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Today, Thursday, March 28, 2024

Posted March 19, 2024

Computational Mathematics Seminar

3:30 pm Digital Media Center 1034

Yue Yu, Lehigh University
Nonlocal operator is all you need

During the last 20 years there has been a lot of progress in applying neural networks (NNs) to many machine learning tasks. However, their employment in scientific machine learning with the purpose of learning physics of complex system is less explored. Differs from the other machine learning tasks such as the computer vision and natural language processing problems where a large amount of unstructured data are available, physics-based machine learning tasks often feature scarce and structured measurements. In this talk, we will take the learning of heterogeneous material responses as an exemplar problem, to investigate the design of neural networks for physics-based machine learning. In particular, we propose to parameterize the mapping between loading conditions and the corresponding system responses in the form of nonlocal neural operators, and infer the neural network parameters from high-fidelity simulation or experimental measurements. As such, the model is built as mappings between infinite-dimensional function spaces, and the learnt network parameters are resolution-agnostic: no further modification or tuning will be required for different resolutions in order to achieve the same level of prediction accuracy. Moreover, the nonlocal operator architecture also allows the incorporation of intrinsic mathematical and physics knowledge, which improves the learning efficacy and robustness from scarce measurements. To demonstrate the applicability of our nonlocal operator learning framework, three typical scenarios in physics-based machine learning will be discussed: the learning of a material-specific constitutive law, the learning of an efficient PDE solution operator, and the development of a foundational constitutive law across multiple materials. As an application, we learn material models directly from digital image correlation (DIC) displacement tracking measurements on a porcine tricuspid valve leaflet tissue, and show that the learnt model substantially outperforms conventional constitutive models.

Monday, April 1, 2024

Posted January 22, 2024
Last modified March 4, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (click here to join)

Boris Kramer, University of California San Diego
Scalable Computations for Nonlinear Balanced Truncation Model Reduction

Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems on nonlinear manifolds and preserves either open- or closed-loop observability and controllability aspects of the nonlinear system. Two computational challenges have so far prevented its deployment on large-scale systems: (a) the computation of Hamilton-Jacobi-(Bellman) equations that are needed for characterization of controllability and observability aspects, and (b) efficient model reduction and reduced-order model (ROM) simulation on the resulting nonlinear balanced manifolds. We present a novel unifying and scalable approach to balanced truncation for large-scale control-affine nonlinear systems that consider a Taylor-series based approach to solve a class of parametrized Hamilton-Jacobi-Bellman equations that are at the core of balancing. The specific tensor structure for the coefficients of the Taylor series (tensors themselves) allows for scalability up to thousands of states. Moreover, we will present a nonlinear balance-and-reduce approach that finds a reduced nonlinear state transformation that balances the system properties. The talk will illustrate the strength and scalability of the algorithm on several semi-discretized nonlinear partial differential equations, including a nonlinear heat equation, vibrating beams, Burgers' equation and the Kuramoto-Sivashinsky equation.

Monday, April 1, 2024

Posted March 26, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett Hall 233

Wei Li, DePaul University
TBA

Tuesday, April 2, 2024

Posted March 28, 2024

Faculty Meeting Questions or comments?

2:30 pm – 3:20 pm Zoom

Meeting of Tenured Faculty

Tuesday, April 2, 2024

Posted November 14, 2023
Last modified March 26, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom

Micah Milinovich, University of Mississippi
Biases in the gaps between zeros of Dirichlet L-functions

We describe a family of Dirichlet L-functions that provably have unusual value distribution and experimentally have a significant and previously undetected bias in the distribution of gaps between their zeros. This has an arithmetic explanation that corresponds to the nonvanishing of a certain Gauss-type sum. We give a complete classification of the characters for when these sums are nonzero and count the number of corresponding characters. It turns out that this Gauss-type sum vanishes for 100% of primitive Dirichlet characters, so L-functions in our newly discovered family are rare (zero density set amongst primitive characters). If time allows, I will also describe some newly discovered experimental results concerning a "Chebyshev-type" bias in the gaps between the zeros of the Riemann zeta-function. This is joint work with Jonathan Bober (Bristol) and Zhenchao Ge (Waterloo).

