Recent Developments on Numerical Tensor Methods for High-dimensional PDEs

Dr. Daniele Venturi

Abstract: There has been a growing interest in efficient numerical algorithms based on tensor networks and low-rank techniques to approximate high-dimensional functions and compute the numerical solution of high-dimensional partial differential equations (PDEs). In this talk, we present two recent developments in this area: The first one is a new tensor rank reduction method that leverages coordinate flows. The idea is very simple: given a multivariate function, determine a coordinate transformation so that the function in the new coordinate system has a smaller tensor rank. We restrict our analysis to linear coordinate transformations, which give rise to a new class of functions that we refer to as tensor ridge functions. By leveraging coordinate flows and tensor ridge functions, we develop a Riemannian optimization algorithm for determining quasi-optimal linear coordinate transformations for tensor rank reduction, and apply this method to approximate multivariate functions and to compute the numerical solution of a high-dimensional PDE, i.e., the Liouville equation of classical statistical mechanics. The results we present for tensor rank reduction via linear coordinate transformations can be generalized to larger classes of nonlinear transformations. The second development is a new class of implicit step-truncation algorithms for temporal integration of high-dimensional PDEs on tensor manifolds. These algorithms can overcome the limitations of explicit step-truncation tensor methods, in particular the time-step restrictions arising from stability constraints when integrating stiff PDEs. Implicit step-truncation algorithms are based on executing one time step with a conventional time integration scheme, followed by an implicit fixed point iteration step involving a rank-adaptive truncation operation onto a tensor manifold. These operations are straightforward to implement, and can be executed by scalable parallel algorithms. We demonstrate implicit step-truncation tensor integrators in numerical applications involving the Allen-Cahn equation, the Fokker-Planck equation, and the nonlinear Schrödinger equation.

Speaker’s Bio: Dr. Venturi is a professor of Applied Mathematics in the Baskin School of Engineering at the University of California- Santa Cruz. He received his Ph.D. in Applied Physics from the University of Bologna (Italy) in 2006. From 2010 to 2015 he joined the Division of Applied Mathematics at Brown University as Research Assistant Professor. His research activity has been recently focused on numerical approximation of high-dimensional PDEs using hierarchical tensor methods, including PDEs arising from discretization of functional differential equations (infinite-dimensional PDEs).



Last Updated: September 6, 2022 - 2:24 pm