Residual Recombination Methods as Anderson-like Acceleration: An Algebraic Interpretation of BoostConv

TL;DR

This paper provides a rigorous theoretical analysis of BoostConv, an acceleration technique for nonlinear iterative processes, establishing convergence guarantees and interpreting it as a residual recombination method akin to Anderson acceleration.

Contribution

It introduces a robust formulation of BoostConv, offers the first convergence proof, and frames it within a mathematical context as a nonlinear acceleration method.

Findings

Proves convergence of BoostConv under certain conditions

Demonstrates effectiveness on linear, benchmark, and Navier-Stokes problems

Bridges empirical performance with theoretical understanding

Abstract

BoostConv has been introduced in earlier works as an effective acceleration technique for nonlinear iterative processes and has been successfully employed in a variety of applications to enhance convergence rates or to compute unstable fixed points that are otherwise inaccessible through standard approaches. Despite its demonstrated practical effectiveness, the theoretical properties of the method have not yet been fully characterized. In this work, we present a more robust formulation of the BoostConv algorithm and, for the first time, provide a rigorous proof of its convergence. The proposed analysis places BoostConv within a precise mathematical framework, clarifying its interpretation as a nonlinear convergence accelerator and establishing sufficient conditions under which convergence to a fixed point is guaranteed. The theoretical findings are illustrated through several numerical examples, spanning from a linear problem to a low-dimensional benchmark and a large-scale incompressible Navier-Stokes simulation. These results demonstrate the robustness and practical relevance of the proposed method and bridge the gap between empirical performance and rigorous analysis, paving the way for further developments and applications to complex nonlinear problems.