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

Vincenzo Citro, Davide Palitta

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.

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

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.
Paper Structure (12 sections, 3 theorems, 55 equations, 3 figures, 1 algorithm)

This paper contains 12 sections, 3 theorems, 55 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background material
  3. Anderson acceleration
  4. BoostConv
  5. BoostConv as Anderson-like acceleration
  6. Robust BoostConv
  7. Convergence properties
  8. Numerical results
  9. A linear problem
  10. Convergence acceleration for the 1D Burgers' equation
  11. Stabilisation of Navier--Stokes equations
  12. Conclusions

Key Result

Lemma 3.5

\newlabellemma5.60 At iteration $k$ of the form eq:iteration_boostconv_stabilized, if $\widehat{N}$ denotes the cardinality of $\mathcal{I}_k$, let the $\widehat{N}$ iterates $x_{k-d}$, $d=0,\ldots,\widehat{N}-1$ all belong to $U_\kappa(x_*)$. Then, it holds where $\widehat{\Delta X}_k^{\mathcal{I}_k}:={\Delta X}_k^{\mathcal{I}_k}R_k^{-1}$, $q_j$ denotes the $j$th column of the orthogonal matrix

Figures (3)

  • Figure 1: Relative residual norm history achieved by the plain version (solid line) of Richardson (left) and Jacobi (right) and their counterparts enhanced by Algorithm \ref{['alg:stabilizedBoostConv']} with $N=3$ and $\tau=10^{-10}$ (dotted line) for the solution od $Ax=b$.
  • Figure 2: Comparison between standard Explicit Euler integration and BoostConv acceleration for the 1D Burgers' equation. Panel (a) shows the temporal evolution of the solution $u(x,t)$ obtained using the standard Explicit Euler scheme. The system progressively relaxes toward the trivial steady solution $u(x,t)=0$. Panels (b) and (c) illustrate the effect of applying BoostConv in its classical and robust variants, respectively, showing a substantial enhancement in the convergence toward the zero solution. To further quantify the performance of the two algorithms, panel (d) reports the decay of the residual norm $\|\mathcal{R}(u)\|_\infty$, while panel (e) shows the decay of the solution energy $\|u\|_{L^2}$. The results clearly indicate that the robust formulation provides a noticeable improvement in the convergence behaviour.
  • Figure 3: Comparison between an unstable flow realization and the corresponding base flow, together with the stabilization behaviour of two algorithms. Panel (a) shows an instantaneous velocity field associated with an unstable solution, characterized by amplified perturbations and complex spatial structures. Panel (b) displays the base flow, obtained as the converged steady solution of the incompressible Navier--Stokes equations. Panel (c) illustrates the evolution of the stabilization process, comparing the behaviour of the original algorithm with that of the proposed robust version, highlighting the improved convergence. The BoostConv parameters are $N = 15$ and $\tau = 10^{-10}$ (threshold for linear dependency detection). Overall, the comparison emphasizes the qualitative differences between unstable dynamics and the underlying base state, as well as the enhanced stabilization properties of the robust algorithm. Parameter setting (see citroPOFBoostconv for further details): $Re_k=500$, $k/\delta^*=2.5$.

Theorems & Definitions (10)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.5
  • Proof 1
  • Lemma 3.6
  • Proof 2
  • Theorem 3.7
  • Proof 3