Anderson acceleration with approximate calculations: applications to scientific computing

TL;DR

This paper develops theoretical bounds and heuristics for Anderson acceleration with approximate calculations, enabling efficient convergence in linear and nonlinear problems while reducing computational costs.

Contribution

It introduces rigorous error bounds, adaptive heuristics, and a reduced-dimensional variant of Anderson acceleration for improved efficiency and automation.

Findings

Maintains convergence with approximate calculations under error bounds.

Reduces computational time through adaptive heuristics.

Effective on linear systems and nonlinear Boltzmann equations.

Abstract

We provide rigorous theoretical bounds for Anderson acceleration (AA) that allow for approximate calculations when applied to solve linear problems. We show that, when the approximate calculations satisfy the provided error bounds, the convergence of AA is maintained while the computational time could be reduced. We also provide computable heuristic quantities, guided by the theoretical error bounds, which can be used to automate the tuning of accuracy while performing approximate calculations. For linear problems, the use of heuristics to monitor the error introduced by approximate calculations, combined with the check on monotonicity of the residual, ensures the convergence of the numerical scheme within a prescribed residual tolerance. Motivated by the theoretical studies, we propose a reduced variant of AA, which consists in projecting the least-squares used to compute the Anderson mixing onto a subspace of reduced dimension. The dimensionality of this subspace adapts dynamically at each iteration as prescribed by the computable heuristic quantities. We numerically show and assess the performance of AA with approximate calculations on: (i) linear deterministic fixed-point iterations arising from the Richardson's scheme to solve linear systems with open-source benchmark matrices with various preconditioners and (ii) non-linear deterministic fixed-point iterations arising from non-linear time-dependent Boltzmann equations.