Adaptive Low-Rank Approximations for Operator Equations: Accuracy Control and Computational Complexity

TL;DR

This paper reviews recent advances in adaptive low-rank approximation methods for high-dimensional operator equations, emphasizing accuracy control, certification, and computational efficiency in complex numerical problems.

Contribution

It introduces adaptive low-rank techniques that integrate accuracy certification and address computational challenges in high-dimensional operator equations.

Findings

Enhanced accuracy control in low-rank approximations

Effective adaptivity improves computational efficiency

Framework supports uncertainty quantification

Abstract

The challenge of mastering computational tasks of enormous size tends to frequently override questioning the quality of the numerical outcome in terms of accuracy. By this we do not mean the accuracy within the discrete setting, which itself may also be far from evident for ill-conditioned problems or when iterative solvers are involved. By accuracy-controlled computation we mean the deviation of the numerical approximation from the exact solution of an underlying continuous problem in a relevant metric, which has been the initiating interest in the first place. Can the accuracy of a numerical result be rigorously certified - a question that is particularly important in the context of uncertainty quantification, when many possible sources of uncertainties interact. This is the guiding question throughout this article, which reviews recent developments of low-rank approximation methods for problems in high spatial dimensions. In particular, we highlight the role of adaptivity when dealing with such strongly nonlinear methods that integrate in a natural way issues of discrete and continuous accuracy.