Christo Christov, University of Louisiana at Lafayette

Stochastic Functional Expansions for Heterogeneous Continuous Media and Chaotic Regimes of Nonlinear Dynamical Systems

Applied Analysis Seminar
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Posted September 20, 2009

Last modified September 22, 2009

Truyen Nguyen, University of Akron

Hamilton--Jacobi equation in the space of measures associated with a system of conservation laws

Abstract. We introduce a class of action functional defined over the set of continuous paths in the Wasserstein space of probability measures on $R^d$. We show that minimizing path for such action exists and satisfies compressible Euler equation in a weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton-Jacobi equations are well--posed and their unique viscositysolutions are given by the dynamic programming principle. The characteristics of these Hamilton--Jacobi equations in the space of probability measures are solutions of the compressible Euler equation in $R^d$. This is joint work with Jin Feng of the University of Kansas.

Applied Analysis Seminar
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Posted September 8, 2009

Last modified October 23, 2009

Rachael Neilan, Department of Oceanography and Coastal Sciences, LSU

Optimal control in disease modeling

Optimal control theory in disease models is used to determine cost- effective disease prevention and treatment strategies. When disease dynamics are governed by ordinary differential equations, Pontryagin's Maximum Principle is used to characterize an optimal control (i.e. optimal treatment strategy). However, many disease models use partial differential equations to describe the spread of infection in space and time. No extension of Pontryagin's Maximum Principle exists for systems of PDEs, but similar techniques are employed to derive optimal spatio-temporal control characterizations. In this talk, we will provide theoretical optimal control results for a system of advection- diffusion equations describing the spread of rabies through a raccoon population. Numerical solutions will illustrate the optimal vaccine distribution on homogeneous and heterogeneous spatial domains.

Applied Analysis Seminar
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Posted September 30, 2009

Last modified October 23, 2009

Scott McKinley, Department of Mathematics, Duke University

Anomalous Diffusion of Distinguished Particles in Bead-Spring Networks. (This is a joint Applied Analyisis & Probability Seminar)

Abstract: Due to recent and compelling experimental observations using passive microrheology there is theoretical interest in anomalous sub-diffusion -- stochastic processes whose long-term mean-squared displacement satisfies $E{x2(t)} sim t^nu$ where $nu leq 1$. The canonical example of a sub-diffusive process is fractional Brownian motion, but for reasons we will discuss, this project focuses on a touchstone model from polymer kinetic theory -- the Rouse chain -- and its natural generalizations. Our interest is in studying the dynamics of a distinguished particle in a network of thermally fluctuating beads that interact with each other through linear springs. Such processes can be expressed as the sum of a Brownian motion with a large number of Ornstein-Uhlenbeck processes. We introduce a single parameter which can be tuned to produce any sub-diffusive exponent $nu in (0,1)$ for the generic sum-of-OU structure and demonstrate the relationship between this parameter and the geometric structure of the bead-spring connection network in which the distinguished particle resides. This development provides a basis to prove a conjecture from the physics community that the Rouse exponent $nu = 1/2$ is universal among a wide class of models.

Applied Analysis Seminar
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Posted September 14, 2009

Last modified October 21, 2009

Fadil Santosa, Director, Institute for Mathematics and its Applications and School of Mathematics, University of Minnesota

The mathematics of progressive lens design

Abstract: Progressive addition lenses are prescribed to patients who need correction for both far and near visions. A progressive lens needs to have power that gradually changes from the far vision zone, used for example in driving, and the near vision zone, used for example in reading a map. The basics of optics and lens design will be described. In particular, it will be shown that the problem can be reduced to one of surface design. The surface design problem itself is solved by a variational approach, which can be further simplified by linearization, leading to a fourth order elliptic partial differential equations. Analysis of the resulting equations and development of a computational method are described. Numerical results are presented to illustrate the process of lens design.

Applied Analysis Seminar
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Posted September 30, 2009

Last modified November 10, 2009

Xiaoliang Wan, Louisiana State University

A note on stochastic elliptic models

In this talk we will look at two strategies that introduce randomness into elliptic models. One is to treat the coefficient as an spatial random process, which results in an stochastic elliptic model widely used in engineering applications; the other one is to define the stochastic integral using Wick product, which can be regarded as a generalization of Ito integral. The statistics given by these two strategies can be dramatically different. I will compare these two strategies using a one-dimensional problem and present a new stochastic elliptic model to makes them more comparable. Numerical methods will also be discussed.

Applied Analysis Seminar
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Posted November 17, 2009

3:40 pm - 4:30 pm Room 233 Lockett Hall
Santiago Fortes, Department of Mathematics, LSU

Electromagnetic wave propagation in Plasmonic Crystals

The possibility of engineering composite materials with unusual electromagnetic properties (a.k.a. metamaterials) has generated much interest lately. Devices such as invisibility cloaks and perfect lenses could, in principle, be constructed using such materials. I will present a method for obtaining convergent power series representations for the fields and associated dispersion relations of electromagnetic waves propagating in a species of metamaterial known as plasmonic crystal. The technology provided by these series lead to a rich scenario in which to explore effective properties in a mathematically rigorous setting. This has allowed us give definite answers regarding the negative index behavior of plasmonic crystals.

Applied Analysis Seminar
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Posted January 19, 2010

3:40 pm - 4:30 pm Room 233 Lockett Hall
Anna Zemlyanova, Department of Mathematics, LSU

Method of Riemann surfaces in modelling of cavitating flow

Cavitation is the formation of a vapor filled area in the liquid which usually appears due to low pressures and high velocities. Riemann surfaces are used in fluid mechanics both for mathematical modeling of the cavity closure and for solution of the resulting mathematical problems. In this talk I will discuss most commonly used cavity closure models and present a detailed solution to the problem of a supercavitating wedge in a jet or under a free surface using Tulin's single- or double-spiral-vortex cavity closure model. The solution involves the application of Riemann-Hilbert problems on the elliptic Riemann surface. The numerical results will be presented.

Applied Analysis Seminar
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Posted January 21, 2010

Last modified January 29, 2010

Phuc Nguyen, Department of Mathematics, Louisiana State University

Capacitary inequalities and quasilinear Riccati type equations with critical or super-critical growth

An abstract is available.

Applied Analysis Seminar
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Posted January 25, 2010

Last modified February 27, 2010

Alexander Barnett, Department of Mathematics, Dartmouth College

Robust and accurate computation of photonic crystal band structure using a new integral equation representation of quasi-periodic fields

Host: Stephen Shipman

Photonic crystals are dielectric structures with periodicity on the scale of the wavelength of light. They have a rapidly growing range of applications to signal processing, sensing, negative-index materials, and the exciting possibility of integrated optical computing. Calculating their `band structure' (propagating Bloch waves) is an elliptic PDE eigenvalue problem with (quasi-)periodic boundary conditions on the unit cell, i.e. eigenmodes on a torus. Since the material is piecewise homogeneous, boundary integral equations (BIE) are natural for high-accuracy solution.

In such geometries BIEs are usually periodized by replacement of the free space Greens function kernel by its quasi-periodic cousin. We show why this approach fails near the (spurious) resonances of the empty torus. We introduce a new approach which cures this problem: imposing the boundary conditions on the unit-cell walls using layer potentials, and a finite number of neighboring images, resulting in a second-kind integral equation with smooth data. This couples to existing BIE tools (including high-order quadratures and Fast Multipole acceleration) in a natural way, allowing accuracies near machine precision. We also discuss inclusions which intersect the unit cell walls, and how we use a small number of evaluations to interpolate over the Brillouin zone to spectral accuracy. Joint work with Leslie Greengard (NYU).

Applied Analysis Seminar
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Posted January 18, 2010

Last modified February 10, 2010

Diego Maldonado, Kansas State University

Bilinear pseudo-differential operators: motivations and recent developments

Abstract: During the 70's, driven by some problems posed by A. Calderon, R. Coifman and Y. Meyer pioneered a theory of bilinear pseudo-differential operators. These operators later found further applications in topics of analysis and PDEs such as compensated compactness, regularity of solutions to PDEs, boundedness properties of commutators, bilinear singular integrals, and paraproducts, and pointwise multipliers for functional spaces. Departing from the definition of the Fourier transform, in this talk we will tour the theory of bilinear pseudo-differential operators and some of its applications to finally arrive at the latest results and some open problems.

Applied Analysis Seminar
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Posted March 21, 2010

Last modified April 11, 2010

Peter Wolenski, LSU Department of Mathematics
Russell B. Long Professor

Impulsive systems

Abstract: An impulsive system is a dynamical system that may exhibit "jumps" in the state variable. We shall introduce a model of such systems driven by a measure, and discuss solution concepts and recent results. A model of synaptic dynamics will be given as an example, which has been introduced in the neuroscience literature to describe neuronal population activity.

Applied Analysis Seminar
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Posted February 27, 2010

Last modified April 7, 2010

Vladislav Kravchenko, Dept. of Mathematics, CINVESTAV del IPN, Unidad Querétaro

Solution of boundary and eigenvalue problems for second order elliptic operators in the plane using pseudoanalytic function theory

Abstract: We propose a method for solving boundary value and eigenvalue problems for the elliptic operator D=divpgrad+q in the plane using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be constructed following a simple algorithm consisting in recursive integration. This system of solutions is used for solving boundary value and spectral problems for the operator D in bounded simply connected domains. We study theoretical and numerical aspects of the method.

Host: Stephen Shipman

Applied Analysis Seminar
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Posted March 21, 2010

Last modified April 11, 2010

M. Gregory Forest, Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina
Grant Dahlstrom Distinguished Professor of Mathematics & Biomedical Engineering

Dynamic defect morphology and hydrodynamics of sheared nematic polymers in physically confined geometries

Abstract: Nematic polymers consist of rigid rod or platelet dispersions where the particles are macromolecules, i.e., larger than liquid crystals but still Brownian. Depending on the properties of the rods or platelets, materials are targeted with extreme barrier, electrical, thermal, mechanical, dielectric or energy storage properties. Unlike fiber processing which yields highly uniform alignment of the rod or platelet phase, film and mold filling processes of nematic polymers typically possess dynamic particle orientational morphology even in steady processing conditions, accompanied by unsteady flow. Furthermore, defects are generic. In this talk we present model equations and boundary conditions, and results from numerical simulations for shear cell and driven cavity experiments of nematic polymers. We use novel defect detection and tracking diagnostics to show defect spawning mechanisms and morphology and flow evolution in these two types of experiments, and sensitivity to boundary conditions as well as initial data. Finally, we report some progress on post-processing of the simulation data to infer the underlying mechanisms for various property enhancements due to the particle phase. This is joint work with several collaborators who will be acknowledged during the lecture.

