Localized Reduced Basis Additive Schwarz Methods

Reduced basis methods build low-rank approximation spaces for the solution sets of parameterized PDEs by computing solutions of the given PDE for appropriately selected snapshot parameters. Localized reduced basis methods reduce the offline cost of computing these snapshot solutions by instead constructing a global space from spatially localized less expensive problems. In the case of online enrichment, these local problems are iteratively solved in regions of high residual and correspond to subdomain solves in domain decomposition methods. We show in this note that indeed there is a close relationship between online-enriched localized reduced basis and domain decomposition methods by introducing a Localized Reduced Basis Additive Schwarz method (LRBAS), which can be interpreted as a locally adaptive multi-preconditioning scheme for the CG method.