Time integration of tensor trains

TL;DR

This paper introduces a new robust and efficient time integrator for dynamical tensor approximation in tensor train format, enabling better handling of high-dimensional tensor differential equations.

Contribution

It presents a novel splitting-based time integrator for tensor trains, with theoretical analysis and practical implementation for high-dimensional tensor differential equations.

Findings

Effective in quantum molecular dynamics simulations

Handles high-dimensional tensor differential equations efficiently

Supported by numerical experiments demonstrating robustness

Abstract

A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train / matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formulation, implementation and theoretical properties of the proposed integrator are studied, and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the tensor train format are included.