Applied Analysis Seminar
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Posted September 3, 2010

Last modified September 10, 2010

Robert Lipton, Mathematics Department, LSU

Multi-scale analysis and optimal local basis functions for Generalized Finite Element Methods

Abstract: Modern structures such as airplane wings exhibit complicated

sub structures and make use of composite materials in their construction.

The high cost of experimental tests for these hierarchical structures is

driving a trend toward virtual testing. This requires the development of

multi-scale numerical methods capable of handling large degrees of

freedom spread across different length scales. In this talk we review

multi-scale numerical methods and introduce the theory of the Kolmogorov

n-width as a means to identify optimal local basis functions for use in

multi-scale finite element methods. We are able to identify a spectral

basis with nearly exponential convergence with respect to the dimension

of the approximation space. The convergence result is shown to hold in a

very general setting. This is joint work with Ivo Babuska.

Applied Analysis Seminar
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Posted September 3, 2010

Last modified September 20, 2010

Blaise Bourdin, Department of Mathematics and Center for Computation & Technology, LSU

Reservoir stimulation: an approach based on variational fracture.

The topic of this talk is to present a first step towards the predictive understanding of the mechanisms used in the creation of the highly connected crack networks required for Enhanced Geothermal Systems and oil shale mining. I will focus on thermal stimulation, where thermal stresses induced by a cold fluid circulating through a hot reservoir lead to nucleation of many short cracks. I will consider the limiting cases of purely diffusive and purely advective heat transfer, corresponding to extreme porosity limits in the reservoir. I will present a mechanistically faithful yet mathematically sound model, based on Francfort and Marigo's generalization of Griffith's idea of competition between bulk and surface energies. I will discuss the virtues of the model, its approximation, and its numerical implementation. Finally, I will present some numerical experiments in 2 and 3 dimensions.

Applied Analysis Seminar
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Posted September 10, 2010

Last modified September 20, 2010

Hongchao Zhang, Louisiana State University

A Derivative-free Regularized Trust Region Approach for Least-squares Minimization

Abstract: We will introduce a class of derivative-free algorithms for the nonlinear least-squares minimization problem. These algorithms are based on polynomial interpolation models and are designed to take advantages of the problem structure. Global and local quadratic convergence properties of the algorithms will be addressed. Promising numerical results compared with other state-of-art software packages indicate the algorithm is very efficient and robust for finding both low and high accuracy solutions.

Applied Analysis Seminar
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Posted September 10, 2010

Last modified October 13, 2010

Tadele Mengesha, Louisiana State University

Weighted and regularity estimates for nonlinear PDEs over rough domains

Global weighted Lp estimates are obtained for the gradient of solutions to nonlinear elliptic Dirichlet boundary value problems over a bounded nonsmooth domain. As an application, Morrey and Holder regularity of solutions are established. These results generalize various existing estimates for nonlinear equations. The nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation, i.e can have mild discontinuity. The boundary of the domain, on the other hand, may exhibit roughness but assumed to be sufficiently flat in the sense of Reifenberg. Our approach is a perturbation argument that uses maximal function estimates, Vitali covering lemma, and known regularity results of solutions to nonlinear homogeneous equations. This is a joint work with Nguyen Cong Phuc.

Applied Analysis Seminar
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Posted October 23, 2010

3:40 pm - 4:30 pm Lockett 223
Shawn Walker, LSU

Shape Optimization of Chiral Propellers in 3-D Stokes Flow

Abstract: Locomotion at the micro-scale is important in biology and in industrial applications such as targeted drug delivery and micro-fluidics. We present results on the optimal shape of a rigid body locomoting in 3-D Stokes flow. The actuation consists of applying a fixed moment and constraining the body to only move along the moment axis; this models the effect of an external magnetic torque on an object made of magnetically susceptible material. The shape of the object is parametrized by a 3-D centerline with a given cross-sectional shape. No a priori assumption is made on the centerline. We show there exists a minimizer to the infinite dimensional optimization problem in a suitable infinite class of admissible shapes. We develop a variational (constrained) descent method which is well-posed for the continuous and discrete versions of the problem. Sensitivities of the cost and constraints are computed variationally via shape differential calculus. Computations are accomplished by a boundary integral method to solve the Stokes equations, and a finite element method to obtain descent directions for the optimization algorithm. We show examples of locomotor shapes with and without different fixed payload/cargo shapes.

Applied Analysis Seminar
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Posted October 10, 2010

3:40 pm - 4:30 pm Lockett 233
Michael Neilan, Louisiana State University

A unified approach to construct and analyze finite element methods for the Monge-Ampere equation

Abstract: The Monge-Ampere equation is a fully nonlinear second order PDE that arises in various application areas such as differential geometry, meteorology, reflector design, economics, and optimal transport. Despite its prevalence in many applications, numerical methods for the Monge- Ampere equation are still in its infancy. In this talk, I will first discuss the inherent difficulty of approximating this equation and briefly review the numerical literature. I will then discuss a new approach to construct and analyze several finite element methods for the Monge-Ampere equation. As a first step, I will show that a key feature in developing convergent discretizations is to construct schemes with stable linearizations. I will then describe a methodology for constructing finite elements that inherits this trait and provide two examples: C^0 finite element methods and discontinuous Galerkin methods. I will briefly show how to prove the well-posedness of such methods as well as derive optimal order error estimates.

Applied Analysis Seminar
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Posted October 25, 2010

Last modified November 15, 2010

Benjamin Jaye, University of Missouri-Columbia

Quasilinear operators with natural growth terms

The abstract can be found here.

Applied Analysis Seminar
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Posted October 10, 2010

Last modified November 28, 2010

Young-Ju Lee, Department of Mathematics Rutgers, The State University of New Jersey

Self-Sustaining Oscillations of the Falling Sphere Through the Johnson-Segalman Fluids

Abstract: In this talk, we review a novel numerical method that can handle the rate-type non-Newtonian equations in a unified fashion and validate the methods in terms of various benchmark solutions as well as theoretical results. We then apply it to the real physical problems. In particular, we present our investigations and attempts to identify a mathematical model for the unusual phenomenon observed in motion of the sphere falling through the wormlike micellar fluids by Jayaraman and Belmonte; a sphere falling in a wormlike micellar fluids undergoes non-transient and continual oscillations. We tackle the Johnson-Segalman models in the parameter regimes that have been unexplored previously for the flow past a sphere and reproduce the self-sustaining, continual, (ir)regular and periodic oscillations. Our results show that the flow instability can be correlated with the critical value of the velocity gradient, as observed in experiments by Jayaraman and Belmonte in 2003. If time permits, we also present recent works on the boundary conditions for the diffusive complex fluids models as well as the fast stokes solvers implemented in a full parallel fashion.

Applied Analysis Seminar
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Posted October 27, 2010

4:30 pm Lockett 233
Dmitry Golovaty, University of Akron

Coarse-graining in Atomistic Models of Dislocations

Dislocations and their dynamics play a major role in the response of materials to mechanical and thermal loading. Extensive work has been done on diﬀerent scales of the problem from atomistic level, to dislocation level to macroscopic level. Yet the behavior of material under plastic deformation is still a source of many challenging mathematical problems. In this talk we focus on models at the atomistic level and deal with questions of coarse graining

where higher level models are sought. We focus on distribution functions characterizing the atomic arrangement and discuss energy representation and dislocation motion in terms of these statistical properties. The evolution is

formulated as a gradient ﬂow.

Applied Analysis Seminar
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Posted November 19, 2010

Last modified November 30, 2010

Andrés León Baldelli, Université Pierre et Marie Curie

Variational Approach to Fracture Mechanics: Multifissuration and Delamination of Thin Films

Thin film materials such as multilayered composite materials, coating films etc plays a key role in modern engineering applications. The different physical characteristics of the various layers, the production and assembly procedures and the tensile stresses that develop in such systems may induce deformations that can lead to damage and failure. Variational energetic approaches to fracture mechanics [1] has been proved to give a reliable and physically consistent description of these complex phenomena, accurately predicting the experimental results. An extension of this approach to thin film/substate systems is presented for a film under thermal loads, accounting for the possibility for the film to undergo multifissuratio and debonding processes. Analytic results are obtained for the 1D case and compared to those obtained by a FEM approximation.

[1] B. Bourdin, G.A. Francfort, and J.-J. Marigo, "The Variational Approach to Fracture", Springer, 2007.

Applied Analysis Seminar
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Posted April 15, 2011

Last modified April 25, 2011

Aleksandra Gruszka, LSU Department of Mathematics
PhD Student of Prof. Malisoff

On tracking for the PVTOL model with bounded feedbacks

We study a class of feedback tracking problems for the planar vertical takeoff and landing (PVTOL) aircraft dynamics, which is a benchmark model in aerospace engineering. After a survey of the literature on the model, we construct new feedback stabilizers for the PVTOL tracking dynamics. The novelty of our contribution is in the boundedness of our feedback controllers and their applicability to cases where the velocity measurements may not be available, coupled with the uniform global asymptotic stability and uniform local exponential stability of the closed loop tracking dynamics, the generality of our class of trackable reference trajectories, and the input-to-state stable performance of the closed loop tracking dynamics with respect to actuator errors. Our proofs are based on a new bounded backstepping result. We illustrate our work in a tracking problem along a circle.

