[Paper Review] A universal matrix-free split preconditioner for the fixed-point iterative solution of non-symmetric linear systems

Tom Vettenburg, Ivo M. Vellekoop|arXiv (Cornell University)|Jul 28, 2022

Matrix Theory and Algorithms4 citations

TL;DR

This paper introduces a universal matrix-free split preconditioner that ensures monotonic convergence of fixed-point iterations for all non-symmetric, accretive linear systems $A\bm{x} = \bm{y}$ by exploiting any splitting $A = L + V$ with $\|V\| < 1$. The unique preconditioner form $(L + I)(I - V)^{-1}$ enables matrix-free evaluation without inner iterations and allows elimination of the forward problem, halving per-iteration cost while guaranteeing convergence across diverse PDEs and delay equations.

ABSTRACT

We present an efficient preconditioner for linear problems $A x=y$. It guarantees monotonic convergence of the memory-efficient fixed-point iteration for all accretive systems of the form $A = L + V$, where $L$ is an approximation of $A$, and the system is scaled so that the discrepancy is bounded with $\lVert V Vert&lt;1$. In contrast to common splitting preconditioners, our approach is not restricted to any particular splitting. Therefore, the approximate problem can be chosen so that an analytic solution is available to efficiently evaluate the preconditioner. We prove that the only preconditioner with this property has the form $(L+I)(I - V)^{-1}$. This unique form moreover permits the elimination of the forward problem from the preconditioned system, often halving the time required per iteration. We demonstrate and evaluate our approach for wave problems, diffusion problems, and pantograph delay differential equations. With the latter we show how the method extends to general, not necessarily accretive, linear systems.

Motivation & Objective

• Address the instability and high memory cost of fixed-point iteration in non-symmetric linear systems by ensuring monotonic convergence.
• Overcome the limitations of traditional preconditioners that require iterative inversion of large operators.
• Develop a matrix-free preconditioner applicable to any splitting $A = L + V$ with $\|V\| < 1$, enabling efficient, closed-form evaluation.
• Eliminate the computational cost of the forward problem in the preconditioned system, significantly reducing per-iteration time.
• Extend the method to general, non-accretive systems via transformation to an equivalent accretive form.

Proposed method

• Propose a universal preconditioner of the form $\Gamma = (L + I)(I - V)^{-1}$, where $A = L + V$ and $\|V\| < 1$, ensuring $\|I - \Gamma^{-1}A\| < 1$ for monotonic convergence.
• Leverage the fact that $L$ can be chosen to admit a closed-form inverse, enabling matrix-free evaluation without iterative solvers.
• Derive a transformation that eliminates the forward problem from the preconditioned system, reducing iteration cost by up to 50%.
• Prove that the proposed preconditioner is the only one that guarantees monotonic convergence for arbitrary splittings of accretive systems.
• Apply the method to wave, diffusion, and delay differential equations, demonstrating robustness and efficiency across diverse PDEs.
• Use operator norm bounds and spectral analysis to derive convergence rate estimates, including $\|M\| \leq 1 - \frac{1}{S}$ for general case and tighter bounds for Hermitian cases.

Experimental results

Research questions

• RQ1 Can a matrix-free preconditioner be constructed that guarantees monotonic convergence for any splitting $A = L + V$ with $\|V\| < 1$?
• RQ2 Is there a unique preconditioner form that ensures convergence for all accretive systems regardless of splitting choice?
• RQ3 Can the computational cost of the forward problem be eliminated from the preconditioned iteration without sacrificing accuracy?
• RQ4 How does the convergence rate of the universal split preconditioner compare to shift-splitting and other standard preconditioners across different PDEs?
• RQ5 Can the method be extended to non-accretive systems through transformation to an equivalent accretive form?

Key findings

• The preconditioner $(L + I)(I - V)^{-1}$ is the only form that guarantees monotonic convergence of fixed-point iteration for all accretive systems with $\|V\| < 1$.
• The method enables matrix-free evaluation of the preconditioner without inner iterations, achieving high efficiency and low memory usage.
• The forward problem can be eliminated from the preconditioned system, reducing per-iteration cost by approximately 50% in practice.
• Convergence rate bounds are derived: $\|M\| \leq 1 - \frac{1}{S}$ for general case and $\|M\| \leq 1 - \frac{1}{S}$ for Hermitian case, with $S = \|A^{-1}\|\|V\|$.
• Numerical experiments show effective convergence with GMRES, BiCGSTAB, and fixed-point iteration across Helmholtz, diffusion, and pantograph delay equations.
• The method successfully extends to non-accretive systems by transforming them into equivalent accretive forms, broadening applicability.