Wednesday, April 3, 2024

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233

Huong Vo, Louisiana State University
TBA

Wednesday, April 3, 2024

Posted December 1, 2023
Last modified March 18, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233

Neal Stoltzfus, Mathematics Department, LSU
TBA

Thursday, April 4, 2024

Posted March 28, 2024

Combinatorics Seminar Questions or comments?

2:00 pm – 3:00 pm Lockett 233 (Simulcasted via Zoom)

Laszlo Szekely, University of South Carolina
Tanglegrams with the largest crossing number

A tanglegram consists of two binary trees with the same number of leaves, a left binary tree and a right binary tree, and a perfect matching between the leaves of the two trees. The size of a tanglegram is the number of matching edges. Tanglegrams are drawn in a special way. Leaves of the left tree must be on the line $x=0$, leaves of the right tree must be on the line $x=1$, the left binary tree is a plane tree in the halfplane $x\leq 0$, the right binary tree is a plane tree in the halfplane $x\geq 1$, and the perfect matching must be drawn in straight line segments. Such a drawing is called a layout of the tanglegram. The crossing number of a layout is the number of unordered pairs of matching edges that cross, while The crossing number of a tanglegram is the least number of crossings in layouts of this tanglegram. It is easy to see that the crossing number of a size $n$ tanglegram is at most $\binom{n}{2}$. Anderson, Bai, Barrera-Cruz, Czabarka, Da Lozzo, Hobson, Lin, Mohr, Smith, Sz\'ekely, and Whitlatch [Electronic J. Comb. {25}(4) (2018) \#P4.24] observed that the crossing number of any tanglegram is strictly less than $\frac{1}{2}\binom{n}{2}$, but some $n$, some tanglegrams have crossing number at least $\frac{1}{2}\binom{n}{2}-\frac{n^{3/2}-n}{2}$. In the current work we show on the one hand that the crossing number of any tanglegram is at most $\frac{1}{2}\binom{n}{2} -\Omega(n)$, and on the other hand that for some $n$, some tanglegrams have crossing number at least $\frac{1}{2}\binom{n}{2}-O(n\log n)$.

Friday, April 5, 2024

Posted March 27, 2024

Graduate Student Colloquium

11:30 am – 1:00 pm Lockett 284

PDE Structures: Finite Elements, Data Science and the Search for Efficient Solutions

Recent advancements in smart materials have significantly influenced the complexity of partial differential equation (PDE) structures, which frequently exhibit material discontinuities and intricate boundary conditions, especially with PDE systems. As we transition further into the age of artificial intelligence, researchers are increasingly exploring machine and deep learning methodologies to derive PDE solutions. However, success has been limited when considering control of distributed parameter systems which is supported by finite element theory. This presentation will present recent findings in generating PDE solutions utilizing both finite elements with adapted bases and hybrid techniques while striving to uphold infinite-dimensional distributed parameter control theory. The discussion will include results of one and two-dimensional clamped structures employing Euler-Bernoulli beams and isotropic plates. Computational methodologies such as modified higher-order bases and neural finite elements will be elaborated upon.

Friday, April 5, 2024

Posted March 28, 2024

Combinatorics Seminar Questions or comments?