Applied Analysis Seminar
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Posted April 29, 2011

3:40 pm Lockett 277
Michael Malisoff, LSU

Uniform global asymptotic stability of adaptive cascaded nonlinear systems with unknown high-frequency gains

We study adaptive tracking problems for nonlinear systems with unknown control gains. We construct controllers that yield uniform global asymptotic stability for the error dynamics, and hence tracking and parameter estimation for the original systems. Our result is based on a new explicit, global, strict Lyapunov function construction. We illustrate our work using a brushless DC motor turning a mechanical load. We quantify the effects of time-varying uncertainties on the motor electric parameters.

Note: This talk will be understandable to faculty, staff, students, and visitors who are familiar with the material in Math 7320 (Ordinary Differential Equations) at LSU. No background in control is needed.

Applied Analysis Seminar
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Posted April 29, 2011

Last modified May 7, 2011

Cody Pond, Department of Mathematics Tulane University

Effective boundary conditions on insulated bodies

ABSTRACT : The temperature of perfectly insulated body can be modeled by the heat equation with Neumann (or no-flux) boundary condition. In reality there are no perfect insulators and the actual boundary condition on the body may be only approximately Neumann. In this talk we will see how properties of a layer of insulation affect the boundary condition experienced by the insulated body. We we also see how ignoring physical restrictions in the model can produce some exotic boundary conditions.

Applied Analysis Seminar
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Posted August 31, 2011

Last modified September 14, 2011

Catalin Turc, Case Western Reserve University

Fast, high-order solvers based on regularized integral equations for acoustic and electromagnetic scattering problems

Abstract: We present a class of solvers based on Nystrom discretizations to produce fast and very accurate solutions of acoustic and electromagnetic scattering problems in small numbers of Krylov-subspace iterations. At the heart of our approach is a general methodology that uses certain regularizing operators to deliver integral equation formulations that possess excellent spectral properties for scattering problems, including smooth and non-smooth geometries and a variety of boundary conditions. Our computational methodology relies on a novel Nystrom approach based on use of a overlapping/non-overlapping-patch technique, Chebyshev discretizations and an acceleration method based on equivalent sources and 3D FFT's.

Applied Analysis Seminar
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Posted September 30, 2011

3:40 pm - 4:30 pm Lockett 233
Hanna Terletska, Department of Physics and Astronomy, LSU and Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory

Towards a multiscale formalism for disordered systems

Applied Analysis Seminar
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Posted August 23, 2011

Last modified October 23, 2011

Rustum Choksi, Department of Mathematics and Statistics, McGill University, Montréal, Canada

Self-assembly of Diblock Copolymers and Variational Problems with Long-Range Interactions

Energy-driven pattern formation induced by competing short and long-range interactions is common in many physical systems. This talk will address mathematical and physical paradigms for periodic pattern formation induced by these energetic competitions. The mathematical paradigm consists of nonlocal perturbations to the well-studied Cahn-Hilliard and isoperimetric problems. The physical paradigm is self-assembly of diblock copolymers. Via a combination of analysis and numerics, I will address the structure of minimizers across the phase diagram.

Applied Analysis Seminar
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Posted September 12, 2011

Last modified November 1, 2011

Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

On the distance between homotopy classes of maps taking values in manifolds

Applied Analysis Seminar
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Posted August 29, 2011

Last modified November 7, 2011

Leonid Berlyand, Department of Mathematics, Penn State University

Modeling of collective swimming of bacteria.

Applied Analysis Seminar
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Posted November 30, 2011

3:40 pm - 4:30 pm Lockett 233
Michel Jabbour, University of Kentucky

On step dynamics and related morphological instabilities during epitaxial growth of thin crystalline films.

Abstract:

Thin crystalline films are often bounded by surfaces consisting

of flat terraces separated by atomic steps. During epitaxial growth in the step-flow regime, adsorbed atoms (from a vapor or beams) diffuse on the terraces until they attach to

steps, causing them to advance. For a train of steps, two modes of morphological instability can occur: bunching, which leads to regions of high step density separated by wide terraces, and meandering, whereby steps become wavy. Experiments indicate that bunching and meandering can coexist on some stepped surfaces, in contrast to the predictions of the standard Burton-Cabrera-Frank (BCF) model. In this talk, I will review the BCF theory and present a thermodynamically consistent (TC) generalization of it that resolves this apparent paradox. In particular, I will show that step bunching and meandering can occur simultaneously, provided that the adatom equilibrium coverage exceeds a critical value. I will also compare the TC model with various extensions of the BCF paradigm that attempt to reconcile theory with experiments.

Applied Analysis Seminar
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Posted February 10, 2012

Last modified March 28, 2012

Michael Borden, Institute for Computational Engineering and Sciences, University of Texas at Austin

Isogeometric Analysis and Computational Fracture Mechanics

Abstract:

I will begin my presentation with an overview of isogeometric analysis, emphasizing its application to problems in nonlinear solid mechanics. The basic idea of the isogeometric concept is to use the same basis for analysis as is used to describe the geometry in, for example, a CAD representation. The smoothness of typical geometric representations (e.g., NURBS and T-splines) has been shown to have computational advantages over standard finite elements in many solid mechanics problems.In the second part of my presentation I will discuss our recent work on the numerical implementation of variational, or phase-field, models of fracture. The phase-field approach to predicting fracture uses a scalar-valued field to indicate that the material is in some state between complete undamaged or completed fractured with a smooth transition between the two states. This allows cracks to be modeled without explicit tracking of discontinuities in the geometry or displacement fields. In this part of my presentation I will also discuss work in which we make use of the smoothness provided by isogeometric analysis to explore the effect of adding higher-order terms to the phase-field model. Several numerical examples will be shown for both two and three-dimensional problems that demonstrate the ability of these models to capture complex crack behavior.

Applied Analysis Seminar
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Posted February 5, 2012

Last modified April 30, 2012

Shari Moskow, Mathematics Department, Drexel University

Scattering and Resonances of Thin High Contrast Dielectrics

We study the scattered field from a thin high contrast dielectric volume of finite extent. We examine both the Helmholtz model and the full three dimensional time-harmonic equations. For the case of the Helmholtz model, we derive an asymptotic expansion and show error estimates. We also consider the problem of calculating resonance frequencies by using these asymptotics and compare it with using finite elements and perfectly matched layers. For Maxwell equations, we derive a formulation of Lippmann-Schwinger type which has an additional surface term to account for the discontinuities. We analyze this surface term and present the limiting equations that result.

(based on joint work with collaborators D. Ambrose, J. Gopalakrishnan, F. Santosa and J. Zhang)

Applied Analysis Seminar
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Posted March 5, 2012

3:40 pm - 4:30 pm Lockett 233
Richard Lehoucq, Sandia National Laboratories

A NEW APPROACH FOR A NONLOCAL, NONLINEAR CONSERVATION LAW

Abstract: My presentation describes an approach to nonlocal, nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux. The spatially nonlocal operators introduced do not involve derivatives. Instead, the spatial operator involves an integral that, in a distributional sense, reduces to a conventional nonlinear advective operator. In particular, we examine a nonlocal inviscid Burgers equation, which gives a basic form with which to characterize well-posedeness. We describe the connection to a nonlocal viscous regularization, which mimics the viscous Burgers equation in an appropriate limit. We present numerical results that compare the behavior of the nonlocal Burgers formulation to the standard local case. The developments presented in this paper form the preliminary building blocks upon which to build a theory of nonlocal advection phenomena consistent within the peridynamic theory of continuum mechanics. This is joint work with Qiang Du (PSU), Jim Kamm (SNL) and Mike Parks (SNL)

Applied Analysis Seminar
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Posted August 8, 2012

3:40 pm - 4:30 pm 233 Lockett
Raul Tempone, King Abdullah University of Science and Technology, KAUST

Strategies for Optimal Polynomial Approximation of Elliptic PDEs with Stochastic Coefficients

Partial differential equations with stochastic coefficients are a suitable tool to describe systems whose parameters are not completely determined, either because of measurement errors or intrinsic lack of knowledge on the system. In the case of elliptic PDEs, an effective strategy to approximate the state variables and their statistical moments is to introduce high order polynomial approximations like Stochastic Galerkin or Stochastic Collocation method, exploiting the fact that the state variables may exhibit high regularity in their dependence with respect to the random parameters. When the number of parameters is moderate, these methods can be remarkably more effective than classical sampling methods. However, contrary to the latter, the performance of polynomial approximations deteriorates as the number of random variables increases (\emph{curse of dimensionality}); to prevent this, care has to be put in the construction of the approximating polynomial space. In this talk we will propose strategies to construct optimal spaces and propose some particular polynomial spaces and generalized sparse grids that are optimal for particular problems. We will also support our claims with some simple numerical examples. This work is a joint collaboration with J. Beck, F. Nobile and L. Tamellini.

Applied Analysis Seminar
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Posted August 7, 2012

Last modified August 8, 2012

Mohammad Motamed, Visiting Scholar, Institute for Engineering and Computational Science UT Austin

Analysis and Computation of Linear Hyperbolic Problems with Random Coefficients

ABSTRACT: In this talk, in particular, we consider the second-order acoustic and

elastic wave equations. In the first part of this talk, we propose and analyze a

stochastic collocation method for solving the acoustic wave equation with a random

wave speed and subjected to deterministic boundary and initial conditions. The speed

is piecewise smooth in the physical space and depends on a finite number of random

variables. The numerical scheme consists of a finite difference or finite element

method in the physical space and a collocation in the zeros of suitable tensor

product orthogonal polynomials (Gauss points) in the probability space. This

approach leads to the solution of uncoupled deterministic problems as in the Monte

Carlo method. We consider both full and sparse tensor product spaces of orthogonal

polynomials. We demonstrate different types of convergence of the ?probability

error? with respect to the number of collocation points for full and sparse tensor

product spaces and under some regularity assumptions on the data. In the second

part of the talk, we present extensions to the elastic wave equation with random

coefficients and random boundary conditions.

