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