Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations

TL;DR

This paper introduces an adaptive iterative method combining low-rank tensor approximations and basis adaptivity for high-dimensional operator equations, with proven convergence and complexity bounds, supported by computational experiments.

Contribution

It presents a novel framework for solving high-dimensional operator equations using adaptive low-rank tensor methods with rigorous convergence analysis.

Findings

Convergence of the proposed iterative schemes is rigorously established.

Complexity bounds depend only on the infinite-dimensional problem, not on discretization.

Computational experiments demonstrate effectiveness in high-dimensional settings.

Abstract

We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a concrete discretization. Under certain assumptions on the rates for the involved low-rank approximations and basis expansions, we can also give bounds on the computational complexity of the iteration as a function of the prescribed target error. Our theoretical findings are illustrated and supported by computational experiments. These demonstrate that problems in very high dimensions can be treated with controlled solution accuracy.