Applied Analysis Seminar
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Posted September 4, 2012

Last modified October 11, 2012

Stephanos Venakides, Department of Mathematics, Duke University

Higher breaking in the focusing nonlinear Schrödinger equation

**Abstract:** The focusing nonlinear Schrödinger equation iε du/dt + ε^{2} d^{2}u/dx^{2} + 2|u|^{2}u = 0 (NLS) appears dominantly in nonlinear optical transmission, together with its many variants. Mathematically, the initial value problem of the NLS on the line is integrable. It can be linearized with the aid of a Lax operator pair, produced by Zakharov and Shabat. Determining the evolution of an NLS waveform becomes possible with the aid of Riemann-Hilbert problems (RHP), the conceptual nature of which is simple and will be explained in the talk.
The development of the steepest descent method for oscillatory RHP provided rigorous asymptotic procedures, that make the solution of NLS and nonlinear integrable systems in general, explicit or nearly explicit. The method applies to asymptotics of RHP in the same spirit as the classic methods of stationary phase and steepest descent apply in the asymptotic evaluation of Fourier type integrals arising from the solution of linear differential equations. In both the linear and the nonlinear cases, there is a separation of space-time scales over similar parameter regimes.
Employing initial data of the form u(x,0)=A(x)exp(iS(x)/ε) in the asymptotic limit ε→0, we describe the solution over a large domain of space-time and the mechanism of the break-down of the method beyond this domain. Using a combination of analytic and numerical considerations, we establish the boundary beyond which the asymptotic solution is still unknown. The spatial component of this domain is bounded.

Applied Analysis Seminar
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Posted October 7, 2012

3:40 pm - 4:30 pm Room 233 Lockett
Stewart Silling, Sandia National Laboratories

Multiscale Modeling of Fracture with Peridynamics

The peridynamic theory is an extension of traditional solid mechanics that treats discontinuous media, including the evolution of discontinuities such as fracture, on the same mathematical basis as classically smooth media. Since it is a strongly nonlocal theory, peridynamic material models contain a length scale that characterizes the interaction distance between material points. By changing this length scale in a way that preserves the bulk elastic properties, greater spatial resolution in a simulation can be focused on a growing crack tip or other evolving singularity. This leads to a consistent way to treat fracture at the smallest physically relevant length scale within a larger model, without remeshing or coupling dissimilar methods. The method works within a meshless discretization of the peridynamic equations similar to that used in the Emu code. The grid has multiple elves of resolution. The high resolution portions of the grid supply material properties, including damage, to the coarser levels. The displacement field in the coarsest level is determined by the equation of hte motion at that level, using these coarse-grained material properties. The resulting coarse displacements are applied as boundary conditions on the finer levels of the grid. The equation of the motion in the finer levels is solved only where the damage is ongoing or large deformations are occurring. In this way, the greatest computational power is focused only on those parts of the region, such as growing crack tips, where it is required. This peridynamic multiscale method appears to provide a promising approach to understanding the evolution of material failure, including the interaction of small defects with each other and with heterogeneities. This talk will first review the basics of the peridynamic theory. The new multi scale method will then be discussed, with computational examples drawn from the mechanics of contact and from damage progression in heterogeneous media.

Applied Analysis Seminar
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Posted September 19, 2012

Last modified October 29, 2012

Anna Zemlyanova, Department of Mathematics, Texas A&M

Elimination of oscillating singularities at the crack-tips of an interface crack with a help of a curvature-dependent surface tension

A new model of fracture mechanics incorporating a curvature-dependent

surface tension acting on the boundaries of a crack is considered. The

model is studied on the example of a single straight interface crack

between two elastically dissimilar semi-planes. Linear elasticity is

assumed for the behavior of the material of the plate in a bulk. A

non-linear boundary condition with a consideration for a

curvature-dependent surface tension is given on the crack boundary. It is

well known from linear elastic fracture mechanics (LEFM) that oscillating

singularities exist at the crack tips and lead to non-physical

interpenetration and wrinkling of the crack boundaries. Using the methods

of complex analysis, such as Dirichlet-to-Neumann mappings, the problem is

reduced to a system of six singular integro-differential equations. It is

proved that the introduction of the curvature-dependent surface tension

eliminates both classical power singularities of the order 1/2 at the

tips of the crack and oscillating singularities, thus resolving the

classical contradictions of LEFM. Numerical computations are presented.

Applied Analysis Seminar
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Posted October 29, 2012

3:40 pm - 4:30 pm Room 233 Lockett
Michael Malisoff, LSU

Asymptotic Stabilization for Feedforward Systems with Delayed Feedbacks

Abstract: We study a problem of state feedback stabilization of time-varying feedforward systems with a pointwise delay in the input. Our approach relies on a time-varying change of coordinates and Lyapunov-Krasovskii functionals. Our result applies for any given constant delay, and provides uniformly globally asymptotically stabilizing controllers of arbitrarily small amplitude. The closed-loop systems enjoy input-to-state stability properties with respect to additive uncertainty on the controllers. We illustrate our work using a tracking problem for a model for high level formation flight of unmanned air vehicles. We will review all of the necessary background on control theory, so no prior exposure to controls will be needed to understand this talk. This work is joint with Frederic Mazenc from INRIA in France.

Applied Analysis Seminar
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Posted November 26, 2012

Last modified February 10, 2013

Timothy Healey, Cornell University
Professor of Mathematics and Mechanical and Aerospace Engineering

Nonlinear problems for thin elastic structures and the ubiquitous isola bifurcation

Abstract: We begin with a simple 1-dimensional, 2-phase solid under "hard" tensile end loading in the presence of inter-facial effects. This is equivalent to a phase-field model, of the van der Waals-Cahn-Hilliard type, that illustrates well the concept of an isola bifurcation. Roughly speaking, the latter corresponds to the nucleation, growth, decay and eventual disappearance of a stable, inhomogeneous solution (representing here a phase mixture) as the loading parameter is monotonically increased. We then present results for three ostensibly distinct problems (models) - all exhibiting this same isola-bifurcation phenomenon: (i) twining in shape-memory solids; (ii) two-phase configuration of GUV's (fluid-elastic shell models of lipid-bilayer vesicles); (iii) wrinkling of highly stretched, finely thin rectangular sheets.

Applied Analysis Seminar
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Posted March 8, 2013

Last modified April 15, 2013

Stefan Llewellyn Smith, University of California, San Diego

Hollow Vortices

Hollow vortices are vortices whose interior is at rest. They posses vortex sheets

on their boundaries and can be viewed as a desingularization of point vortices.

After giving a history of point vortices, we obtain exact solutions for hollow

vortices in linear and nonlinear strain and examine the properties of streets of

hollow vortices. The former can be viewed as a canonical example of a hollow vortex

in an arbitrary flow, and its stability properties depend. In the latter case, we

reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its

stability to arbitrary disturbances, and then investigate the double hollow vortex

street. Implications and extensions of this work are discussed.

Applied Analysis Seminar
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Posted February 23, 2013

Last modified April 9, 2013

Daniel Onofrei, University of Houston

Active control of acoustic and electromagnetic fields

The problem of controlling acoustic or electromagnetic fields is at the core of many important applications such as, energy focusing, shielding and cloaking or the design of supper-directive antennas. The current state of the art in this field suggests the existence of two main approaches for such problems: passive controls, where one uses suitable material designs to control the fields (e.g., material coatings of certain regions of interest), and active control techniques, where one employs active sources (antennas) to manipulate the fields in regions of interest. In this talk I will first briefly describe the main mathematical question and its applications and then focus on the active control technique for the scalar Helmholtz equation in a homogeneous environment. The problem can be understood from two points of view, as a control question or as an inverse source problem (ISP). This type of ISP questions are severely ill posed and I will describe our results about the existence of a unique minimal energy solution. Stability of the solution and extensions of the results to the case of nonhomogeneous environment and to the Maxwell system are part of current work and will be described accordingly.

Applied Analysis Seminar
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Posted May 8, 2013

Last modified May 10, 2013

Guillermo Ferreyra, Mathematics Department, LSU

The Future of Analysis at LSU

SCI Data (pdf)

Letter from Provost (docx)

Applied Analysis Seminar
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Posted October 15, 2013

Last modified October 17, 2013

Stephen Shipman, Mathematics Department, LSU

Efficient Evaluation of 2D-periodic Green functions in 3D

I will describe the analytical basis behind a fast method of computing periodic Green functions, ultimately for the purpose of efficiently solving problems of scattering by periodic structures. The Poisson summation formula provides super-algebraic convergence away from frequencies for which one of the Rayleigh-Bloch modes is grazing. At grazing (cutoff) frequencies, the periodic Green function ceases to exist, and a more complicated method is needed. This involves introducing several sheets of periodic sources to create a half-space Green function. This is work with Oscar Bruno, Catalin Turc, and Stephanos Venakides.

Applied Analysis Seminar
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Posted October 24, 2013

3:30 pm - 4:20 pm Room 233 Lockett Hall
Robert Lipton, Mathematics Department, LSU

Dynamics in Materials Far From Equilibrium

In this talk we address the role of local instability in the precipitation and propagation of failure in macroscopic samples of material. We work with non-locally interacting systems, eg. peridynamics. A class of scaled nonlinear interaction potentials are identified for which dynamic instability localizes and fracture surfaces appear in the scaling limit.

Applied Analysis Seminar
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Posted October 25, 2013

3:30 pm - 4:20 pm Lockett Hall 233
Yaniv Almog, Department of Mathematics, LSU

Global stability of the normal state of superconductors under the effect of strong electric current

Consider a superconducting wire whose temperature is lower than the critical one. When one flows a sufficiently strong current through the wire, it is well known from experimental observation that the wire becomes resistive, behaving like a normal metal. We prove that the time-dependent Ginzburg-Landau model anticipates this behaviour. We first prove that, for sufficiently strong currents, the semi-group associated with the model, becomes a contraction semi-group. Then, we obtain an upper bound for the critical current where the semi-group becomes stable. We relate this current to the resolvent of the linearized elliptic operator. Joint work with Bernard Helffer

Applied Analysis Seminar
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Posted October 24, 2013

3:00 pm - 4:00 pm Lockett Hall 233
Amit Acharya, Carnegie Melon University

PDE dynamics of dislocations

The talk will describe a PDE framework to deal with the dynamics of dislocations leading to plasticity in solids. Dislocations are defects of deformation compatibility/integrability in elastic response. The presented framework will be shown to be capable of representing discrete defect dynamics as well as present a natural setting for asking questions related to macroscopic plasticity arising from the underlying dislocation dynamics.

