On Fractional Moment Estimation from Polynomial Chaos Expansion

TL;DR

This paper introduces an analytical method to estimate fractional statistical moments directly from polynomial chaos expansions, improving accuracy over traditional sampling methods in uncertainty quantification tasks.

Contribution

The paper presents a novel approach for estimating fractional moments analytically from PCE coefficients, bypassing the need for extensive sampling.

Findings

The method accurately estimates probability distributions in numerical examples.

It outperforms Latin hypercube sampling in the presented cases.

The approach is effective for models of increasing complexity.

Abstract

Fractional statistical moments are utilized for various tasks of uncertainty quantification, including the estimation of probability distributions. However, an estimation of fractional statistical moments of costly mathematical models by statistical sampling is challenging since it is typically not possible to create a large experimental design due to limitations in computing capacity. This paper presents a novel approach for the analytical estimation of fractional moments, directly from polynomial chaos expansions. Specifically, the first four statistical moments obtained from the deterministic PCE coefficients are used for an estimation of arbitrary fractional moments via H\"{o}lder's inequality. The proposed approach is utilized for an estimation of statistical moments and probability distributions in three numerical examples of increasing complexity. Obtained results show that the proposed approach achieves a superior performance in estimating the distribution of the response, in comparison to a standard Latin hypercube sampling in the presented examples.