Adaptive integration of nonlinear evolution equations on tensor manifolds

TL;DR

This paper introduces adaptive step-truncation algorithms for efficiently solving high-dimensional nonlinear evolution equations on tensor manifolds, ensuring stability and accuracy with scalable tensor operations.

Contribution

It develops and proves convergence of new adaptive algorithms that combine traditional time-stepping with tensor manifold truncation, suitable for high-dimensional PDEs.

Findings

Algorithms are easy to implement and scalable.

Numerical experiments demonstrate effectiveness on high-dimensional PDEs.

Adaptive methods maintain stability and accuracy in complex simulations.

Abstract

We develop 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. By selecting the rank of the tensor manifold adaptively to satisfy stability and accuracy requirements, we prove convergence of a wide range of step-truncation methods, including explicit one-step and multi-step methods. These methods are very easy to implement as they 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 optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a Fokker-Planck equation with spatially dependent drift on a flat torus of dimension two and four.