Applied Analysis Seminar
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Posted October 24, 2013

3:00 pm - 4:00 pm Lockett Hall 233
Amit Acharya, Carnegie Melon University

PDE dynamics of dislocations

The talk will describe a PDE framework to deal with the dynamics of dislocations leading to plasticity in solids. Dislocations are defects of deformation compatibility/integrability in elastic response. The presented framework will be shown to be capable of representing discrete defect dynamics as well as present a natural setting for asking questions related to macroscopic plasticity arising from the underlying dislocation dynamics.

Applied Analysis Seminar
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Posted February 5, 2014

Last modified February 12, 2014

Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

Asymptotics of eigenstates of elliptic problems with mixed boundary data in domains tending to infinity

We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to innity. We identify the correct limiting problem and show, in particular, that in general the limiting behavior is very different from the one with Dirichlet boundary conditions. This is a joint work with Michel Chipot and Prosenjit Roy from the University of Zurich.

Applied Analysis Seminar
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Posted February 19, 2014

3:30 pm - 4:20 pm Lockett Hall Room 233
Michael Mascagni, Department of Computer Science, Florida State University

Random Number Generation Using Normal Numbers

Normal numbers are irrationals with perfect digit distribution, and thus they are potentially sources of computational random numbers. Among provably normal numbers are the Stoneham numbers, which are special not only in their digit distribution, but in the fact that finite segments of the digits can be quickly computed. Thus, we examine random numbers produced by periodic sections of the digits of Stoneham numbers. We show how they are equivalent to a linear congruential generator with special parameters, and we investigate this generator as a linear congruential generator. This is joint work with Steve F. Brailsford.

Applied Analysis Seminar
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Posted February 5, 2014

Last modified February 12, 2014

Bernard Helffer, University of Paris South, Orsay

Introduction to spectral minimal partitions, Aharonov-Bohm's operators and Pleijel's theorem

Applied Analysis Seminar
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Posted April 3, 2014

Last modified April 21, 2014

Yuri Antipov, Mathematics Department, LSU

Diffraction of an obliquely incident electromagnetic wave by an impedance wedge

Applied Analysis Seminar
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Posted March 26, 2014

Last modified April 21, 2014

Mark Wilde, LSU Department of Physics/CCT

Renyi generalizations of the conditional quantum mutual information

Abstract: The conditional quantum mutual information I(A;B|C) of a tripartite quantum state on systems ABC is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A;B|C)=I(A;B|D) for a four-party pure state on systems ABCD. It has been an open question to find Renyi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different Renyi generalizations of the conditional mutual information that all converge to the conditional mutual information in a limit. Furthermore, we prove that many of these generalizations satisfy the aforementioned properties. As such, the quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Renyi conditional mutual informations defined here with respect to the Renyi parameter. We prove that this conjecture is true in some special cases and when the Renyi parameter is in a neighborhood of one. Finally, we discuss how our approach for conditional mutual information can be extended to give Renyi generalizations of an arbitrary linear combination of von Neumann entropies, particular examples including the multipartite information and the topological entanglement entropy. This is joint work with Mario Berta (Caltech) and Kaushik Seshadreesan (LSU). This is based on the recent paper http://arxiv.org/abs/1403.6102

Applied Analysis Seminar
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Posted August 21, 2014

3:30 pm - 4:20 pm Lockett 233
Phuc Nguyen, Department of Mathematics, Louisiana State University

The Navier-Stokes equations in nonendpoint borderline Lorentz spaces

Applied Analysis Seminar
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Posted September 8, 2014

Last modified September 25, 2014

Cristi Guevara, LSU Department of Mathematics

Characterization of finite-energy solutions to the focusing 2-dimensional quintic NLS equation

Abstract. In this lecture we will focus on the mass-supercritical and energy-subcritical nonlinear Schroedinger equation or 2 dimensional quintic NLS. Using the concentration-compactness and rigidity method developed by Kenig-Merle, we characterize global behavior of solutions with H1 (finite energy) initial data. In particular, we will discuss an application of the concentration-compactness to the existence of weak blowup solutions for infinite-variance initial data. In addition, we will describe extensions on the conditions for scattering of globally existing solutions for the d-dimensional case.

Applied Analysis Seminar
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Posted October 22, 2014

4:30 pm Lockett 223
Martin Adler, University of Tuebingen

Perturbations of generators of C_0-semigroups

Abstract: The theory of strongly continuous semigroups is an elegant method to investigate the wellposedness of abstract Cauchy problems. After introducing the basic theory of C_0-semigroups needed for this approach, I provide an overview of bounded and unbounded perturbation results. Finally, we will apply this theory to a delay equation.

Applied Analysis Seminar
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Posted November 1, 2014

3:30 pm - 4:20 pm Room 233 Lockett
Jacob Grey, Department of Mathematics LSU

A qualitative analysis of some Nonlinear Dispersive Evolution Equations

Applied Analysis Seminar
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Posted November 12, 2014

3:30 pm - 4:20 pm Room 233 Lockett
Michael Malisoff, LSU

Designs and Theory for State-Constrained Nonlinear Feedback Controls for Delay Systems: An Infomercial

This talk will discuss some of my research that is being supported by my two new research grants from the US National Science Foundation Directorate for Engineering. The first grant project is entitled "Robustness of Networked Model Predictive Control Satisfying Critical Timing Constraints" and focuses on resolving contentions in a class of communication networks that are common in automobiles and other real-time control applications, and is joint with the Georgia Institute of Technology School of Electrical and Computer Engineering. The second project, "Designs and Theory of State-Constrained Nonlinear Feedback Controls for Delay and Partial Differential Equation Systems," covers control designs for classes of ordinary and hyperbolic partial differential equations that arise in oil production and rehabilitation engineering, and is joint with the University of California, San Diego Department of Mechanical and Aerospace Engineering. In the first 10 minutes, I will provide a brief description of the basic ideas of control theory. Then, I will present a 25 minute summary of my research on neuromuscular electrical stimulation (or NMES), which is a biomedical approach for helping to restore movement in patients with mobility disorders. My NMES research designed controls for NMES of the human knee under delays and subject to a constraint on the allowable knee position, and is joint with my PhD student Ruzhou Yang and with Prof. Marcio de Queiroz, who are with the LSU Department of Mechanical and Industrial Engineering. In the last 10 minutes, I will advertise for my open PhD student positions on my grants, by providing a brief nontechnical summary of the problems to be addressed and discussing the role PhD students would play in the research. This talk will be accessible to students and others who are familiar with basic differential equations. No background in controls is needed.

Applied Analysis Seminar
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Posted February 23, 2015

3:30 pm
Itai Shafrir, Department of Mathematics, Technion - Israel Institute of Technology

Asymptotic behavior of critical points of an energy involving a "circular-well" potential

We study the singular limit of critical points of an energy with a penalization term depending on a small parameter. The energy involves a potential which is a nonnegative function on the plane, vanishing on a closed curve. We generalize to this setting results obtained by Bethuel, Brezis and Helein for the Ginzburg-Landau energy. This is a joint work with Petru Mironescu (Lyon I).

Applied Analysis Seminar
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Posted February 15, 2015

3:30 pm - 4:20 pm Lockett Hall Room 233
Robert P. Viator, Jr., LSU

Perturbation Theory of High-Contrast Photonic Crystals

Applied Analysis Seminar
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Posted February 15, 2015

Last modified March 8, 2015

Anthony Polizzi, LSU

An asymptotic formula for solutions to a heterogeneous logistic equation with small diffusion rate

Of concern is a semilinear elliptic boundary value problem on a bounded domain with a smooth boundary. Its solution u(x) represents the steady state population density of a species in an insulated habitat with a given resource profile a(x) and a given nonnegative coefficient representing the species' rate of random diffusion. We study asymptotic expansions for solutions in the form of Taylor series in the diffusion coefficient. It turns out that, in the presence of diffusion, u(x) depends analytically on the diffusion coefficient, which trivializes the convergence of such a series. We therefore focus our attention primarily on the more delicate case of zero diffusion, in which u(x) is not analytic in the coefficient. It is known that u(x) tends to a(x) as the coefficient tends to zero. We generalize this result by rigorously establishing the desired expansions under suitable assumptions on a(x). Our main result is their convergence on the closure of the domain in this case. We also give an explicit formula for each coefficient of the expansions.

Applied Analysis Seminar
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Posted February 13, 2015

3:30 pm - 4:30 pm 233 Lockett Hall
Aleksandr Smirnov, Department of Mathematics, LSU

A discrete model of a fracture in an inhomogeneous strip

Applied Analysis Seminar
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Posted February 3, 2015

3:30 am - 4:30 pm 233 Lockett Hall
Yuri Antipov, Mathematics Department, LSU

Singular integral equations in a segment with two fixed singularities and applications

Applied Analysis Seminar
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Posted April 29, 2015

3:30 pm - 4:20 pm Room 233 Lockett Hall
Kaushik Dayal, Carnegie Mellon University

A Dynamic Phase-field Model for Structural Transformations and Twinning: Regularized Interfaces with Transparent Prescription of Complex Kinetics and Nucleation

Phase-field models enable easy computations of microstructure because they regularize sharp interfaces. In addition, the nucleation of new interfaces and the kinetics of existing interfaces occurs "automatically" using only the energy and a gradient descent dynamics. This automatic nucleation and kinetics is often cited as an advantage of these models, and is not present in sharp interface approaches where nucleation and kinetics must be explicitly prescribed. However, this is not necessarily an advantage. Rather, it does not allow us to use nucleation and kinetic insights that may be gained from experiment and/or molecular simulations. Hence, this feature is actually a disadvantage because it breaks the multiscale modeling hierarchy of feeding information through the scales. Motivated by this, we have developed a phase-field model (i.e., with regularized interfaces) that allows for easy and transparent prescription of kinetics and nucleation. We present the formulation of the model, and characterization through various examples.