2:00 pm – 3:00 pm Lockett Hall 233 (Simulcasted via Zoom)

Eva Czabarka, University of South Carolina
Maximum diameter of $k$-colorable graphs

Between 1965 and 1989 several people showed that the diameter of an $n$-vertex connected graph $G$ with minimum degree $\delta$ is at most $\frac{3n}{\delta+1}-1$. In 1989 Erd\H{o}s, Pach, Pollack and Tuza posed the following conjecture: For fixed integers $r,\delta\geq 2$, for any connected graph $G$ with minimum degree $\delta$ and order $n$ we have (1) If $G$ is $K_{2r}$-free and $\delta$ is a multiple of $(r-1)(3r+2)$ then, as $n\rightarrow \infty$, $$ \operatorname{diam}(G) \leq \frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}{\delta} + O(1)=\left(3-\frac{2}{2r-1}-\frac{1}{(2r-1)(2r^2-1)}\right)\frac{n}{\delta}+O(1). $$ (2) If $G$ is $K_{2r+1}$-free and $\delta$ is a multiple of $3r-1$, then, as $n\rightarrow \infty$, $$\operatorname{diam}(G) \leq \frac{3r-1}{r}\cdot \frac{n}{\delta} + O(1)=\left(3-\frac{2}{2r}\right)\frac{n}{\delta}+O(1). $$ Erd\H{o}s, Pach, Pollack and Tuza also created examples that show that the above conjecture, if true, is tight. Not much progress was made till 2009, when Czabarka, Dankelman and Sz\'ekely showed that for $r=2$ a weaker version of (2) holds: For every connected $4$-colorable graph $G$ of order $n$ and minimum degree $\delta\ge 1$, $ \operatorname{diam}(G) \leq \frac{5n}{2\delta}-1.$ This suggests a weakening of the conjecture by replacing the condition $K_{k+1}$-free with $k$-colorability. With Inne Singgih and L\'aszl\'o A. Sz\'ekely we provided conterexamples of part (1) of the conjecture in both versions (forbidden clique size or colorability) for every $r\ge 2$ for large enough $\delta$. These examples give that, if we are to bound the diameter of a $K_{k+1}$-free $n$-vertex graph with minimum degree $\delta$ by $C_k\cdot\frac{n}{\delta}$, then $C\ge 3-\frac{2}{k}$ regardless of the parity of $k$. With Stephen Smith and L\'aszl\'o A. Sz\'ekely we showed that this modified conjecture holds for both $3$- and $4$-colorable graphs (the latter result is an alternative and shorter proof to the 2009 result).

Monday, April 8, 2024

Posted January 23, 2024
Last modified March 4, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (click here to join)

Luca Zaccarian, LAAS-CNRS and University of Trento
Lyapunov-Based Reset PID for Positioning Systems with Coulomb and Stribeck Friction

Reset control systems for continuous-time plants were introduced in the 1950s by J.C. Clegg, then extended by Horowitz twenty years later and revisited using hybrid Lyapunov theory a few decades ago, to rigorously deal with the continuous-discrete interplay stemming from the reset laws. In this talk, we provide an overview a recent research activity where suitable reset actions induce stability and performance of PID-controlled positioning systems suffering from nonlinear frictional effects. With the Coulomb-only effect, PID feedback produces a nontrivial set of equilibria whose asymptotic (but not exponential) stability can be certified by using a discontinuous Lyapunov-like function. With velocity weakening effects (the so-called Stribeck friction), the set of equilibria becomes unstable with PID feedback and the so-called ''hunting phenomenon'' (persistent oscillations) is experienced. Resetting laws can be used in both scenarios. With only Coulomb friction, the discontinuous Lyapunov-like function immediately suggests a reset action providing extreme performance improvement, preserving stability and increasing the convergence speed. With Stribeck, a more sophisticated set of logic-based reset rules recovers the global asymptotic stability of the set of equilibria, providing an effective solution to the hunting instability.

Monday, April 8, 2024

Posted February 21, 2024

Probability Seminar Questions or comments?

