Welcome to the homepage of the Applied Analysis Graduate Student Seminar, formerly the Student Seminar on Control Theory and Optimization. The seminar is a series of informal and interdisciplinary talks by graduate students on research interests and recent work. Listed below are the schedule of talks and files associated with those talks. If you are interested in giving a talk or being added to the contact list, please contact Rick Barnard at rbarnard@math.lsu.edu. Talks will be in Prescott 203 from 3:00pm and on Mondays, unless otherwise noted.

Spring 2009

1. February 26, 2009, 4:00pm, Lockett 233
   Optimal lower bounds on the strain and stress inside prestressed, random two-phase elastic composites Yue Chen I will present the optimal lower bounds for the L^p norm of pre stress inside random media, and also give the microstructure which attains some optimal lower bounds. This is the research work under the direction of Prof. Lipton.

Fall 2008

1. November 26, 2008 Strong Approximation of Local Fields in Nonlinear Power Law Materials and Application Silvia Jimenez We focus on developing a corrector theory that provides a strong approximation for local fields in Nonlinear Power Law Materials. The approximations are used to derive a lower bound that can be used to assess the singularity strength inside micro-structured materials.

1. December 3, 2008 Multigrid Methods for a Class of Discontinuous Galerkin Methods on Graded Meshes by Jintao Cui In this talk we discuss multigrid solvers for systems resulting from the discretization of second order elliptic boundary value problems by a class of stable and consistent discontinuous Galerkin (DG) methods on graded meshes. Quasi-optimal error estimates in both the energy norm and the L_2 norm for this class of DG methods are derived and uniform convergence of the W-cycle multigrid algorithm for the resulting discrete problem is proved. We will present both theoretical and numerical results. This is joint work with Susanne C. Brenner, Thirupathi Gudi and Li-yeng Sung.

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2. December 5, 2008 Mathematical methods in kinesiology and voice analysis: two case studies by Alvaro Guevara at 1:30 in Lockett 233A growing number of research projects have used mathematics as a tool to integrate approaches from many disciplines. Two case studies of this type will be discussed, namely, (1) local stability properties of human gait, and (2) voice data analysis of populations at risk for developing schizophrenia-related disorders. In these projects, mathematical ideas from nonlinear dynamical systems were used in (1), and Shannon entropy and information theory in (2). We will describe our contributions that implemented the theory, and generated effective quantitative tools that provided fresh insights to the researchers of these studies. This research was conducted in the context of the Mathematical Consultation Clinic at LSU. Finally, I will briefly address my dissertation work, involving impulsive solutions to optimal control problems.

Spring 2008

1. January 23, 2008 Survey on Estimation Algorithms for Networked Control Systems using UDP-like Communication by Laurentiu Marinovici Nowadays, many applications demand remote control of applications over unreliable networks. Therefore they require communication and control fields to become tightly coupled since issues like communication delay, data loss or time synchronization between components cannot be addressed independently. The surveyed papers introduce and test an effective algorithm for estimating the states of a plant and controlling them through a UDP-like communication.

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2. January 30, 2008 A Proof of the Uniform Boundedness Principle using Continuous Functions by Santiago Fortes I will give a short talk presenting Osgood's very cool proof of the UBD using a boundedness property of pointwise bounded families of real-valued continuous functions defined on the real line.

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3. February 13, 2008 Introduction to Nonsmooth Analysis and Control Theory I by Lingyan Huang

