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An adaptive ANOVA stochastic Galerkin method for partial differential equations with high-dimensional random inputs

Guanjie Wang, Smita Sahu, Qifeng Liao

TL;DR

This work tackles the curse of dimensionality in solving PDEs with high-dimensional random inputs by combining an adaptive ANOVA decomposition with generalized polynomial chaos (gPC) within a stochastic Galerkin framework. By representing each ANOVA component $u_{\mathbb{T}}(\mathbf{x},\boldsymbol{\mu}_{\mathbb{T}})$ via a concise gPC expansion and selecting active terms based on relative variance, the method dramatically reduces the stochastic space dimension while preserving accuracy. The approach yields substantial computational gains over Monte Carlo and anchored collocation, especially in high-dimensional diffusion problems and Helmholtz problems, and provides a straightforward surrogate model. The results demonstrate that the adaptive ANOVA SG method can efficiently handle up to 50-dimensional diffusion and 10-dimensional Helmholtz problems, with performance advantages tied to the sparsity of the resulting stochastic system and the adaptive activation of gPC bases.

Abstract

It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method for both diffusion and Helmholtz problems.

An adaptive ANOVA stochastic Galerkin method for partial differential equations with high-dimensional random inputs

TL;DR

This work tackles the curse of dimensionality in solving PDEs with high-dimensional random inputs by combining an adaptive ANOVA decomposition with generalized polynomial chaos (gPC) within a stochastic Galerkin framework. By representing each ANOVA component via a concise gPC expansion and selecting active terms based on relative variance, the method dramatically reduces the stochastic space dimension while preserving accuracy. The approach yields substantial computational gains over Monte Carlo and anchored collocation, especially in high-dimensional diffusion problems and Helmholtz problems, and provides a straightforward surrogate model. The results demonstrate that the adaptive ANOVA SG method can efficiently handle up to 50-dimensional diffusion and 10-dimensional Helmholtz problems, with performance advantages tied to the sparsity of the resulting stochastic system and the adaptive activation of gPC bases.

Abstract

It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method for both diffusion and Helmholtz problems.
Paper Structure (18 sections, 2 theorems, 81 equations, 9 figures, 6 tables, 2 algorithms)

This paper contains 18 sections, 2 theorems, 81 equations, 9 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Stochastic Galerkin method and ANOVA decomposition
  4. Variational formulation
  5. Discretization
  6. ANOVA decomposition
  7. Adaptive ANOVA decomposition
  8. Adaptive ANOVA stochastic Galerkin method
  9. Generalized polynomial chaos expansion of component functions
  10. Adaptive ANOVA stochastic Galerkin method
  11. Numerical results
  12. Test problem 1
  13. Case I: a $10$ dimensional diffusion problem
  14. Case II: a 50 dimensional diffusion problem
  15. Test problem 2
  16. ...and 3 more sections

Key Result

theorem 1

Given $\bm{x}\in D$ and $\mathbb{T}\subseteq \mathbb{U}$ with $|\mathbb{T}|>0$, assuming that the $|\mathbb{T}|$-th order component function $u_{\mathbb{T}}(\bm{x},\bm{\mu}_{\mathbb{T}})$ belongs to $L^2_{\rho}(\Gamma)$, then the generalized polynomial chaos expansion of $u_{\mathbb{T}}(\bm{x},\bm{\ where $u_{\bm{i}}(\bm{x})$ is the coefficient of $\Phi_{\bm{i}}(\bm{\mu})$ defined by

Figures (9)

  • Figure 1: Multi-indices of the gPC basis functions corresponding to the component functions of each order in $3$ dimensions with the total degree up to $6$, arranged according to the order of the component functions (from left to right): $0$-th, first, second, and third order.
  • Figure 2: Matrix block-structure (each block has dimension $n_{x}\times n_{x}$) for test problem 1 with $N=10$.
  • Figure 3: Comparison of errors with respect to CPU times and stochastic degrees of freedom for test problem 1 with $N=10$, where both the total degree of gPC in the AASG method and the grid level in the AASC method are set to $5$.
  • Figure 4: Matrix block-structure (each block has dimension $n_{x}\times n_{x}$) for test problem 1 with $N=50$.
  • Figure 5: Comparison of errors with respect to CPU times and stochastic degrees of freedom for test problem 1 with $N=50$, where both the total degree of gPC in the AASG method and the grid level in the AASC method are set to $5$.
  • ...and 4 more figures

Theorems & Definitions (4)

  • theorem 1
  • proof
  • theorem 2
  • proof