Applied Analysis Seminar
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Posted May 3, 2015

3:30 am Lockett 233
Rainer Nagel, University of Tübingen

Some Operator Theoretic Aspects of Ergodic Theory

Abstract: We discuss some techniques and results on linear operators in Banach spaces as, e.g., appearing in the proof of the Green-Tao Theorem on arithmetic progressions in the primes. The main object is the so called Koopman operator yielding a linear model of a nonlinear dynamical system. It is joint work with Tanja Eisner, Balint Farkas and Markus Haase appearing as Springer Graduate Text in Mathematics.

Applied Analysis Seminar
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Posted October 4, 2015

3:30 pm Lockett 223
Bálint Farkas, University of Wuppertal

The periodic decomposition problem for semigroups

Given commuting power-bounded linear operators T_{1},...,T_{n} on a Banach space the periodic decomposition problems, originally due to I.Z. Ruzsa, asks whether and under which conditions the equality ker (T_{1}-I) ··· (T_{n}-I) = ker(T_{1}-I)+···+ker (T_{n}-I) holds true. In this talk we focus also on the case when T_{j}=T(t_{j}), t_{j} >0, j=1,..., n for some (strongly continuous) one-parameter semigroup (T(t))_{t≥0}. Moreover, we look at a generalization of the periodic decomposition problem when instead of the cyclic semigroups {T_{j}^{n}:n ∈ **N**} more general semigroups of bounded linear operators are considered.

Applied Analysis Seminar
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Posted October 9, 2015

Last modified October 13, 2015

Ko-Shin Chen, U. Conn.

Ginzburg-Landau and Gross-Pitaevskii Vortices on Surfaces

We consider the Ginzburg-Landau energy on compact and simply-connected surfaces. The first result is the instability of critical points of the Ginzburg-Landau energy. We show on a surface without boundary, any non-constant critical points must be unstable for small epsilon if at least one limiting vortex is located at a point of positive Gauss curvature. The second is the vortex dynamics for the Ginzburg-Landau heat flow, both in the asymptotic regime where the parameter 'epsilon' attends to zero and for a fixed epsilon. We show the vortices of a solution evolve according to the gradient flow of the renormalized energy. Then we establish vortex annihilation results for both ODE and PDE settings. The third is a similar analysis of vortex motion for the Gross-Pitaevskii equation. We show the vortices of a solution follow the Hamiltonian point-vortex flow associated with the renormalized energy. Then on surfaces of revolution, we find rotating periodic solutions to the generalized point-vortex problem and seek a rotating solution to the Gross-Pitaevskii equation having vortices that follow those of the point-vortex flow for small epsilon.

Applied Analysis Seminar
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Posted October 26, 2015

3:30 pm Lockett 223
Boris Baeumer, University of Otago, New Zealand

Anomalous Reaction-Diffusion Equations

Abstract: We show how a simple random walk model can be build up step by step to lead to a Volterra integral equation problem whose kernel depends on its solution. The build-up includes fractional differential equations, continuous time random walk limits, and surprising reaction effects. Variants or special cases of the model have been used to describe phenomena in cell dynamics, ecology, epidemiology, and hydrology.

Applied Analysis Seminar
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Posted March 21, 2016

Last modified April 5, 2016

Gianni Royer Carfagni, Università degli Studi Di Parma, Department of Civil Engineering, Environmental Engineering, and Architecture

Phase-field description of structured deformations in plasticity

Abstract: A variational approach to determine the deformation of an ideally plastic substance is proposed by solving a sequence of energy minimization problems under proper conditions to account for the irreversible character of plasticity. The flow is driven by the local transformation of elastic strain energy into plastic work on slip surfaces, once that a certain energetic barrier for slip activation has been overcome. The distinction of the elastic strain energy into spherical and deviatoric parts can also be used to incorporate in the model the idea of von Mises plasticity and isochoric plastic strain. This is a "phase field mode" because the matching condition at the slip interfaces are substituted by the evolution of an auxiliary phase field that, similarly to damage theory, is unitary on the elastic phase and null on the yielded phase. The slip lines diffuse in bands whose width depends upon a material length-scale parameter. Numerical experiments on representative problems in plane strain give solutions with striking similarities with the results from classical slip-line field theory of plasticity, but the proposed model is much richer because, accounting for elastic deformations, it can describe the formation of slip bands at the local level, which can nucleate, propagate, widen and diffuse by varying the boundary conditions.

Applied Analysis Seminar
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Posted May 2, 2016

1:30 pm - 2:30 pm Lockett 233
Kim Pham, ENSTA ParisTech

Construction of a macroscopic model of phase-transformation for the modeling of superelastic Shape Memory Alloys

Abstract: Shape Memory Alloys (SMA) e.g. NiTi display a superelastic behavior at high temperature. Initially in a stable austenite phase, SMA can transform into an oriented martensite phase under an applied mechanical loading. After an unloading, the material recovers its initial stress-free state with no residual strain. Such loading cycle leads to an hysteresis loop in the stress-strain diagram that highlights the dissipated energy for having transformed the material. In a rate-independent context, we first show how a material stability criterion allows to construct a local one-dimensional phase transformation model. Such models relies on a unique scalar internal variable related to the martensite volume fraction. Evolution problem at the structural scale is then formulated in a variational way by means of two physical principles: a stability criterion based on the local minima of the total energy and an energy balance condition. We show how such framework allows to handle softening behavior and its compatibility with a regularization based on gradient of the internal variable. We then extend such model to a more general three dimensional case by introducing a tensorial internal variable. We derive the evolution laws from the stability criterion and energy balance condition. Second order conditions are presented. Illustrations of the features of such model are shown on different examples.

Applied Analysis Seminar
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Posted September 23, 2016

3:30 pm - 4:30 pm Lockett 233
Viktoria Kuehner, University of Tuebingen

Semiflows and Koopmansemigroups

We characterize Koopman semigroups $(T(t))_{t\geq 0}$ on $\mathrm{L}^1(X,\Sigma,\mu)$, where $(X,\Sigma,\mu)$ is a standard probability space, induced by a measurable semiflow $(\varphi_t)_{t\geq 0}$ on $X$, by means of their generator $(A,D(A))$. We then construct a topological model $(\psi_t)_{t\geq 0}$ of that semiflow on a compact space $K$ such that the Koopman semigroup induced by the continuous semiflow $(\psi_t)_{t\geq 0}$ is isomorphic to the original semigroup.

Applied Analysis Seminar
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Posted December 2, 2016

Last modified January 15, 2017

Malcolm Brown, Department of Computer Science & Informatics, Cardiff University

Scattering and inverse scattering for a left-definite Sturm-Liouville problem

This talk reports on recent work which develops a scattering and an inverse scattering theory for the Sturm-Liouville equation u'' + qu = λ w u, where w may change sign but q is positive. Thus the left-hand side of the equation gives rise to a positive quadratic form, and one is led to a left-definite spectral problem. The crucial ingredient of the approach is a generalised transform built on the Jost solutions of the problem and hence termed the "Jost transform" and the associated Paley-Wiener theorem linking growth properties of transforms with support properties of functions. One motivation for this investigation comes from the Camassa-Holm equation for which the solution of the Cauchy problem can be achieved by the inverse scattering transform for u'' + qu = λ w u.

This is joint work with Christer Bennewitz (Lund, Sweden) and Rudi Weikard (Birmingham, AL).

Applied Analysis Seminar
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Posted September 28, 2016

Last modified February 1, 2017

David Shirokoff, New Jersey Institute of Technology

Approximate global minimizers to pairwise interaction problems through a convex/non-convex energy decomposition: with applications to self-assembly

Abstract: A wide range of particle systems are modeled through energetically driven interactions, governed by an underlying non-convex and often non-local energy. Although numerically finding and verifying local minima to these energies is relatively straight-forward, the computation and verification of global minimizers is much more difficult. Here computing the global minimum is important as it characterizes the most likely self-assembled arrangement of particles (in the presence of low thermal noise) and plays a role in computing the material phase diagram. In this talk I will examine a general class of model functionals: those arising in non-local pairwise interaction problems. I will present a new approach for computing approximate global minimizers based on a convex/non-convex splitting of the energy functional that arises from a convex relaxation. The approach provides a sufficient condition for global minimizers that may in some cases be used to show that lattices are exact, and also be used to estimate the optimality of any candidate minimizer. Physically, the approach identifies the emergence of new length scales seen in the collective behavior of interacting particles. (This is a joint Applied Analysis/Computational Mathematics seminar.)

Applied Analysis Seminar
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Posted September 1, 2017

3:30 pm Lockett 233
Yaniv Almog, Department of Mathematics, LSU

On a Schrödinger operator with a purely imaginary potential in the semiclassical limit

We consider the operator ${\mathcal A}_h=-h^2\Delta+iV$ in the semi-classical limit $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.

Applied Analysis Seminar
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Posted August 25, 2017

Last modified September 8, 2017

Changfeng Gui, University of Texas at San Antonio

The Sphere Covering Inequality and its applications

In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two {it distinct} surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least $4 pi$. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. Other applications of this inequality include the classification of certain Onsager vortices on the sphere, the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. The resolution of several open problems in these areas will be presented. The talk is based on joint work with Amir Moradifam from UC Riverside.

Applied Analysis Seminar
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Posted August 9, 2017

Last modified September 21, 2017

Giles Auchmuty, University of Houston

The SVD of the Poisson kernel

The Poisson kernel provides a representation for the solution operator for the Dirichlet problem for Laplace's equation on a bounded region. It is usually treated as an integral operator and this talk will describe spectral representations of this operator when the boundary data is in L^2(partialOmega). For this problem Fichera (1955) proved, under strong regularity conditions on the boundary, that the Poisson kernel is a continuous linear transformation of L^2(partialOmega) to L^2(Omega) and that it has norm related to the first eigenvalue of a Steklov eigenproblem for the biharmonic operator on Omega. In this talk two quite different representations of this operator using Steklov eigenfunctions and Hilbert space theory will be outlined. The first is based on the use of harmonic Steklov eigenfunctions. They may be used to develop a different theory of boundary trace spaces such as H^s(partialOmega). This yields spectral representation of solutions of Robin and Neumann boundary value problems for Laplace's equation as well as the Dirichlet problem. There are associated approximation theories and generalizations of results such as the mean value theorem to rectangles and boxes. When the domain is a ball, the results provide an analysis in terms of classical spherical harmonics. A weak version of the Dirichlet Biharmonic Steklov eigenproblem that Fichera studied will be described using Hilbert-Sobolev space methods. It can be shown that the normal derivatives of these eigenfunctions provide an orthonormal basis of L^2(partial Omega) while their Laplacians provide an L^2 orthogonal basis of harmonic functions on Omega. This yields an SVD of the Poisson kernel and the norm of the operator is related to the first Steklov eigenvalue of the problem.

