Departmental Colloquium

Title

Lagrangian and Game Theoretic Methods for Multi-scale and Multi-Dimensional Problems

Abstract

In this talk, we discuss some recent development of Lagrangian and game theoretic (i.e. stochastic and two-player control generalizations of the method of characteristics) approaches for multi-scale and multi-dimensional reaction-diffusion-advection equations. Through two case studies, we show how stochastic interacting particle methods (IPM) work out as a mesh-free and self-adaptive computational tool. The first case, dated back to Kolmogorov 1937, is concerned with entropy production of reverse-time diffusion processes, and the resulting principal eigenvalue problem of a non-self-adjoint advection-diffusion operator. At a linear complexity rate, the IPM, derived from the Feynman-Kac formula with a genetic interpretation, computes the eigenfuction as a concentrated invariant measure of particle population evolution up to dimension 16. In the second case study of a haptotaxis advection-diffusion system modeling cancer cell spreading, an IPM with a field coupling captures cell merging and expanding dynamics in 3 space dimensions.

The third study aims to address a fundamental problem in turbulent combustion by analyzing a curvature dependent level set Hamilton-Jacobi equation (a.k.a. curvature G-equation), and proving the existence of effective front speeds in a cellular flow. To overcome non-coercivity and non-convexity of the Hamiltonian, we combine a one-sided reachability estimate based on the Kohn-Serfaty deterministic two player game characterization, the streamline structure of the flow and a minimum value principle.