3:30 pm

Jessica Lin, McGill University
TBA

TBA

Monday, April 8, 2024

Posted February 19, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232

Jessica Lin, McGill University
TBA

Wednesday, April 10, 2024

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233

Nilangshu Bhattacharyya, Louisiana State University
Characteristic Classes

Wednesday, April 10, 2024

Posted December 6, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233

Joseph Breen, University of Iowa
TBA

Monday, April 15, 2024

Posted January 27, 2024
Last modified March 4, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (click here to join)

Sergey Dashkovskiy , Julius-Maximilians-Universität Würzburg
Stability Properties of Dynamical Systems Subjected to Impulsive Actions

We consider several approaches to study stability and instability properties of infinite dimensional impulsive systems. The approaches are of Lyapunov type and provide conditions under which an impulsive system is stable. In particular we will cover the case, when discrete and continuous dynamics are not stable simultaneously. Also we will handle the case when both the flow and jumps are stable, but the overall system is not. We will illustrate these approaches by means of several examples.

Tuesday, April 16, 2024

Posted March 19, 2024

Computational Mathematics Seminar

until 3:30 pm Digital Media Center 1034

Quoc Tran-Dinh, UNC Chapel Hill
Boosting Convergence Rates for Fixed-Point and Root-Finding Algorithms

Approximating a fixed-point of a nonexpansive operator or a root of a nonlinear equation is a fundamental problem in computational mathematics, which has various applications in different fields. Most classical methods for fixed-point and root-finding problems such as  fixed-point or gradient iteration, Halpern's iteration, and extragradient methods have a convergence rate of at most O(1/square root k) on the norm of the residual, where k is the iteration counter. This convergence rate is often obtained via appropriate constant stepsizes. In this talk, we aim at presenting some recent development to boost the theoretical convergence rates of many root-finding algorithms up to O(1/k). We first discuss a connection between the Halpern fixed-point iteration in fixed-point theory and Nesterov's accelerated schemes in convex optimization for solving monotone equations involving a co-coercive operator (or equivalently, fixed-point problems of a nonexpansive operator). We also study such a connection for different recent schemes, including extra anchored gradient method to obtain new algorithms. We show how a faster convergence rate result from one scheme can be transferred to another and vice versa. Next, we discuss various variants of the proposed methods, including randomized block-coordinate algorithms for root-finding problems,which are different from existing randomized coordinate methods in optimization. Finally, we consider the applications of these randomized coordinate schemes to monotone inclusions and finite-sum monotone inclusions. The algorithms for the latter problem can be applied to many applications in federated learning.

Wednesday, April 17, 2024

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233

Megan Fairchild, Louisiana State University
TBA

Monday, April 22, 2024

Posted January 6, 2024
Last modified March 4, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (click here to join)

Madalena Chaves, Centre Inria d'Université Côte d'Azur
Coupling, Synchronization Dynamics, and Emergent Behavior in a Network of Biological Oscillators

Biological oscillators often involve a complex network of interactions, such as in the case of circadian rhythms or cell cycle. Mathematical modeling and especially model reduction help to understand the main mechanisms behind oscillatory behavior. In this context, we first study a two-gene oscillator using piecewise linear approximations to improve the performance and robustness of the oscillatory dynamics. Next, motivated by the synchronization of biological rhythms in a group of cells in an organ such as the liver, we then study a network of identical oscillators under diffusive coupling, interconnected according to different topologies. The piecewise linear formalism enables us to characterize the emergent dynamics of the network and show that a number of new steady states is generated in the network of oscillators. Finally, given two distinct oscillators mimicking the circadian clock and cell cycle, we analyze their interconnection to study the capacity for mutual period regulation and control between the two reduced oscillators. We are interested in characterizing the coupling parameter range for which the two systems play the roles "controller-follower".

Monday, April 22, 2024

Posted January 28, 2024

Mathematical Physics and Representation Theory Seminar

2:30 pm – 3:20 pm Lockett 233

Greg Parker, Stanford University
TBA

Monday, April 22, 2024

Posted February 21, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232

Ben Seeger, The University of Texas at Austin
TBA

Wednesday, April 24, 2024

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233

Krishnendu Kar, Louisiana State University
TBA

Wednesday, April 24, 2024

Posted January 31, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233

Morgan Weiler, Cornell University
TBA