The power of randomness – random solvers for forward and inverse PDE problems

One key feature of modern scientific computing is the large scale of data that may result from a fine grid to resolve multiple scales in the PDE solutions, a large number of measurements for an inverse problem, and high dimensional problems with tensor structures. These problems are difficult to solve because the solution variable lives in a high dimensional ambient space, even if the solution space itself admits low dimensional features. In this talk, I will present how one can use randomness to exploit the low dimensional feature of the problem, and develop fast solvers and accuracy guarantees for PDE problems both in the forward and inverse settings. In particular, I talk about random generalized finite element methods for multi-scale PDEs, random sketching methods for linear inverse problems, and related quadratic regression optimization problems.