Reduced basis stochastic Galerkin methods for partial differential equations with random inputs

TL;DR

This paper introduces a reduced basis stochastic Galerkin method for PDEs with random inputs, significantly lowering computational costs by integrating reduced basis techniques and secant method optimization.

Contribution

It develops a novel framework combining reduced basis and stochastic Galerkin methods, enhancing efficiency in solving PDEs with randomness.

Findings

Validated accuracy through numerical experiments

Demonstrated significant computational cost reduction

Effective integration of secant method for basis selection

Abstract

We present a reduced basis stochastic Galerkin method for partial differential equations with random inputs. In this method, the reduced basis methodology is integrated into the stochastic Galerkin method, resulting in a significant reduction in the cost of solving the Galerkin system. To reduce the main cost of matrix-vector manipulation involved in our reduced basis stochastic Galerkin approach, the secant method is applied to identify the number of reduced basis functions. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments.