Coordinate-adaptive integration of PDEs on tensor manifolds

TL;DR

This paper presents a novel tensor integration method for time-dependent PDEs that adaptively controls tensor rank through coordinate transformations, improving efficiency and accuracy in solving PDEs on tensor manifolds.

Contribution

The paper introduces a new coordinate-adaptive tensor integration technique that optimizes PDE solutions via diffeomorphic transformations and convex functionals, enhancing previous non-convex approaches.

Findings

Effective in solving Liouville and Fokker-Planck equations

Significantly improves tensor rank control and solution accuracy

Compatible with existing coordinate-adaptive algorithms

Abstract

We introduce a new tensor integration method for time-dependent PDEs that controls the tensor rank of the PDE solution via time-dependent diffeomorphic coordinate transformations. Such coordinate transformations are generated by minimizing the normal component of the PDE operator relative to the tensor manifold that approximates the PDE solution via a convex functional. The proposed method significantly improves upon and may be used in conjunction with the coordinate-adaptive algorithm we recently proposed in JCP (2023) Vol. 491, 112378, which is based on non-convex relaxations of the rank minimization problem and Riemannian optimization. Numerical applications demonstrating the effectiveness of the proposed coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.