Reduced Krylov Basis Methods for Parametric Partial Differential Equations

TL;DR

This paper introduces a reduced basis method leveraging preconditioned Krylov subspace techniques to efficiently solve parametric PDEs, significantly lowering computational costs by generating basis vectors from a single high-fidelity solve.

Contribution

It presents a user-friendly approach that combines Krylov subspace methods with reduced basis techniques for parametric PDEs, enabling efficient approximation of large-scale problems.

Findings

Few Krylov iterations suffice for accurate solutions

Reduced basis construction requires only one high-fidelity solve

Computational cost is significantly reduced

Abstract

This work is on a user-friendly reduced basis method for solving a family of parametric PDEs by preconditioned Krylov subspace methods including the conjugate gradient method, generalized minimum residual method, and bi-conjugate gradient method. The proposed methods use a preconditioned Krylov subspace method for a high-fidelity discretization of one parameter instance to generate orthogonal basis vectors of the reduced basis subspace. Then large-scale discrete parameter-dependent problems are approximately solved in the low-dimensional Krylov subspace. As shown in the theory and experiments, only a small number of Krylov subspace iterations are needed to simultaneously generate approximate solutions of a family of high-fidelity and large-scale systems in the reduced basis subspace. This reduces the computational cost dramatically because (1) to construct the reduced basis vectors, we only solve one large-scale problem in the high-fidelity level; and (2) the family of large-scale problems restricted to the reduced basis subspace have much smaller sizes.