Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems

TL;DR

This paper presents a new family of rank-adaptive algorithms for solving high-dimensional linear systems in tensor train format, combining alternating directions with steepest descent ideas, and analyzes their convergence and complexity.

Contribution

Introduces a novel class of tensor train algorithms that are rank-adaptive and combine alternating directions with steepest descent, improving convergence analysis and computational efficiency.

Findings

Algorithms have linear complexity in mode size and dimension.

Convergence rate is geometrically related to steepest descent.

Numerical experiments show comparable performance to DMRG algorithms.

Abstract

We introduce a family of numerical algorithms for the solution of linear system in higher dimensions with the matrix and right hand side given and the solution sought in the tensor train format. The proposed methods are rank--adaptive and follow the alternating directions framework, but in contrast to ALS methods, in each iteration a tensor subspace is enlarged by a set of vectors chosen similarly to the steepest descent algorithm. The convergence is analyzed in the presence of approximation errors and the geometrical convergence rate is estimated and related to the one of the steepest descent. The complexity of the presented algorithms is linear in the mode size and dimension and the convergence demonstrated in the numerical experiments is comparable to the one of the DMRG--type algorithm.