1. February 20, 2008 Introduction to Nonsmooth Analysis and Control Theory II by Jacob Blanton

1. February 27, 2008 Introduction to Nonsmooth Analysis and Control Theory III by Jacob Blanton

1. March 12, 2008 Local boundary behavior of harmonic and analytic functions: Abelian theorems for quasiasymptotics of distributions by Jasson Vindas The aims of this talk are to give a brief introduction to the concept of quasiasymptotic behavior of distributions and present some new abelian results for harmonic and analytic functions on the upper semiplane admitting distributional boundary values. The plan of the talk is as follows. I will only consider the one dimensional case. We review the concept of harmonic and analytic representations of distributions. The usual methods for constructing explicit harmonic and analytic representations are discussed. These methods include representations via the Poisson kernel, the Cauchy transform and the Fourier transform. I then proceed to introduce the local concept of quasiasymptotic of Schwartz distributions at points. Finally, the abelian theorems are presented. For analytic and harmonic functions on the upper semiplane admitting distributional boundary values, these abelian theorems describe the angular asymptotic behavior of the functions at approaching boundary points where the boundary distribution has quasiasymptotic behavior.

1. March 26, 2008 Distributed Fast Marching Methods by Cristina Tugurlan Fast Marching Methods are efficient algorithms for solving problems of front evolution where the front speed is monotonic. They are theoretically optimal in terms of operation count. They are also highly sequential and hence not straightforward to parallelize. I will present several parallel implementations of the Fast Marching Method. In these implementations one combines fast sweeping with fast marching, in such a way that allows fast convergence. I will illustrate the power of these approaches on some numerical examples, show the monotonicity and stability properties of the algorithms, and study the convergence and the error estimates.

Fall 2007

1. September 5, 2007 Introduction to Convex Analysis I by Alvaro Guevara

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2. September 12, 2007 Introduction to Convex Analysis II by Alvaro Guevara

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3. September 19, 2007 Lower Bounds on Field Concentrations by Silvia Jimenez

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4. September 26, 2007 Introduction to Differential Inclusions by Rick Barnard In this talk we look at the basic selection problem for differential inclusions. Differential inclusions are similar to differential equations, but with multifunctions as the right hand side. We present several selection results, relaxation techniques, and finish with the basic existence of measurable selections for such problems under mild hypotheses. References
   1. Aubin, J.P. and Cellina, A. Differential Inclusions.Springer-Verlag, Berlin, 1984.
   2. Clarke, F.H. Optimization and Nonsmoorth Analysis. Centre de Recherches Mathematiques, Montreal, 1989.
   3. Smirnov, G.S. Introduction to the Theory of Differential Inclusions. AMS, Providence, 2002.

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5. October 17, 2007 Introduction to Multi-objective optimal control problem by Qinqxia Li In the first part of my talk, I discuss the Hamiltonian necessary conditions for a nonsmooth function with endpoint constraints involving a general preference. Then I also briefly discuss the Hamiltonian necessary conditions for a nonsmooth function with both endpoint and differential inclusion constraints.

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6. October 24, 2007 Dispersion Relations for Layered Media by Santiago Fortes I will present a problem solved by Lord Kelvin which introduces the concept of a dispersion relation.

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7. October 31, 2007 A Survey of Hybrid Control Systems by Rick Barnard We look at several different frameworks for hybrid systems. Following an introduction to basic terminology from optimal control theory, we look at multiprocesses from [1], hybrid systems using the structure from [4], and finally stratified domains in [2]. Each of these problems have necessary conditions for optimality proven that are analogous to the Pontryagin Maximum Principle and/or the familiar Hamilton Jacobi Theory results from standard optimal control theory. We finish by looking at the framework of [3]. References
   1. Clarke, F.H.; Vinter, R.B., Applications of Optimal Multiprocesses, SIAM J. Control and Optimization 27 no. 5 (1989) 1048-1071.
   2. Bressan, A.; Hong, Y. Optimal Control Problems on Stratified Domains, Networks and Heterogeneous Media 2 no. 2 (2007) 313-331.
   3. Goebel, R.; Teel, A. R., Solutions to hybrid inclusions via set and graphical convergence with stability theory applications, Automatica J. IFAC 42 no. 4, (2006) 573-587.
   4. Garavello, M; Piccoli, B, Hybrid necessary principle, SIAM J. Control Optim. 43 no. 5, (2005) 1867-1887.

