Rank-adaptive tensor methods for high-dimensional nonlinear PDEs

TL;DR

This paper introduces a novel rank-adaptive tensor method combining FTT expansions and operator splitting to efficiently solve high-dimensional nonlinear PDEs, dynamically adjusting tensor ranks during time integration.

Contribution

The paper proposes a new rank-adaptive algorithm for tensor methods that improves efficiency and robustness in solving high-dimensional nonlinear PDEs.

Findings

Successfully applied to 2D advection problems.

Effective in 4D Fokker-Planck equations.

Overcomes low-rank modeling errors and matrix inversion issues.

Abstract

We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker-Planck equation.