Descriptive GMRES Convergence Bounds for Nonnormal Systems

Develop a convergence bound for GMRES applied to nonnormal linear systems that accurately describes the transient behavior early in the iteration, incorporating effects of nonnormality beyond eigenvalue locations.

Background

While GMRES behavior is well understood for normal (or Hermitian) problems, nonnormality introduces transient phenomena not captured by classical eigenvalue-based bounds.

Field-of-values and pseudospectral tools provide partial insights, but a descriptive, widely applicable early-iteration bound remains missing.

References

Despite numerous existing approaches, the following problem is largely open (cf. also): Develop a descriptive convergence bound for GMRES applied to nonnormal linear algebraic systems, that particularly describes the (transient) behavior early in the iteration.

— Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop  (2602.05394 - Amsel et al., 5 Feb 2026) in Subsection "Preconditioning and convergence of iterative solvers" (Section 2)

A rigorous GMRES convergence theory for the active-set-dependent preconditioned systems remains open.

— A Function-Space Framework for BDDC Preconditioning in Control- and State-Constrained Sparse Optimal Control  (2609.09541 - Antil et al., 8 Sep 2026) in Section 1, immediately following the list of contributions; reiterated in Section 6 (Conclusion)