[Paper Review] Approximation of probability density functions for SPDEs using truncated series expansions

Giacomo Capodaglio, Max Gunzburger+1 more|arXiv (Cornell University)|Oct 2, 2018

Probabilistic and Robust Engineering DesignDecision Sciences3 citations

TL;DR

This paper proposes approximating the probability density function (PDF) of functionals from stochastic partial differential equations (SPDEs) using truncated Gram-Charlier and Edgeworth series expansions. By leveraging cumulants derived from solutions computed via Monte Carlo and stochastic Galerkin methods, the approach efficiently models non-Gaussian PDFs with polynomial expansions, offering a computationally efficient alternative to sampling-based density estimation.

ABSTRACT

The probability density function (PDF) of a random variable associated with the solution of a stochastic partial differential equation (SPDE) is approximated using a truncated series expansion. The SPDE is solved using two stochastic finite element (SFEM) methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the SPDE, 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 SPDEs has not yet been explored.

Motivation & Objective

• To address the challenge of accurately approximating the probability density function (PDF) of functionals derived from solutions of stochastic partial differential equations (SPDEs), which often exhibit non-Gaussian behavior.
• To explore the application of classical series expansions—Gram-Charlier and Edgeworth—to SPDEs, a domain where such methods remain underexplored despite their theoretical potential.
• To compare the performance and accuracy of truncated series expansions against traditional sampling-based methods in capturing the shape and tail behavior of SPDE solution PDFs.
• To evaluate the effectiveness of cumulant-based approximations using results from two stochastic finite element methods: Monte Carlo sampling and stochastic Galerkin with global polynomials.
• To provide a computationally efficient alternative to Monte Carlo sampling for PDF estimation in SPDEs, particularly in cases where sampling is prohibitively expensive.

Proposed method

• The method employs truncated Gram-Charlier and Edgeworth series expansions to approximate the PDF of a random variable that is a functional of the SPDE solution, such as the spatial average over the domain.
• Cumulants of the random variable are computed from the SPDE solution using two stochastic finite element methods: Monte Carlo sampling and the stochastic Galerkin method with global polynomial chaos expansions.
• The series expansions express the target PDF as a correction to a known reference PDF (typically the normal distribution), with correction terms based on cumulants of the random variable.
• The number of terms in the series is truncated to a finite number, balancing accuracy and computational cost, with higher-order terms capturing skewness and kurtosis effects.
• The resulting approximated PDFs are evaluated for accuracy by comparing them to reference PDFs obtained via Monte Carlo sampling.
• The approach is validated across test cases where the true solution PDF is known or well-approximated, enabling quantitative assessment of the approximation error.

Experimental results

Research questions

• RQ1 Can truncated Gram-Charlier and Edgeworth series effectively approximate the PDF of functionals derived from SPDE solutions with non-Gaussian characteristics?
• RQ2 How do the accuracy and convergence of these series approximations compare to Monte Carlo sampling in capturing the shape and tails of the PDF?
• RQ3 What is the impact of using cumulants from different stochastic finite element methods—Monte Carlo versus stochastic Galerkin—on the quality of the PDF approximation?
• RQ4 In what scenarios do higher-order terms in the series expansions significantly improve the approximation, and when are lower-order expansions sufficient?
• RQ5 To what extent can these series expansions serve as a computationally efficient alternative to full sampling for PDF estimation in SPDEs?

Key findings

• The truncated Gram-Charlier and Edgeworth series provide a viable and computationally efficient method for approximating non-Gaussian PDFs of SPDE functionals without requiring extensive sampling.
• The approximation quality improves with increasing number of cumulants used, particularly in capturing skewness and kurtosis, which are critical for non-Gaussian distributions.
• The stochastic Galerkin method yields more accurate cumulant estimates than Monte Carlo sampling for the same computational cost, leading to better PDF approximations.
• The method effectively captures the PDF shape in regions of high density and exhibits reasonable performance in the tails, though accuracy diminishes for highly skewed or heavy-tailed distributions.
• The approach demonstrates significant computational savings compared to Monte Carlo sampling, especially when high-resolution PDF estimation is required.
• The use of global polynomial chaos in the stochastic Galerkin framework enables smooth and stable cumulant computation, which is essential for reliable series expansion coefficients.