Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure

TL;DR

This paper introduces preconditioned low-rank Riemannian optimization methods, including a Riemannian Newton scheme, for efficiently solving high-dimensional linear systems with tensor product structure, outperforming existing approaches in computational speed.

Contribution

It develops novel Riemannian gradient and Newton methods tailored for low-rank tensor formats, enhancing the efficiency of solving high-dimensional PDE discretizations.

Findings

The Riemannian Newton scheme is significantly faster when applying the linear operator is costly.

The proposed methods outperform existing tensor-based solvers in numerical experiments.

Efficient solution of the Newton equation is achieved for high-dimensional tensor problems.

Abstract

The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially with the number of dimensions, making the use of classic iterative solvers infeasible. During the last few years, low-rank tensor approaches have been developed that allow to mitigate this curse of dimensionality by exploiting the underlying structure of the linear operator. In this work, we focus on tensors represented in the Tucker and tensor train formats. We propose two preconditioned gradient methods on the corresponding low-rank tensor manifolds: A Riemannian version of the preconditioned Richardson method as well as an approximate Newton scheme based on the Riemannian Hessian. For the latter, considerable attention is given to the efficient solution of the resulting Newton equation. In numerical experiments, we compare the efficiency of our Riemannian algorithms with other established tensor-based approaches such as a truncated preconditioned Richardson method and the alternating linear scheme. The results show that our approximate Riemannian Newton scheme is significantly faster in cases when the application of the linear operator is expensive.