Any decreasing cycle-convergence curve is possible for restarted GMRES

Abstract

Given a matrix order $n$, a restart parameter $m$ ($m<n$), a decreasing positive sequence $f(0)>f(1)>...>f(q)\ge 0$, where $q<n/m$, it is shown that there exits an $n$-by-$n$ matrix $A$ and a vector $r_0$ with $\Vert r_0\Vert =f(0)$ such that $\Vert r_k\Vert =f(k)$, $k=1,...,q$, 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 $r_0=b-A{x}_0$. Moreover, the matrix $A$ can be chosen to have any desired eigenvalues. We can also construct arbitrary cases of stagnation; namely, when $f(0)>f(1)>...>f(i)=f(i+1)\ge 0$ for any $i<q$. The restart parameter can be fixed or variable.

References in corpus (1)

• How Descriptive are GMRES Convergence Bounds?