Session II — Computational and Probabilistic Methods in Applied Mathematics

Monday, October 26 · 9:00–10:30 a.m.
Ballroom B

Chair: Yunpeng Shi

Bistatic GPR-SAR imaging of low-contrast targets below a rough interface

Joseph Simpson · UC Merced

We study a recently proposed bistatic ground penetrating radar-synthetic aperture radar (GPR-SAR) imaging system for landmine detection to understand how well it can detect low-contrast targets. To simulate measurements we combine the method of fundamental solutions for scattering by a rough interface and a boundary integral equation method for scattering by a target. We use Kirchhoff migration with appropriate illuminations to form an image. We find that when signals scattered by targets are not completely overwhelmed by ground bounce signals or measurement noise, we can image those targets without loss of resolution. When ground bounce signals and measurement noise are significant, we apply principal component analysis to approximately remove their dominant effects thereby allowing for effective imaging of targets. We show that we can detect and locate low-contrast targets in soil as long as there is a sufficient portion of signals scattered by the target in measurements.

Recent development on Bayesian optimization with expected improvement

Jingyi "Frank" Wang · Lawrence Livermore National Laboratory

Bayesian optimization with expected improvement is one of the most widely used global black-box optimization methods. In this talk, we will discuss our recent work on the simple and cumulative regret bounds of Bayesian optimization with expected improvement. In particular, we seek to answer whether some parameter choices of the algorithm can lead to no-regret behavior. Further, we will present convergence rates on constrained Bayesian optimization methods with constrained expected improvement. We will present the practical implications of the theoretical results and discuss some of the limitations.

Numerical solutions to nonlinear stochastic systems with financial, population growth and neuroscience applications

Coşkun Çetin · California State University, Sacramento

Many dynamical systems involve randomness in their parameters or with additive noise terms. In continuous time, they can be represented with a random ordinary differential equation (RODE) or a stochastic differential equation (SDE), e.g. of Ito-type. Their PDE and jump variations are also common. By focusing on certain SDE systems with nonlinear and locally Lipschitz drift terms, this talk focuses on the numerical solutions of such systems including partially implicit, split-step, tamed, and semi-implicit split-step Euler and Milstein methods. Some applications include financial (e.g. stochastic volatility with random interest rate) as well as biological/medical (population growth, epidemic, tumor growth, predator-prey and neural) models. A few theoretical and empirical convergence results will be presented.

Image Statistics in Random Cellular Automata

Travis Kulhanek · UC Davis

Discrete, translation-invariant dynamical systems arise naturally across systems biology, spatial pattern formation, and computer science. While predicting individual trajectories in such systems is generally intractable, universal statistical behaviors often emerge across large random ensembles. We focus on one-dimensional cellular automata governed by local interactions. When the local update rule is chosen at random, the system acts as a discrete analog of a random differential operator. We transform questions about its image set into the study of rare events in an edge-labeled directed graph equipped with an Erdős-Rényi-like probability measure. We derive characterizations of both the probability that a given periodic configuration lies in the image of the operator and the distribution of its periodic preimages. We show this in two asymptotic regimes, long-range interaction and large alphabet size, using Poisson approximation and other combinatorial techniques.

Defect-correction methods for convection-diffusion problems on layer-adapted meshes

Thai Nhan · Santa Clara University

Defect-correction methods are well established for solving one-dimensional convection-diffusion problems on layer-adapted meshes, including Shishkin and Bakhvalov meshes. These methods enhance solution accuracy by coupling a stable, low-order upwind discretization with a higher-order, unstabilized central scheme. In this talk, we revisit these classical approaches and present recent advances, including novel higher-order defect-correction methods in one dimension. Furthermore, we outline our progress toward extending these techniques to two-dimensional problems, with a particular focus on Bakhvalov-type meshes.