Speaker: Stefan Schnake

Affiliation: ORNL

Title: Title Residual based rank adaptive algorithms for dynamic low-rank approximation 

Abstract: Numerical discretizations of kinetic equations often suffer from the curse of dimensionality which makes full resolution computations intractable. A recent and popular technique to reduce the order of complexity for these systems is Koch and Lubich’s Dynamic Low-Rank Approximation (DLRA) which seeks to evolve the PDE on a low-rank manifold. Numerical integrators utilizing DLRA accurately capture low-rank features of kinetic systems by updating the global basis functions in time, but few give indicators for when to adaptively increase the rank of the solution. In this talk, we first present a formulation of standard discontinuous Galerkin methods as a semi-discrete matrix-valued ODE which is low memory, provides computationally efficient evaluations of the discrete operator and easily fits the target framework of DLRA. Second, we present residual-based indicators for when to increase the rank of the DLRA solution and which basis vectors should be added in order to optimally increase accuracy. These residual-based estimators are also low memory and can be cheaply computed. Low- and high-rank numerical examples are given at the end of the talk.0