Localized Reduced Basis Additive Schwarz Methods

TL;DR

This paper introduces a Localized Reduced Basis Additive Schwarz (LRBAS) method that combines reduced basis and domain decomposition techniques to efficiently solve parameterized PDEs with localized problem solving.

Contribution

It establishes a novel connection between localized reduced basis methods and domain decomposition, proposing an adaptive multi-preconditioning scheme for iterative solvers.

Findings

LRBAS effectively reduces offline computational costs.

The method provides a locally adaptive preconditioning approach.

Numerical experiments demonstrate improved convergence rates.

Abstract

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.