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Step Truncation Methods for Nonlinear Evolution Equations on Tensor Manifolds

AMS Seminar Series

Speaker: Abram Rodgers, Computational Aerosciences Branch, NASA Ames Research Center

Presentation

Seminar Slide Deck (PDF-4.6MB)

Abstract

We present new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. In particular, we present a mathematical framework for the analysis of these algorithms which encompasses both explicit and implicit time stepping. With this framework we demonstrate convergence of a wide range of step-truncation methods, including one-step and multi-step methods. These methods rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which have become central to many new areas of application such as optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a class of advection-diffusion equations, the nonlinear Schrodinger, and a Burgers' equation with uncertain initial condition.

Biography

Photo of Abram Rodgers
Abram Rodgers is a Mathematician in the Computational Aerosciences Branch at NASA Ames Research Center. Broadly, his interest is in the development and analysis of high-performance parallel numerical algorithms for solutions to nonlinear problems. More specifically, his interests are in numerical solutions to differential equations, numerical linear algebra, computational methods in differential geometry, tensor compression algorithms, and low-rank dynamic modeling. Since joining NASA Ames, he has also contributed to the unstructured Voronoi grid generation software for fluid dynamics simulation. He has attended the University of California, Santa Cruz, since 2014 where he earned his BA in Mathematics in 2017, MS in Applied mathematics in 2018, and defended his PhD in Applied Mathematics in December 2023. His doctoral research centered around conditions for highly accurate low-rank tensor modeling for high dimensional time dependent partial differential equations.