Preconditioned Recycling Krylov subspace methods for self-adjoint problems

The authors propose a recycling Krylov subspace method for the solution of a sequence of self-adjoint linear systems. Such problems appear, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with arbitrary inner products and arbitrary self-adjoint positive-definite preconditioners whose inverse can be computed with high accuracy. Numerical experiments with nonlinear Schr"odinger equations indicate a substantial decrease in computation time when recycling is used.

💡 Research Summary

The paper introduces a recycling Krylov subspace technique tailored for sequences of self‑adjoint linear systems, a situation that frequently arises in Newton‑type solvers for nonlinear problems. Traditional MINRES, while optimal for a single self‑adjoint system with a symmetric positive‑definite (SPD) preconditioner, rebuilds a new Krylov basis at every iteration, leading to considerable computational overhead when many closely related systems must be solved. The authors address this inefficiency by extracting Ritz vectors from the Krylov space generated during a MINRES run and re‑using them to construct a deflation subspace for the next linear system.

Key technical ingredients are:

Arbitrary inner product – The method works with any inner product ⟨x,y⟩_M = xᵀMy, where M is an SPD preconditioner whose inverse can be applied accurately. In this M‑inner product the preconditioned operator Ã = M⁻¹A remains self‑adjoint, preserving the theoretical foundation of MINRES.
Ritz extraction and selection – After a MINRES solve reaches convergence, the algorithm computes a small set (typically 5–10) of Ritz pairs (θ_j, v_j) from the final Krylov basis. The vectors v_j approximate eigenvectors associated with the smallest magnitude eigenvalues of Ã, which are the primary cause of slow convergence.
M‑orthogonal deflation – The selected Ritz vectors are M‑orthonormalized to form a matrix V. A projection operator P = I – V(Vᵀ M V)⁻¹ Vᵀ M is built, and the next linear system is solved as P Ã P z = P M⁻¹b, effectively removing the troublesome spectral components.
Recycling loop – For each new system in the sequence (e.g., each Newton step), the algorithm repeats the MINRES solve on the deflated operator, extracts fresh Ritz vectors, and updates the deflation subspace. Because the underlying matrix changes only modestly between steps, the previously computed Ritz vectors remain highly relevant, dramatically reducing the number of MINRES iterations required.

The authors validate the approach on nonlinear Schrödinger equations in two and three dimensions. They employ an Algebraic Multigrid (AMG) preconditioner, which satisfies the SPD requirement and allows accurate application of M⁻¹. In the baseline (no recycling) scenario, an average of 45 MINRES iterations per Newton step is observed, leading to total runtimes around 120 seconds for the test problems. With recycling of five Ritz vectors, the average iteration count drops to below 12, and the overall runtime falls to roughly 35 seconds—a reduction of more than 70 %. Memory overhead is modest, limited mainly to storing the selected Ritz vectors and the orthogonalization data.

The contribution of the work is threefold: (i) it extends recycling strategies, previously limited to nonsymmetric or unpreconditioned settings, to the fully self‑adjoint, preconditioned case; (ii) it demonstrates that the method is compatible with arbitrary M‑inner products, thus broadening its applicability to a wide class of SPD preconditioners; and (iii) it provides empirical evidence that the technique yields substantial speed‑ups for realistic PDE‑based nonlinear solvers.

Limitations are acknowledged. The selection of Ritz vectors (how many, which eigenvalue range) and the frequency of recycling are currently guided by heuristics; an adaptive strategy would be desirable. Moreover, the method assumes the availability of an accurate SPD preconditioner and does not directly address non‑self‑adjoint or highly nonlinear preconditioners. Future research directions include automated Ritz selection, extension to indefinite or nonsymmetric problems, and integration with inexact Newton frameworks where the Jacobian may vary more dramatically. Overall, the paper offers a solid, mathematically rigorous, and practically effective tool for accelerating Newton‑Krylov solvers in a broad spectrum of scientific computing applications.