On the theoretical foundations of stochastic reduced basis methods
Prasanth B. Nair · 19th AIAA Applied Aerodynamics Conference · 2001
Stochastic reduced basis methods (SRBMs) are a class of numerical techniques for approximately computing the response of stochastic systems. The basic idea is to approximate the response using a linear combination of stochastic basis vectors with undetermined coefficients. In this paper, we examine the theoretical foundations of SRBMs by exploring their relationship with Krylov subspace methods for deterministic systems. The mathematical justification for employing the terms of the stochastic Krylov subspace as basis vectors is presented. It is shown that SRBMs are a stochastic generalization of preconditioned Krylov subspace methods. Subsequently, some approaches for stochastic generalization of the Bubnov-Galerkin scheme are analyzed. We also address the issue of computing a posteriori error estimates of SRBMs. Some preliminary numerical studies are presented for examining the accuracy of the error estimates. The paper concludes with a discussion of ongoing work on algebraic random eigenvalue problems.