Any decreasing cycle-convergence curve is possible for restarted GMRES

TL;DR

This paper demonstrates that any decreasing residual convergence curve is achievable with restarted GMRES for any eigenvalues, including cases with stagnation, by constructing specific matrices and initial residuals.

Contribution

It proves the theoretical possibility of realizing any decreasing residual sequence in restarted GMRES, regardless of eigenvalues or stagnation, through explicit matrix and residual construction.

Findings

Any decreasing residual sequence can be realized with restarted GMRES.

The constructed matrix can have any prescribed eigenvalues.

Arbitrary stagnation cases are also achievable.

Abstract

Given a matrix order $n$, a restart parameter $m$ ($m < n$), a decreasing positive sequence $f(0) > f(1) > ... > f(q) \geq 0$, where $q < n/m$, it is shown that there exits an $n$-by-$n$ matrix $A$ and a vector $r_0$ with $\|r_0\|=f(0)$ such that $\|r_k\|=f(k)$, $k=1,...,q$, where $r_k$ is the residual at cycle $k$ of restarted GMRES with restart parameter $m$ applied to the linear system $Ax=b$, with initial residual $r_0=b-Ax_0$. Moreover, the matrix $A$ can be chosen to have any desired eigenvalues. We can also construct arbitrary cases of stagnation; namely, when $f(0) > f(1) > ... > f(i) = f(i+1) \geq 0 $ for any $i <q $. The restart parameter can be fixed or variable.