Mathematics Colloquium

Learning nonlinear operators using deep neural networks for diverse applications

Abstract: It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. In this talk, I will present the deep operator network (DeepONet) to learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. More generally, DeepONet can learn multiscale operators spanning across many scales and trained by diverse sources of data simultaneously. We will demonstrate the effectiveness of DeepONet to multiphysics and multiscale problems.

Lifting of polynomial ideals and of projective schemes: a very useful algebraic and geometric construction

A classical approach to investigate the properties of both a polynomial ideal I and an algebraic variety V (more generally, of a projective scheme) consists in considering a general hyperplane section, i.e. an ideal and a variety of one dimension less, that are conveniently related to the given ideal I and variety V. Many of the properties of I and V are preserved and are more easily studied in the lower dimensional variety. This talk considers the inverse problem, called a lifting problem.

Lifting of ideals and of projective schemes can be obtained with several techniques of both Commutative Algebra and Algebraic Geometry, useful to construct polynomial ideals and projective schemes with specific properties.

The lifting problem has been considered in several interesting contexts. For example, in 1927 Macaulay obtained ideals of distinct points from given monomial ideals, preserving the Hilbert function. In this talk, some of the classical lines of investigations are revisited and some further points of view are described.

BCBSMS Actuarial Presentation

No abstract provided.

Permutations with equal orders

Let P(n) be the probability that two independent, uniformly random permutations in the symmetric group S_n have the same order, and let K(n) be the probability that they are in the same conjugacy class. Answering a question of Thibault Godin, I will show that P(n) = n^−2+o(1) and that lim sup P(n)/K(n) = ∞. In particular, P(n) ≥ 0.5n⁻² lg^∗ n for infinitely many n, where lg^∗n is the height of the tallest tower of twos that is less than or equal to n. (This is based on joint work with Huseyin Acan, Sean Eberhard, Eric Schmutz, and James Thomas.)

Ball Constrained Newton’s Method for Convex Optimization

Theoretical convex optimization has seen tremendous progress in recent years with the development of new algorithms tailored for settings extending the Lipschitz gradient setting of Nesterov's acceleration. In this talk we give a new algorithm, called Ball Constrained Newton, which minimizes convex functions f(x) given access to a 'ball oracle' which minimizes f(x) inside a Euclidean-norm ball of some small radius around an arbitrary center. With this tool, we show how this ball oracle can be efficiently implemented for functions satisfying a certain third-derivative condition and consequently recover state-of-the-art running times for softmax and p-norm regression. We conclude by showing optimality of our algorithm in the ball-constrained setting via standard lower bound instances in convex optimization.

Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

High order strong stability preserving (SSP) time discretizations are advantageous for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. The search for high order strong stability time-stepping methods with large allowable strong stability time-step has been an active area of research over the last two decades. In this talk we will review the theory of Strong Stability Preserving time discretizations and find some optimal SSP multistage two-derivative methods. While explicit SSP Runge–Kutta methods exist only up to fourth order, we show that this order barrier is broken for explicit multi-stage two-derivative methods. Numerical experiments will also be presented.

Gaston Darboux: Monster-maker par excellence

The drama of the rise of rigor in nineteenth century mathematical analysis has now been widely rehearsed. Notable within this saga is the appearance of functions with features so unexpected (e.g., everywhere continuous but nowhere differentiable) that contemporary critics described them as “bizarre”, “ridiculous,” “pathological,” or even “monsters.” Among those who played the part of a “monster-maker,” one of the most talented and influential actors was French mathematician Gaston Darboux (1842–1917).

In this talk, we consider Darboux’s mathematical and “backstage” contributions to the development of nineteenth century analysis. We examine in particular excerpts from letters exchanged by Gaston Darboux and Jules Houël (1823–1886) in which they debated issues related to rigor in analysis, as well as his 1875 Mémoire sur les fonctions discontinues, in which he proved the result now known as “Darboux’s Theorem” (i.e., all derivatives have the intermediate value property). After meeting some of Darboux’s own favorite pet monsters, we examine the role that functions such as these played in setting the scene for the re-shaping of analysis during the latter part of nineteenth century.

Krylov Subspace Spectral Methods for Problems in Acoustics

This talk provides an overview of my work as a Summer Faculty Research Fellow with the Acoustics Branch of the U. S. Naval Research Laboratory at NASA Stennis Space Center. This work includes (1) the use of techniques for approximating bilinear and quadratic forms involving matrix functions to measure the sensitivity of solutions of PDEs, (2) the application of Krylov Subspace Spectral (KSS) Methods to the parabolic equation for acoustic pressure in oceans, and (3) KSS methods for wave propagation problems featuring periodic media and shock waves.

