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SIAC Accuracy Enhancement of Stochastic Galerkin Solutions for Wave Equations with Uncertain Coefficients

Andrés Galindo-Olarte, Jennifer K. Ryan

TL;DR

The paper tackles uncertainty quantification for a wave equation with random coefficients by coupling generalized polynomial chaos (gPC) with a discontinuous Galerkin (DG) discretization of the gPC coefficients. It introduces the Smoothness-Increasing Accuracy-Increasing (SIAC) filter as a post-processing step to DG-gPC, and provides negative-order Sobolev norm based error estimates showing that the filter can boost convergence from $O(h^{k+1})$ to up to $O(h^{2k+1})$ for the computed moments, notably the mean and variance. The authors prove convergence results for both the random-space truncation and the spatial-DG discretization, and validate them through numerical experiments on a periodic wave problem, demonstrating substantial noise reduction and improved accuracy of the first moments. The contributions extend SIAC filtering to hyperbolic PDEs with uncertainty, quantify the impact of filter parameters and moment choices, and enhance the practicality of uncertainty quantification for stochastic wave propagation.

Abstract

This article establishes the usefulness of the Smoothness-Increasing Accuracy-Increasing (SIAC) filter for reducing the errors in the mean and variance for a wave equation with uncertain coefficients solved via generalized polynomial chaos (gPC) whose coefficients are approximated using discontinuous Galerkin (DG-gPC). Theoretical error estimates that utilize information in the negative-order norm are established. While the gPC approximation leads to order of accuracy of $m-1/2$ for a sufficiently smooth solution (smoothness of $m$ in random space), the approximated coefficients solved via DG improves from order $k+1$ to $2k+1$ for a solution of smoothness $2k+2$ in physical space. Our numerical examples verify the performance of the filter for improving the quality of the approximation and reducing the numerical error and significantly eliminating the noise from the spatial approximation of the mean and variance. Further, we illustrate how the errors are effected by both the choice of smoothness of the kernel and number of function translates in the kernel. Hence, this article opens the applicability of SIAC filters to other hyperbolic problems with uncertainty, and other stochastic equations.

SIAC Accuracy Enhancement of Stochastic Galerkin Solutions for Wave Equations with Uncertain Coefficients

TL;DR

The paper tackles uncertainty quantification for a wave equation with random coefficients by coupling generalized polynomial chaos (gPC) with a discontinuous Galerkin (DG) discretization of the gPC coefficients. It introduces the Smoothness-Increasing Accuracy-Increasing (SIAC) filter as a post-processing step to DG-gPC, and provides negative-order Sobolev norm based error estimates showing that the filter can boost convergence from to up to for the computed moments, notably the mean and variance. The authors prove convergence results for both the random-space truncation and the spatial-DG discretization, and validate them through numerical experiments on a periodic wave problem, demonstrating substantial noise reduction and improved accuracy of the first moments. The contributions extend SIAC filtering to hyperbolic PDEs with uncertainty, quantify the impact of filter parameters and moment choices, and enhance the practicality of uncertainty quantification for stochastic wave propagation.

Abstract

This article establishes the usefulness of the Smoothness-Increasing Accuracy-Increasing (SIAC) filter for reducing the errors in the mean and variance for a wave equation with uncertain coefficients solved via generalized polynomial chaos (gPC) whose coefficients are approximated using discontinuous Galerkin (DG-gPC). Theoretical error estimates that utilize information in the negative-order norm are established. While the gPC approximation leads to order of accuracy of for a sufficiently smooth solution (smoothness of in random space), the approximated coefficients solved via DG improves from order to for a solution of smoothness in physical space. Our numerical examples verify the performance of the filter for improving the quality of the approximation and reducing the numerical error and significantly eliminating the noise from the spatial approximation of the mean and variance. Further, we illustrate how the errors are effected by both the choice of smoothness of the kernel and number of function translates in the kernel. Hence, this article opens the applicability of SIAC filters to other hyperbolic problems with uncertainty, and other stochastic equations.

Paper Structure

This paper contains 15 sections, 4 theorems, 57 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Polynomial chaos Galerkin approximation
  5. Eigenvalue decomposition for the gPC system
  6. Convergence in random space
  7. Approximation of coefficients via the discontinuous Galerkin method.
  8. Temporal Discretizations
  9. Convergence of the approximated coefficients.
  10. SIAC filtering
  11. The SIAC Filter
  12. Theory: Superconvergence-Extracting Post-Processing
  13. Numerical Experiments
  14. Affect of moments and smoothness
  15. Conclusion

Key Result

Theorem 1

Consider the hyperbolic equation eqn:model where $y$ is a random variable with beta distribution in $\Omega_y = (-1,1)$. Let $u(x,t;y)$ be the solution of eqn:model whose exact gPC expansion is given by Equation eqn:chaos_expansion. Denote the truncated gPC expansion by $v(x,t;y),$ which is defined where $m$ is given in eqn:decay_condition.

Figures (3)

  • Figure 1: Errors before (solid line) and after post-processing (dashed line) for the mean $\mu_u$ and variance $\sigma^2_u$. $T=1$. $N=5$, $\mathbb{P}^2$ and $20$ elements.
  • Figure 2: Comparison of errors for the mean and variance before and after post-processing for different values B-Spline degree $\ell$.
  • Figure 3: Comparison of errors before (solid line) and after post-processing (dashed line) for mean ($\mu_u$) and variance ($\sigma^2_u$) for enforcing different number of moments $r$. $T=1$. $N=5$, $\mathbb{P}^1$ and $20$ elements.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2: $L^2$-error estimates
  • proof
  • Theorem 3: Negative-order error estimates
  • proof
  • Theorem 4
  • proof