Applied Analysis Seminar
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Posted August 7, 2017

Last modified October 7, 2017

Keng Deng, University of Louisiana at Lafayette

Global existence and blow-up for nonlinear diffusion equations with boundary flux governed by memory

In this talk, we introduce the study of global existence and blow-up in finite time for nonlinear diffusion equations with flux at the boundary governed by memory. Via a simple transformation, the memory term arises out of a corresponding model introduced in previous studies of tumor-induced angiogenesis. The study is also in the spirit of extending work on models of the heat equation with local, nonlocal, and delay nonlinearities present in the boundary flux. Specifically, we establish an identical set of necessary and sufficient conditions for blow-up in finite time as previously established in the case of local flux conditions at the boundary.

Applied Analysis Seminar
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Posted August 25, 2017

Last modified October 10, 2017

Ryan Hynd, University of Pennsylvania

Partial regularity for doubly nonlinear parabolic systems

We will present a regularity result for solutions of a PDE system which is a model for general doubly nonlinear evolutions. The system we focus on a particular case of a general class of flows that arise in the study of phase transitions.

Applied Analysis Seminar
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Posted September 13, 2017

2:30 pm - 3:20 pm Lockett 233
Jiahong Wu, Oklahoma State University

Partial differential equations related to fluids with partial or fractional dissipation

There have been substantial recent developments on several partial differential equations from fluid dynamics with partial or fractional dissipation. This talk summarizes results on the global existence and regularity problem for the 3D Navier-Stokes equations with partial hyperdissipation, the surface quasi-geostrophic equation, the 2D Boussinesq equations with partial or fractional dissipation and the 2D magnetohydrodynamic equations with partial or fractional dissipation.

Applied Analysis Seminar
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Posted August 15, 2017

Last modified October 11, 2017

Kun Zhao, Tulane University

Analysis of a System of Parabolic Conservation Laws Arising From Chemotaxis

In contrast to random diffusion without orientation, chemotaxis is the biased movement of organisms toward the region that contains higher concentration of beneficial or lower concentration of unfavorable chemicals. The former often refers to the attractive chemotaxis and latter to the repulsive chemotaxis. Chemotaxis has been advocated as a leading mechanism to account for the morphogenesis and self-organization of a variety of biological coherent structures such as aggregates, fruiting bodies, clusters, spirals, spots, rings, labyrinthine patterns and stripes, which have been observed in many laboratory experiments. Mathematical modeling of chemotaxis was initiated more than half a century ago. The Keller-Segel type model has provided a corner for much of the works investigating chemotaxis, its success being its intuitive simplicity, analytical tractability and capability of modeling the basic phenomena in chemotactic populations. In this talk, I will present a group of results concerning the rigorous analysis of a system of parabolic conservation laws derived from a Keller-Segel type chemotaxis model with singular sensitivity. In particular, global well-posedness, long-time asymptotic behavior, zero chemical diffusion limit and boundary layer formation of classical solutions will be discussed.

Applied Analysis Seminar
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Posted August 15, 2017

Last modified October 10, 2017

Zhifu Xie, University of Southern Mississippi

Variational method with SPBC and Broucek-Henon orbit and Schubart orbit

N-body problem concerns the motion of celestial bodies under universal gravitational attraction. Although it has been a long history to apply variational method to N-body problem, it is relatively new to make some important progress in the study of periodic solutions. We develop the Variational Method with Structural Prescribed Boundary Conditions (SPBC) and we apply it to study periodic solutions in the 3-body problem with equal masses. We show that under an appropriate topological constraint, the action minimizer must be either the Schubart orbit (1956) or the Broucke-Henon orbit (1975). One of the main challenges is to prove that the Schubart orbit coincides with the action minimizer connecting a collinear configuration with a binary collision and an isosceles configuration which must be collinear. A geometric property of the action minimizer is introduced to overcome this challenge. The action minimizer without collisions can be extended to the Broucke-Henon orbit.

Applied Analysis Seminar
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Posted November 30, 2017

Last modified December 3, 2017

Michael Malisoff, LSU

Stability And Robustness Analysis For A Multispecies Chemostat Model With Delays

Abstract: The chemostat is a laboratory device and a mathematical model for the continuous culture of microorganisms. Chemostat models have been studied extensively, because of their importance in biotechnology and ecology. This talk will discuss a chemostat model with an arbitrary number of competing species, one substrate, and constant dilution rates. We allow delays in the growth rates and additive uncertainties. Using constant inputs of certain species as controls, we derive bounds on the sizes of the delays that ensure asymptotic stability of an equilibrium when the uncertainties are zero, which can allow persistence of multiple species. Under delays and uncertainties, we provide bounds on the delays and on the uncertainties that ensure input-to-state stability with respect to uncertainties. No prerequisite background in biology or control theory will be necessary to understand and appreciate this talk.

Applied Analysis Seminar
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Posted January 11, 2018

Last modified February 20, 2018

Wei Li, LSU

Fluorescence ultrasound modulated optical tomography in diffusive regime

Fluorescence optical tomography (FOT) is an imaging technology that localizes fluorescent targets in tissues. FOT is unstable and of poor resolution in highly scattering media, where the propagation of multiply-scattered light is governed by the smoothing diffusion equation. We study a hybrid imaging modality called fluorescent ultrasound-modulated optical tomography (fUMOT), which combines FOT with acoustic modulation to produce high-resolution images of optical properties in the diffusive regime. The principle of fUMOT is to perform multiple measurements of photon currents at the boundary as the optical properties undergo a series of perturbations by acoustic radiation, in which way internal information of the optical field is obtained. We set up a Mathematical model for ufUMOT, prove well-posedness for certain choices of parameters, and present reconstruction algorithms and numerical experiments for the well-posed cases.

Applied Analysis Seminar
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Posted January 10, 2018

Last modified February 12, 2018

Masato Kimura, Kanazawa University, Japan

A phase field model for crack propagation and some applications

Applied Analysis Seminar
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Posted January 16, 2018

Last modified February 20, 2018

Tadele Mengesha, The University of Tennessee, Knoxville

Sobolev regularity estimates for solutions to spectral fractional elliptic equations

Abstract: Global Calderon-Zygmund type estimates are obtained for solutions to fractional elliptic problems over smooth domains. Our approach is based on the "extension problem" where the fractional elliptic operator is realized as a Dirichlet-to-Neumann map corresponding to a degenerate elliptic PDE in one more dimension. This allows the possibility of deriving estimates for solutions to the fractional elliptic equations from that of degenerate elliptic equations. We will confirm this first by obtaining weighted estimates for the gradient of solutions to a class of linear degenerate/singular elliptic problems over a bounded, possibly non-smooth, domain. The class consists of those with coefficient matrix that symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a particular weight that belongs to a Muckenhoupt class. The weighted estimates are obtained under a smallness condition on the mean oscillation of the coefficients with a weight. This is a joint work with T. Phan.

Applied Analysis Seminar
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Posted January 10, 2018

Last modified February 5, 2018

Prashant Kumar Jha, LSU

Numerical analysis of finite element approximation of nonlocal fracture models

We discuss nonlocal fracture model and present numerical analysis of finite element approximation. The peridynamic potential considered in this work is the regularized version of the bond-based potential generally considered in peridynamic literature (Silling 2000). In the limit of vanishing nonlocality, peridynamic model behaves like a elastodynamic model away from a crack zone and has a finite fracture energy associate to crack set (Lipton 2014, 2016).Using this property we relate the parameters in a peridynamic potential with given elastic constant and fracture toughness. Before we consider finite element approximation, we show that the problem is well posed. We show the existence of evolutions in H^2 space. We consider finite element discretization in space and central difference in time to approximate the problem. Approximation is shown to converge in L^2 norm at the rate Cttriangle t+C_sh^2/s^2. Here triangle t is the size of time step, h is the mesh size, and is the size of horizon (nonlocal scale). Constants C_t and C_s are independent of h and triangle t. In the absence of nonlinearity, stability of approximation is shown. Numerical results are presented to verify the convergence rate. This is a joint work with Robert Lipton.

Applied Analysis Seminar
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Posted March 18, 2018

3:30 pm - 4:30 pm Lockett 233
Stephen Shipman, Mathematics Department, LSU

Reducibility of the Fermi surface for periodic quantum-graph operators

The Fermi, or Floquet, surface for a periodic operator at a given energy level is an algebraic variety that describes all complex wave vectors admissible by the periodic operator at that energy. Its reducibility is intimately related to the construction of embedded eigenvalues supported by local defects. The rarity of reducibility is reflected in the fact that a generic polynomial in several variables cannot be factored. The "easy" mechanism for reducibility is symmetry. However, reducibility ensues in much more general and interesting situations. This work constructs a class of non-symmetric periodic Schrodinger operators on metric graphs (quantum graphs) whose Floquet surface is reducible. The graphs in this study are obtained by coupling two identical copies of a periodic quantum graph by edges to form a bilayer graph. Reducibility of the Floquet surface for all energies ensues when the coupling edges have potentials belonging to the same asymmetry class, that is, when their "spectral A-functions" are identical. If the potentials of the connecting edges belong to different asymmetry classes, then typically the Floquet surface is not reducible. Bilayer graphene is a notable exception--its Floquet surface is always reducible.

