### Joint Applied Math/Inverse Problems Seminar at Colorado State University

##### Thursday 2:00-3:00PM, Online

Meeting ID: 963 0995 0719
Passcode: 384535

#### Spring 2021

Jab 21   Feb 25   Mar 18   Mar 25   Apr 01   Apr 08   Apr 22   Apr 29

#### Mikyoung Lim

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST)

Title  Reconstruction of a conductivity inclusion using the Faber polynomials

Abstract  In this talk, I present analytical shape recovery methods for the conductivity inclusion in two dimensions, obtained in collaboration with Doosung Choi and Junbeom Kim. For a simply connected planar domain, there exists a function that conformally maps a region outside a circular disk to the region outside the domain. The conformal mapping then defines the so-called Faber polynomials, which form a basis for analytic functions. An electrical inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a far-field expansion, whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from the exterior measurements. As a modification of GPTs, we recently introduced the Faber polynomial polarization tensors (FPTs). We design two analytical approaches for the shape recovery of a conductivity inclusion by employing the concept of FPTs. Numerical experiments demonstrate the validity of the proposed analytical methods.

#### Andrew Gillette

Lawrance Livermore National Laboratory

Title  Delaunay-guided neural network design

Abstract  Neural networks for scientific machine learning provide approximations of an unknown function, based on a set of known data points. Given the wide variety of hyper-parameters involved in training a neural network, how can we assess whether a particular approximation is actually any good? In this talk, I will introduce a set of techniques in development that allow comparison of a trained neural network to a geometric benchmark, based on the Delaunay mesh of the training data points. Simple examples in 2D will motivate the approach, followed by extension to dimensions beyond the reach of classical computational geometry methods. I will present applications of these tools to surrogate modeling problems and variational autoencoder design. In addition, I will describe ongoing opportunities for student, postdoctoral, and faculty research at Lawrence Livermore National Laboratory. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. IM release number LLNL-ABS-819655.

#### Teemo Saksala

Department of Mathematics, North Carolina State University

Title  Stable reconstruction of simple Riemannian manifolds from unknown interior sources

Abstract  Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a Riemannian manifold with boundary? With a finite set of sources we can only hope to get an approximate reconstruction, and we indeed provide a discrete metric approximation to the manifold with explicit data-driven error bounds when the manifold is simple. This is the geometrization of a seismological inverse problem where we measure the arrival times on the surface of waves from an unknown number of unknown interior microseismic events at unknown times. The closeness of two metric spaces with a marked boundary is measured by a labeled Gromov--Hausdorff distance. If measurements are done for infinite time and spatially dense sources, our construction produces the true Riemannian manifold and the finite-time approximations converge to it in the metric sense.

This talk is based on a joint work with Maarten V. de Hoop, Joonas Ilmavirta and Matti Lassas.

#### Mirjeta Pasha

Department of Mathematical and Statistical Sciences, Arizona State University

Title  On the Krylov subspace type methods to solve non-negative, large-scale inverse problems and estimate maximum a posteriori for non-Gaussian noise

Abstract  Ill-posed inverse problems arise in many fields of science and engineering. Their solution, if it exists, is very sensitive to perturbations in the data. The challenge of working with linear discrete ill-posed problems comes from the ill-conditioning and the possible large dimension of the problems. Regularization methods try to reduce the sensitivity by replacing the given problem with a nearby one, whose solution is less affected by perturbations. The methods in this talk are concerned with solving large scale problems by projecting them into a Krylov or generalized Krylov subspace of fairly small dimension. The first type of methods discussed are based on Bregman-type iterative methods that even though the high quality reconstruction that they deliver, they may require a large number of iterations and this reduces their attractiveness. We develop a computationally attractive linearized Bregman algorithm by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. Recently, the use of a $p$-norm to measure the fidelity term and a $q$-norm to measure the regularization term has received considerable attention. For applications such as image reconstruction, where the pixel values are non-negative, we impose a non-negativity constraint to make sure the reconstructed solution lies in the non-negative orthant. The quality of the reconstructed solution depends on a regularization parameter that balances the influence of a data fitting term and a regularization term on the computed solution. We propose techniques to select the regularization parameter without any significant computational cost. This makes the proposed method more efficient and useful especially for large-scale problems. In addition, we explore how to estimate maximum a posteriori when the available data are perturbed with non-Gaussian noise. Numerical examples illustrate the performances of the approaches proposed in terms of both accuracy and efficiency. We consider two-dimensional problems, with a particular attention to the restoration of blurred and noisy images.

#### Justin O'Connor

Department of Mathematics, Colorado State University

Title  Topology Optimization in Elastic Media

Abstract  The field of topology optimization, especially as it pertains to elastic media, has grown quickly in recent years as technological advancements have allowed for many real-life applications in structural engineering. A main issue in common application of these algorithms is the time necessary for computation. This talk begins with a brief history of the field to provide an understanding of current topology optimization methods.We will then discuss how advancements in interior point optimization can be used in this case, before looking into preconditioning and solving the resulting linear system. These advancements allow for faster solves, which lead to the ability to solve 3d, high resolution problems. The talk will conclude with a discussion of future work which aims to continue allowing for larger, higher resolution problems.

#### Colin Jensen

Department of Mathematics, Colorado State University

Title  Solving the continuous optimal transport problem with novel finite element methods

Abstract  The optimal transport problem has received much attention in recent years because of its myriad applications to data science, image processing, mesh generation, and more. Most of the literature focuses on its discrete and semi-discrete forms. Its continuous formulation, and especially its solution via the fully nonlinear Monge-Ampère PDE, has received less attention. This presentation will first review some of the existing numerical solutions to the continuous optimal transport problem. These include methods implementing infinite-dimensional optimization and nonlinear PDE solvers. Then, we will propose a new method that utilizes the novel weak Galerkin finite element method. Initial experiments on a simplified version of the problem suggest that the method holds some promise

#### Sui Tang

Department of Mathematics, University of California, Santa Barbara

Title  Data-driven discovery of interaction laws in multi-agent systems

Abstract  Multi-agent systems are ubiquitous in science, from the modeling of particles in Physics to prey-predator in Biology, to opinion dynamics in economics and social sciences, where the interaction law between agents yields a rich variety of collective dynamics. We consider the following inference problem for a system of interacting particles or agents: given only observed trajectories of the agents in the system, can we learn what the laws of interactions are? We would like to do this without assuming any particular form for the interaction laws, i.e. they might be `any' function of pairwise distances.

In this talk, we consider this problem in the case of a finite number of agents, with an increasing number of observations. We cast this as an inverse problem, and study it in the case where the interaction is governed by an (unknown) function of pairwise distances. We discuss when this problem is well-posed, and we construct estimators for the interaction kernels with provably good statistical and computational properties. We measure their performance on various examples, that include extensions to agent systems with different types of agents, second-order systems, and stochastic systems. We also conduct numerical experiments to test the large time behavior of these systems, especially in the cases where they exhibit emergent behavior. This talk is based on the joint work with Fei Lu, Mauro Maggioni, Jason Miller, and Ming Zhong.