  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.
