Fall 2020

Abstract: Given two probability measures and a transportation cost, the optimal transport problem asks to find the most cost efficient way to transport one measure to the other. Since its introduction in 1781 by Gaspard Monge, the optimal transport problem has found applications in logistics, economics, physics, PDEs, and more recently data science. However, despite sustained attention from the numerics community, solving optimal transport problems has been a notoriously difficult task.

In this talk I will introduce the back-and-forth method, a new algorithm to efficiently solve the optimal transportation problem for a general class of strictly convex transportation costs. Given two probability measures supported on a discrete grid with n points, the method computes the optimal map in O(n log(n)) operations using O(n) storage space. As a result, the method can compute highly accurate solutions to optimal transportation problems on spatial grids as large as 4096 x 4096 and 384 x 384 x 384 in a matter of minutes. If time permits, I will demonstrate an extension of the algorithm to the simulation of a class of gradient flows. This talk is joint work with Flavien Leger.

Abstract: The Rubik's Snake is an interesting toy invented almost 40 years ago. However, very few serious mathematical studies have been found in the literature. In this talk, I will present the work in our recently published papers on some theoretical derivations and the application to the Rubik's Snake shape design.

Abstract: Developing algorithms for solving high-dimensional partial differential equations, controls, and games has been an exceedingly difficult task for a long time, due to the notorious "curse of dimensionality". In this talk we introduce a series of deep learning-based algorithms for these problems. The algorithms exploit the mathematical structure of problems and utilize deep neural networks as efficient approximations to high-dimensional functions. Numerical results of a variety of examples demonstrate the efficiency and accuracy of the proposed algorithms in high-dimensions. This opens up new possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their inter-relationships.

Abstract: In AI and inverse problems, the Markov chain Monte Carlo (MCMC) method is a classical model-free method for sampling target distributions. A fact is that the optimal transport first-order method (gradient flow) forms the MCMC scheme, known as Langevin dynamics. A natural question arises: Can we propose accelerated or high order optimization techniques for MCMC methods? We positively answer this question by applying optimization methods from optimal transport and information geometry, known as transport information geometry. E.g., we introduce a theoretical framework for Newton's flows in probability space w.r.t. the optimal transport metric. Several numerical examples are given to demonstrate the effectiveness of the proposed optimization-based sampling methods.

Abstract: Consider a smooth bounded domain, representing a conductive medium, and suppose its electrical conductivity at any is given by. Ohm's Law states that the electrical potential inside satisfies the elliptic PDE. The goal of Electrical Impedance Tomography is to reconstruct the conductivity function from measurements of voltage and current flux at the boundary. In this talk, we will present an overview of the history of this problem, including some recent results and problems that still remain open.