Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST)
Reconstruction of a conductivity inclusion using the Faber polynomials
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.
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.
Department of Mathematics, North Carolina State University
Stable reconstruction of simple Riemannian manifolds from unknown interior sources
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.
Department of Mathematical and Statistical Sciences, Arizona State University
On the Krylov subspace type methods to solve non-negative, large-scale inverse problems and estimate maximum a posteriori for non-Gaussian noise
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
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
Department of Mathematics, Colorado State University
Topology Optimization in Elastic Media
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.
Department of Mathematics, Colorado State University
Solving the continuous optimal transport problem with novel finite element methods
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
Department of Mathematics, University of California, Santa Barbara
Data-driven discovery of interaction laws in multi-agent systems
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.