Inverse problems involve the reconstruction of hidden objects from possibly noisy indirect measurements and are ubiquitous in a variety of scientific and engineering applications. This kind of problems have two main features that make them interesting yet challenging to solve. First, they tend to be ill-posed: the reconstruction is very sensitive to perturbations in the measurements. Second, real-world applications are often large-scale: resulting in computationally demanding tasks. This colloquium talk will center on discrete linear problems, offering a comprehensive overview of Krylov subspace methods for ill-posed problems —a well-established class of solvers— and delving into their regularization properties. Additionally, I will explore recent advances that enhance their efficacy in solving more complex optimization tasks. The presentation will feature results and examples drawn from various imaging applications, providing insights into the practical implications of these approaches.

The ongoing COVID-19 pandemic offers a unique opportunity to assess the performance of mathematical modeling frameworks in predicting the course of epidemics and pandemics across various spatial and temporal dimensions. In this talk, I will discuss advancements in creating novel ensemble modeling techniques that require minimal data and surpass the predictive accuracy of single models for short-term forecasts without significantly adding to the model complexity. In addition to enhancing forecasting methods, this project is committed to developing toolboxes and tutorials that incorporate these advanced ensemble modeling approaches to encourage their application among public health professionals. This dual focus on toolbox development and student training underscores the project's broader goal of not only advancing the state of pandemic forecasting but also empowering the next generation of public health professionals with the knowledge and tools necessary to tackle future public health crises.

From Calculus we know that a derivative of a function can be approximated using a difference quotient. There are different forms of the difference quotient, such as the forward difference (most common), backward difference and centered difference (more accurate). In this talk, I will discuss several different differences, specifically nonstandard finite differences (NFSD) that can be used to approximate the derivatives that appear in differential equations as a solution technique. Many NSFD schemes have been discovered and promoted by one of my mentors, Ronald E. Mickens, an African-American Emeritus Professor of Physics at Clark Atlanta University, who has written more than 300 research articles and a dozen books. I'll provide a number of examples of how NSFD schemes can be used to solve a wide variety of problems drawn from first-semester Calculus, elementary ordinary differential equations and nonlinear partial differential equations.

Can one recover a structured matrix efficiently from only matrix-vector products? If so, how many are needed? In this talk, we will describe algorithms to recover structured matrices, such as tridiagonal, Toeplitz-like, and hierarchical low-rank, from matrix-vector products. Then, we will use insights from matrix recovery to understand the data-efficiency of operator learning in the context of PDE learning. We will partially explain the success of neural operators, like Fourier Neural Operators and DeepONet, by understanding the data-efficiency of recovery of the solution operators associated with elliptic, parabolic, and hyperbolic PDEs.

Combinatorial Commutative Algebra is rich with connections to other areas of Mathematics, Science, and Engineering. This talk will focus on connections with Coding Theory and Information Theory. (No background on these topics will be assumed for the talk.) Specifically, we investigate the codeword ideal $K(C)$ of a linear code $C$ over a field $F$. This is the ideal in a polynomial ring over $F$ generated by the squarefree monomials given by the nonzero codewords of $C$. Algebraic information about $K(C)$ can be seen in the code $C$, and vice versa. For instance, it is known that the prime decomposition of $K(C)$ is determined by information-theoretic aspects of $C$, and the generalized Hamming weight of $C$ is determined by the minimal free resolution of $K(C)$. The talk will focus primarily on these known connections and results, with appropriate background, and will end with partial progress on open questions about the free resolutions and symbolic powers of these ideals.

There has been a surge of interest in recent years in general-purpose `acceleration' methods that take a sequence of vectors converging to the limit of a fixed point iteration, and produce from it a faster converging sequence. A prototype of these methods that attracted much attention recently is the Anderson Acceleration (AA) procedure. We introduce the nonlinear Truncated Generalized Conjugate Residual (nlTGCR) algorithm, an alternative to AA which is designed from a careful adaptation of the Conjugate Residual method for solving linear systems of equations to the nonlinear context. The various links between nlTGCR and inexact Newton, quasi-Newton, and multisecant methods are exploited to build a method that has strong global convergence properties and that can also exploit symmetry when applicable. Taking this algorithm as a starting point we explore a number of other acceleration procedures including a short-term (`symmetric') version of Anderson Acceleration which we call AndersonAcceleration with Truncated Gram-Schmidt.

In scientific and engineering computing, we encounter time-dependent partial differential equations (PDEs) frequently. When designing high order schemes for solving these time-dependent PDEs, we often first develop semi-discrete schemes paying attention only to spatial discretizations and leaving time $t$ continuous. It is then important to have a high order time discretization to main the stability properties of the semi-discrete schemes. In this talk we discuss several classes of high order time discretization, including the strong stability preserving (SSP) time discretization, which preserves strong stability from a stable spatial discretization with Euler forward, the implicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats the more stiff term (e.g. diffusion term in a convection-diffusion equation) implicitly and the less stiff term (e.g. the convection term in such an equation) explicitly, for which strong stability can be proved under the condition that the time step is upper-bounded by a constant under suitable conditions, the explicit-implicit-null (EIN) time marching, which adds a linear highest derivative term to both sides of the PDE and then uses IMEX time marching, and is particularly suitable for high order PDEs with leading nonlinear terms, and the explicit Runge-Kutta methods, for which strong stability can be proved in many cases for semi-negative linear semi-discrete schemes. Numerical examples will be given to demonstrate the performance of these schemes.

Although many mathematical models in wave theory lead to hyperbolic initial-boundary value problems which are inherently local due to the finite domain-of-dependence of the solution at any point on past values, there are also examples where nonlocal operators, in particular space-time integral op- erators, arise. A primary example is the radiation boundary condition needed to truncate an unbounded domain to a finite one to enable numerical solu- tions, as well as closely-related operators for unidirectional propagation. In this work we show how to leverage results from rational function approxima- tion theory to construct spectrally-convergent local algorithms for evaluating such operators. For an important class of problems - systems equivalent to the scalar wave equation, such as acoustics or Maxwell’s equations in ho- mogeneous, isotropic media, we will explain the construction, analysis, and implementation of our complete radiation boundary conditions, which are in a certain sense optimal. We will also discuss other applications of these approximation methods, as well as the fundamental barriers to extending the succesful methods to other systems.

Abstract: It is well-known that Hilbert's tenth problem, which asks for an algorithm to detect whether a polynomial equation has integer solutions or not, is undecidable, by the works of Matiyasevich and Davis--Putnam--Robinson. On the other hand, a variant of Hilbert's tenth problem, where one searches for rational solutions to a polynomial instead of integer solutions, is still open (that is, we don't know whether this problem is undecidable). In this talk, I will describe some attempts towards Hilbert's tenth problem over Q; in particular, by considering the intermediate problems that could potentially link this problem to the original Hilbert's tenth problem.