Approximation of probability density functions for PDEs with random parameters using truncated series expansions

TL;DR

This paper introduces a novel approach to approximate the probability density functions of solutions to PDEs with random parameters using truncated Gram-Charlier and Edgeworth series expansions, validated through stochastic finite element methods.

Contribution

It is the first to apply Gram-Charlier and Edgeworth series expansions for PDFs of PDE solutions with random parameters, combining stochastic finite element methods.

Findings

Successful approximation of PDFs using truncated series expansions.

Application of Monte Carlo and stochastic Galerkin methods for PDE solutions.

Novel use of cumulant-based series expansions in PDE uncertainty quantification.

Abstract

The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two stochastic finite element methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the random PDE, such as the average over the physical domain. The truncated series are obtained considering a finite number of terms in the Gram-Charlier or Edgeworth series expansions. These expansions approximate the PDF of a random variable in terms of another PDF, and involve coefficients that are functions of the known cumulants of the random variable. To the best of our knowledge, their use in the framework of PDEs with random parameters has not yet been explored.