Applied Analysis Seminar
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Posted April 2, 2018

3:30 pm - 4:30 pm Lockett 233
Ivan Gudoshnikov, The University of Texas at Dallas

Stabilization of quasistatic evolution of elastoplastic systems subject to periodic loading

We consider an arrangement of m elastoplastic springs (elastoplastic system) that are connected according to a given graph. Each spring i is described by both elastic e_i and plastic p_i strains, but only the elastic strains e_i generate stress responses s_i. We develop an analytic framework to design time-periodic loadings which make the evolution s(t) of the stress vector s = (s_1, ..., s_m) converging to a globally asymptotically stable time-periodic regime.
The core of our approach is in converting the problem into a sweeping process with a moving polyhedron, which was earlier proposed by Moreau [C.I.M.E. notes, 1974]. We prove that global stability of a unique periodic regime takes place if the moving polyhedron is a simplex, which we further link to a simple topological property of the elastoplastic system under consideration.
To illustrate the abstract theorem, sample sweeping processes are solved numerically by the so-called catch-up algorithm (which we implement using a constrained quadratic optimization pack-age).

The preprint is available at https://arxiv.org/abs/1708.03084.

Applied Analysis Seminar
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Posted August 13, 2018

Last modified September 2, 2018

Ian Wood, School of Mathematics, University of Kent

Boundary triples and spectral information in abstract M-functions

Abstract: The Weyl-Titchmarsh m-function is an important tool in the study of forward and inverse problems for ODEs, as it contains all the spectral information of the problem. The abstract setting of boundary triples allows the introduction of an abstract operator M-function. It is then interesting to study how much spectral information is still contained in the M-function in this more general setting. Boundary triples allow for the study of PDEs, block operator matrices and many other problems in one framework. We will discuss properties of M-functions, their relation to the resolvent and the spectrum of the associated operator, and connections to the extension theory of operators.

Applied Analysis Seminar
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Posted August 29, 2018

Lockett 232
Malcolm Brown, Department of Computer Science & Informatics, Cardiff University

Spectral problems on star graphs

We report on a two-step reduction method for spectral problems on a star graph with n+1 edges and a self-adjoint matching condition at the central vertex . The first step is a reduction to the problem on a single edge but with an energy depending boundary condition at the vertex. In the second step, by means of an abstract inverse result for m-functions, , a reduction to a problem on a path graph with two edges joined by continuity and Kirchhoff conditions is given. All results are proved for symmetric linear relations in an orthogonal sum of Hilbert spaces. This ensures wide applicability to various different realisations, in particular, to canonical systems and Krein strings which include, as special cases, Dirac systems and Stieltjes strings. Employing two other key inverse results by de Branges and Krein, we answer the question: If all differential operators are of one type, when can the reduced system be chosen to consist of two differential operators of the same type? This is joint work with Heinz Langer and Christine Tretter

Applied Analysis Seminar
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Posted September 12, 2018

Last modified September 21, 2018

Aynur Bulut, LSU

Logarithmically energy-supercritical Nonlinear Wave Equations: axial symmetry and global well-posedness

In nonlinear dispersive PDE, radial symmetry often plays a key role in allowing for more refined analysis of the nonlinear interactions which could lead to possible blowup. We will describe recent work where we have recently introduced a mechanism for relaxing assumptions of radiality by considering symmetry in a subset of the variables (for instance, assuming that the initial data is axially symmetric). We applied this philosophy to show global well-posedness and scattering in for the nonlinear wave equation in the logarithmically energy-supercritical setting, generalizing a result of Tao which was established for the radial case. The uses Morawetz and Strichartz estimates that have been adapted to the new symmetry assumption. These methods in fact bring a new perspective to sharp estimates for the energy-critical problem, along the lines of the influential work of Ginibre, Soffer, and Velo. This is joint work with B. Dodson

Applied Analysis Seminar
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Posted August 31, 2018

Last modified September 12, 2018

Karthik Adimurthi, Seoul National University

Partial existence result for Homogeneous Quasilinear parabolic problems beyond the duality pairing

Applied Analysis Seminar
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Posted August 25, 2018

Last modified August 29, 2018

Jiuyi Zhu, LSU

Nodal sets for Robin and Neumann eigenfunctions

We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the nodal sets is obtained for the Robin eigenfunctions. For the analytic domains, we show a sharp upper bound for the nodal sets on the boundary of the Robin and Neumann eigenfunctions. Furthermore, the sharp doubling inequality and vanishing order are obtained.

Applied Analysis Seminar
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Posted September 13, 2018

Last modified October 18, 2018

Blaise Bourdin, Department of Mathematics, Louisiana State University

Variational phase-field models of fracture

Since their inception, over 20 years ago, variational phase-field models of fracture have become widely popular. Part of their success is undoubtedly due to their ability to be efficiently implemented in two and three space dimension, and to their demonstrated ability to capture complex fracture behavior in a wide range of situations. In this presentation, I will go back to the roots of this family of models, deriving Francfort and Marigo's variational approach to fracture from Griffith's classical theory. I will construct variational phase-field models as a numerical approximation for this approach. I will present numerical simulation highlighting the properties of this approximation, as well as some that cannot be fully explained by the mathematical theory. I will then describe an alternate construction as gradient-damage models can explain this behavior and will show how this dual view can address some of the long standing issues in the modeling of brittle solids, including crack nucleation and size effect. Finally, I will discuss ongoing extensions and open issues.

Applied Analysis Seminar
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Posted August 29, 2018

3:30 pm - 4:30 pm Lockett 233
Robert Lipton, Mathematics Department, LSU

Predicting complex fracture evolution using nonlocal dynamics

The dynamic fracture of brittle solids is a particularly interesting collective interaction connecting both large and small length scales. Apply enough stress or strain to a sample of brittle material and one eventually snaps bonds at the atomistic scale leading to fracture of the macroscopic specimen. We discuss a nonlocal model for calculating dynamic fracture. The force interaction is derived from a double well strain energy density function, resulting in a non- monotonic material model. The material properties change in response to evolving internal forces eliminating the need for a separate phase field to model the fracture set. (However there is no free lunch and the discrete model is posed in terms of a dense matrix and parallel computation must be used to solve fracture problems.) The model can be viewed as a regularized fracture model. In the limit of zero nonlocal interaction, the model recovers a sharp interface evolution characterized by the classic Griffith free energy of brittle fracture with elastic deformation satisfying the linear elastic wave equation off the crack set. We conclude with a numerical analysis of the model which is joint work with Prashant Jha.

Applied Analysis Seminar
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Posted August 29, 2018

3:30 pm - 4:30 pm Lockett 233
Robert Lipton, Mathematics Department, LSU

Understanding nonlocal models for fracture simulation

The peridynamic model is increasingly being used and developed for fracture simulation. In this talk we go "under the hood" to see how nonlocal models can capture the fracture process and to see how they relate to existing fracture models. Along the way we show how the peridynamic energy is related to the Griffiths fracture energy and how the nonlocal evolution satisfies the principle of least action.

Applied Analysis Seminar
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Posted September 10, 2018

Last modified October 30, 2018

Yuri Antipov, Mathematics Department, LSU

Method of automorphic functions for an inverse problem of antiplane elasticity

A nonlinear inverse problem of antiplane elasticity (theory of harmonic functions) for a multiply connected domain is examined. It is required to determine the profile of $n$ uniformly stressed inclusions when the surrounding infinite body is subjected to antiplane uniform shear at infinity. A method of conformal mappings for circular multiply connected domains is employed. The conformal map is recovered by solving consequently two Riemann-Hilbert problems for piecewise analytic symmetric automorphic functions. For domains associated with the first class Schottky symmetry groups a series-form representation of a ($3n-4$)-parametric family of conformal maps solving the problem is discovered. Numerical results for two and three uniformly stressed inclusions are reported and discussed.

Applied Analysis Seminar
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Posted September 12, 2018

Last modified November 18, 2018

Wei Li, LSU

Embedded eigenvalues for the Neumann-Poincare operator

The Neumann-Poincare operator and its adjoint are boundary-integral operators associated with harmonic layer potentials. We proved the existence of eigenvalues in the essential spectrum for the Neumann-Poincare operator for certain Lipschitz curves in the plane with reflectional symmetry, when considered in the functional space in which it is self-adjoint. The proof combines the compactness of the Neumann-Poincare operator for curves of class C^2 with the continuous spectrum generated by a corner. Even (odd) eigenfunctions are proved to lie within the continuous spectrum of the odd (even) component of the operator when a C^2 curve is perturbed by inserting a small corner.

Applied Analysis Seminar
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Posted March 11, 2019

3:30 pm Lockett 233
Alexander Ioffe, Technion

Topics in Variational Analysis

Applied Analysis Seminar
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Posted April 10, 2019

3:30 pm Lockett 233
Peter Wolenski, LSU Department of Mathematics
Russell B. Long Professor

Fully convex Bolza problems with state constraints and impulses

A Fully Convex Bolza (FCB) problem has the appearance of the classical calculus of variations Bolza problem
min ∫_{0}^{T} L(x(t),x'(t))dt + l(x(0),x(T))
where the minimization is over x() belonging to some class of arcs. The distinguishing features of FCB are that the data L(,) and l(,) (i) may take on the value infinitiy and (ii) are convex functions. Allowance of (i) provides great flexibility incorporating constraints so that most standard control problems come under its purlieu. However, broad generality is restrained by (ii), which although quite special, nonetheless includes the classical linear quadratic regulator and many of its generalizations. Furthermore, (ii) opens up the applicabililty of the tools of convex ananlysis. We shall review the Hamilton-Jacobi (HJ) theory for FCB problems when the data has no implicit state constraints and is coercive, in which case the minimizing class of arcs are Absolutely Continuous (AC). When a state constraint x(t) ∈ X is added to the problem formulation, the dual variable may exhibit an impulse or "jump" when the constraint is active. The two properties of a state constraint and noncoercive data (which induce impulsive behavior) are in fact dual to each other, and the minimizing class becomes those of bounded variation. We shall describe Rockafellar's optimality conditions for these problems and a new technique for approximating them by AC problems that utilizes Goebel's self-dual envelope. The approximating AC problems maintain duality and the existing theory can be applied to them. It is proposed that an HJ theory can be developed for BV problems as an appropriate limit of the approximating AC problems. An explicit example will illustrate this.