Admissible and attainable convergence behavior with stagnation mirroring in restarted (block) GMRES

In this work, we describe how to construct matrices and block right-hand sides that exhibit a specified restarted block \gmres convergence pattern, such that the eigenvalues and Ritz values at each iteration can be chosen independent of the specified convergence behavior. This work is a generalization of the work in [Meurant and Tebbens, Num.~Alg.~2019] in which the authors do the same for restarted non-block \gmres. We use the same tools as were used in [Kubínová and Soodhalter, SIMAX 2020], namely to analyze block \gmres as an iteration over a right vector space with scalars from the $^\ast$-algebra of matrices. To facilitate our work, we also extend the work of Meurant and Tebbens and offer alternative proofs of some of their results, that can be more easily generalized to the block setting.

💡 Research Summary

The paper develops a comprehensive theory for prescribing the convergence behavior of restarted block GMRES (Bl‑GMRES). Building on the earlier work of Meurant and Tebbens (2019) for non‑block GMRES, the authors show that for any admissible decreasing sequence of residual norms {f₀≥f₁≥…≥f_{n‑1}>0} one can construct a matrix A and a block right‑hand side B such that restarted Bl‑GMRES reproduces exactly this sequence, while the eigenvalues of A and the Ritz values generated at each cycle can be chosen independently of the prescribed convergence pattern.

The authors first review the classical GMRES framework, the definition of admissible convergence, and the known result that any admissible sequence can be realized by an appropriate (A,b) pair, with Ritz values freely assignable when no stagnation occurs. They then extend this to the restarted setting. A key observation, originally proved for the scalar case, is that if a restart cycle ends with s steps of stagnation (i.e., the residual norm does not decrease for the last s iterations), then the next cycle must begin with exactly s steps of stagnation. This “mirroring of stagnation” follows directly from the polynomial‑minimization characterisation of GMRES: the minimizing polynomial after the stagnating block is the same as the one that generated the stagnation, so the new cycle inherits it as its initial polynomial. Consequently, admissible convergence sequences for restarted GMRES are subject to this symmetry constraint.

To handle the block case (p>1), the paper adopts the ‑algebra framework of Kubínová and Soodhalter (2020). Here scalars are replaced by p×p complex matrices, and Krylov subspaces become right‑vector spaces over this algebra. The block Arnoldi process yields a block Hessenberg matrix H_j∈ℂ^{p j×p j}. The authors show that each H_j admits a factorisation H_j = D_j U_j C_j (D_j U_j)^{-1}, where U_j is unit‑upper‑triangular and D_j is diagonal. Using the first row of (D_j U_j)^{-1} they express the residual at iteration j as r_j = V_{j+1} g_j with g_j = (e₁ᵀ(D_j U_j)^{-1})‖r_j‖², and prove that ‖g_j‖ = ‖r_j‖. This representation mirrors the scalar case and enables direct control of the residual norm through the diagonal entries of D_j.

The construction proceeds as follows. One fixes the restart length m, the block size p, and the desired residual norm sequence. From the sequence the authors compute coefficients h_i = √(‖r_{i‑1}‖²‑‖r_i‖²) which define the columns of a “restarted Krylov matrix” K̂ =