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8. November 7, 2007 Quick introduction to Lagrange multipliers and duality theory by Alvaro Guevara

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9. November 14, 2007 Some asymptotic notions for Schwartz distributions by Jasson Vindas
10. November 28, 2007 Rational approximation of semigroups by Patricio Jara We start this talk by discussing what semigroups of linear operators are, and by showing that natural examples of PDE's can be solved by semigroup theory. Then we look at the functional calculus developed by E.Hille and R. Phillips [4] for strongly continuos semigroups, and we use it in order to show the classical work on rational approximation of strongly continuous semigroups initiated by R.Hersh & T. Kato [3] and continued by P. Brenner & V. Thomée [1]. Then we show how the rational approximations of semigroups translate into novel and fast inversion formulas for the Laplace transform [5]. The figures below show two different approximations for the solution of the stiff ODE of Curtiss-Hirshfelder [2] given by
    
    
    

    Figure 1: Backward Euler	Figure 2: Pade 21

    
    
    For other examples see the Laplace transform inversion website. We finish by discussing some interesting open questions concerning the approximation methods.
    References
    
    1. P. Brenner and V. Thomée. On rational approximations of semigroups. SIAM J. Numer. Anal. 16 (1979), 683-694.
    2. C.F. Curtiss and J.O. Hirshfelder. Integration of Stiff Equations. Proc. Natl. Acad. Sci. 38, 1952, 235-243.
    3. R. Hersh and T. Kato. High-accuracy stable difference schemes for well-posed initial value problems. SIAM J. Numer. Anal. 16 (1979), 670-682.
    4. E. Hille and R.S. Phillips. Functional Analysis and Semigroups. Rev. Ed. Amer. Math. Soc. Coll. Publ., vol 31, Amer. Math. Soc., 1957.
    5. P. Jara, F. Neubrander, and K. Ozer. Rational inversion of the Laplace transform. Preprint.

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11. December 5, 2007 Optimal Lower Bounds on the Stress and Strain Fields Concentrations in Random Media by Bacim Alali We consider a composite material made from two perfectly bonded isotropic linear elastic components. A prescribed uniform stress is applied to the composite. We develop lower bounds on the maximum stress generated inside the composite for all mixtures made with two elastic materials in fixed volume fractions. We outline various cases in which these lower bounds are optimized by simple configurations of the two materials. References
    1. Alali, B. and Lipton, R. (2007). Optimal lower bounds on the stress and strain fields inside random two-phase elastic composites."[in preparation]
    2. Allaire, G. and R.V. Kohn (1993). Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions."Quart. Appl. Math. 51, pp. 675-699.
    3. Grabovsky, Y. and Kohn, R. V., (1995). Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The Vigdergauz microstructure." J. Mech. Phys. Solids. 43, pp. 949-972.
    4. Hashin, Z. and Shtrikman, S., (1963). A variational approach to the theory of the elastic behavior of multiphase materials." J. Mech. Phys. Solids. 11, pp. 127-140.
    5. Kelly, A. and Macmillan, N.H., (1986). Strong Solids. Monographs on the Physics and Chemistry of Materials. Clarendon Press, Oxford.
    6. Lipton R., (2004). Optimal lower bounds on the electric-field concentration in composite media." Journal of Applied Physics, 96, pp. 2821-2827.
    7. Lipton R., (2005). Optimal lower bounds on the hydrostatic stress amplification inside random two-phase elastic composites." Journal of the Mechanics and Physics of Solids, 53, pp. 2471-2481.
    8. Lipton R., (2006). Optimal lower bounds on the dilatational strain inside random two-phase elastic composites subjected to hydrostatic loading." Mechanics of Materials, 38, pp. 833-839.
    9. Milton, G.W., (1980). Bounds on the complex dielectric constant of a composite material." Appl. Phys. Lett., 37, pp. 300-302.

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