Reduced Krylov Basis Methods for Parametric Partial Differential Equations

We present a user-friendly reduced basis method for solving a family of parametric PDEs by preconditioned Krylov subspace methods including the CG method, GMRes, and BiCGStab. 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 the 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 on the high-fidelity level; and (2) the problems for any value in the parameter set have much smaller dimensions they are restricted to the subspace defined during the Krylov iterations. This is a joint work with Yuwen Li (Zhejiang University), Cheng Zuo (Penn State).