Infinite GMRES for Parameterized Linear Systems

We consider linear parameterized systems $A(\mu) x(\mu) = b$ for many different $\mu$, where $A$ is large and sparse depends nonlinearly on $\mu$. Solving such individually each $\mu$ would require great computational effort. In this work we propose to compute a partial parameterization $\tilde{x} \approx x(\mu)$, $\tilde{x}(\mu)$ cheap evaluate Our methods are based the observation that companion linearization can be formed dependence only linear. particular, presented combine well-established Krylov subspace method systems, GMRES, with algorithms nonlinear eigenvalue problems (NEPs) generate basis subspace. Within new approach, matrix constructed in three ways, using tensor structure exploiting certain have low-rank properties. The analyzed analogously standard convergence theory GMRES systems. More specifically, error estimated magnitude of parameter spectrum matrix, which corresponds reciprocal solutions corresponding NEP. Numerical experiments illustrate competitiveness large-scale problems. simulations reproducible publicly available online.