Additive reduced basis preconditioners for large-scale parametrized PDEs
We introduce a class of additive reduced basis preconditioners designed to accelerate the iterative solution of large-scale linear systems arising from discretized parametrized PDEs. The main idea is to regularize the inherently singular reduced-order approximation by adding simple correction terms: either a scaled identity correction or a projected correction based on a basic preconditioner. This yields nonsingular preconditioners under explicit and easily checked conditions. The construction and application of the preconditioners are integrated into an FGMRES framework through an offline strategy that dynamically builds the reduced-basis component by proper orthogonal decomposition at each FGMRES step. We establish sufficient conditions for the nonsingularity of the preconditioners and derive error bounds for the preconditioned Richardson iteration. Numerical results for convection-diffusion, anisotropic vortex, Stokes, and Helmholtz problems are provided to verify the efficiency and convergence of the proposed ARB preconditioners. The method consistently converges in a few iterations and substantially reduces online solve time, supporting its efficiency for multi-query engineering scenarios.