Linear Systems and Eigenvalue Problems — Problem 14

Definitions and mathematical statement

A conventional framework for expressing the question is the following. Let $\mathbb K\in\{{\mathbb{R}},{\mathbb{C}}\}$, let $A\in\mathbb K^{n\times n}$, and consider $Ax=b$ with initial approximation $x_0$ and initial residual $r_0=b-Ax_0$. Define \[ \mathcal K_0(A,r_0)=\{0\},\qquad \mathcal K_k(A,r_0)=\operatorname{span}\{r_0,Ar_0,\ldots,A^{k-1}r_0\}\quad(k\geq1). \] For standard unrestarted GMRES in the Euclidean norm, the residual $r_k=b-Ax_k$ satisfies \[ x_k\in x_0+\mathcal K_k(A,r_0),\qquad \|r_k\|_2=\min_{x\in x_0+\mathcal K_k(A,r_0)}\|b-Ax\|_2. \] These conventions provide notation; the supplied record does not specify the GMRES variant, residual norm, or precise discretized system in the examples it refers to.

Eigenvalue information means the locations of the elements of $\sigma(A)$, the spectrum of $A$. If $A$ admits a diagonalization $A=V\Lambda V^{-1}$, eigenvector conditioning can be expressed, for a specified eigenvector matrix $V$, by \[ \kappa_2(V)=\|V\|_2\,\|V^{-1}\|_2. \] This conditional definition does not assume that the convection--diffusion matrices in the source are diagonalizable. A choice of eigenvector normalization and, for repeated eigenvalues, a choice of eigenbasis would be needed to make such a measure unambiguous.

Problem (supplied source formulation). Explain the acceleration of GMRES convergence after an initial transient phase observed in some convection--diffusion model problems, including the example cited in the source as [LieStr05]. In particular, determine whether, and in what precise sense, the convergence speed can be related to the eigenvalue locations and eigenvector conditioning of the discretized convection--diffusion operators.

The supplied formulation is an explanatory research question, rather than a quantified assertion of a particular convergence estimate. It does not define the transient phase or acceleration, identify the differential operators, domains, coefficients, boundary conditions, discretizations, right-hand sides, initial guesses, or preconditioners, or specify which spectral data should predict which measure of convergence. The bibliographic identity and model problem of [LieStr05] are also absent. Consequently, no universal quantifier over convection--diffusion systems, particular onset criterion, or specific convergence bound is asserted here; those unresolved qualifications require examination of the cited source.

Short English statement

Explain why GMRES sometimes converges faster after an initial transient when applied to discretized convection--diffusion problems, especially the example referenced as [LieStr05]. Determine whether eigenvalue locations and eigenvector conditioning explain that behavior. The supplied record leaves the model, algorithmic details, and meaning of acceleration unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $800505$, source problem code catalogue entry, title "Linear Systems and Eigenvalue Problems --- Problem 14." The supplied record attributes the problem to Amsel, Baumann et al. (2026) and labels its status as open. This is a supplied registry label, not an independently verified mathematical or literature assessment. The supplied upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Amsel, Baumann et al., Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop (2026), Problem 14. Supplied source URL: https://arxiv.org/abs/2602.05394. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• [LieStr05], cited in the supplied statement for a convection--diffusion model problem exhibiting the reported behavior. No authors, title, publication details, or URL for this reference are supplied, so its identity and the precise example remain unresolved.