6 Sep 2019

Dynamic Gröbner basis computation (Part 2)

30 Aug 2019

Dynamic Gröbner basis computation (Part 1)

Applications of Gröbner bases occur wherever multivariate polynomials appear, but first we must compute them! This difficult task lies at the intersection of mathematics and computer science.

We first present an "intuitive" grasp of attractive Gröbner basis properties and their traditional computation, via analogy with linear algebra. An example from cryptanalysis motivates the study.

While traditional algorithms never re-order polynomials, "dynamic" algorithms typically do. First described by Sturmfels and Caboara, this once-forgotten approach has recently enjoyed a revival, described in the second presentation, which concludes by demonstrating the computation of Gröbner bases more useful than those typically obtained.

16 Nov 2018

3D Shape Retrieval Meets Deep Learning

This talk provides an introduction of 3D shape retrieval, as well as some latest research work in applying deep learning, especially Convolutional Neural Networks (CNNs), in 3D shape retrieval. 3D models (typically triangular meshes) consist of 3D data to represent 3D objects. They are widely used in a lot of fields such as industrial product design and 3D entertainment. However, it is still challenging to develop effective and efficient 3D shape retrieval algorithms for related applications. I will introduce some of our 3D shape retrieval techniques as well as several current research directions in 3D scene retrieval in our group.

Two continuous finite element methods on solving FSI problems

Fluid-structure interaction (FSI) problems appear almost everywhere in engineering, sciences, and medicine. It involves the coupling of the solution of momentum equations of both the fluid and the solid. To solve FSI problems, two continuous finite element methods will be presented: One is a kind of immersed boundary method (IBM) and the other is called the shifted boundary method (SBM). Compared to other similar methods in the literature, the key feature is the use of delta function to enforce the constraints over the global domain through Nitsche's technique in IBM and the use of projection scheme in SBM. After discussing the theory and properties of those two methods, some numerical results using the open source package Proteus developed by our group will be shown.

How to compute spectral projectors numerically

For a given square matrix \(A\), there are spectral projectors associated with the eigenvalues of \(A\). The spectral projectors have some important applications. Among them one application is that they can be used to compute \(f(A)\), if the function fulfills certain smoothness conditions at the eigenvalues of \(A\). In this talk, we discuss how to compute the spectral projectors of \(A\) numerically.

On teaching the Chain Rule

(no abstract)

Mesh Warping Algorithms for Use in Dynamic Finite Element Simulations

Dynamic meshes are used to capture deforming geometry in computational simulations involving motion. Mesh warping algorithms employ interpolation and/or extrapolation to transfer the mesh from the source geometry to the target geometry. Using mesh warping for generating dynamic meshes is advantageous with respect to both accuracy and efficiency. In this talk, I will present our parallel log barrier-based tetrahedral mesh warping algorithm. I will also present our high-order curvilinear tetrahedral mesh generation algorithm which deforms the linear tetrahedral mesh into a high-order mesh. This talk represents joint work with Thap Panitanarak, Chulalongkorn University, and Mike Stees, University of Kansas

The method of time integration and approximate fundamental solutions for nonlinear Poisson-type problems

Through a fictitious time approach, a nonlinear Poisson-type problem is converted into a time-dependent quasilinear problem. This is further approximated by a sequence of time-dependent linear nonhomogeneous modified Helmholtz boundary value problems, which are solved by the method of particular solutions of Delta-shaped basis functions and approximate fundamental solutions. Numerical results are provided to show the accuracy and validity of the computational method.

On teaching the Chain Rule

(No abstract)

Under-appreciated uses of the extended euclidean algorithm

The extended euclidean algorithm famously gives us \(\gcd(a, b) = am + bn\). This suffices to produce inverses in finite fields, but the intermediate computations have many uses:

• They solve the key equation \(c/d = s\) to find error positions and magnitudes for Reed-Solomon codes.
• I can reconstruct fractions \(c/d\) by working modulo one large prime or several small primes. This allows me to lift algorithms I wrote for finite fields to ones that work in characteristic 0 as well.
• Generalizations to several inputs allow the production of unimodular matrices describing unimodular transformations, that I use for desingularization of surfaces.

I’ll give at least one simple example for each topic.