Weighted reduced order methods for parametrized partial differential equations with random inputs

TL;DR

This paper develops weighted reduced order methods for stochastic PDEs with random inputs, leveraging parametrized formulations and stochastic weights to improve computational efficiency, demonstrated through an elasticity problem example.

Contribution

It introduces weighted reduced basis and proper orthogonal decomposition methods tailored for stochastic PDEs with random inputs, enhancing reduction efficiency.

Findings

Effective reduction of stochastic PDEs demonstrated

Weighted methods outperform unweighted approaches

Numerical example confirms computational savings

Abstract

In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.