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Finite element analysis of density estimation using preintegration for elliptic PDE with random input

Alexander D. Gilbert

TL;DR

This work analyzes the finite element (FE) component of density estimation for elliptic PDEs with random input, focusing on the FE error in the preintegration-based method that yields smooth cdf and pdf for uncertainty quantification. By proving FE monotonicity, deriving two robust lower bounds, and establishing parametric regularity for the FE approximation, the authors obtain FE error bounds for the cdf and pdf that match the rates of the simpler expected-value problem. They then integrate FE with quasi-Monte Carlo (QMC) quadrature to bound the overall mean-square error of the density estimates, obtaining a convergence rate of order $h^{\tau'+\tau}$ in space and $N^{-1+\varepsilon}$ in the number of QMC points, for any small $\varepsilon>0$, up to constants depending on the problem’s stochastic dimension. The results provide a complete error framework for density estimation in random elliptic PDEs, enabling accurate and efficient UQ via preintegration, FE discretization, and QMC integration.

Abstract

This paper analyses the finite element component of the error when using preintegration to approximate the cdf and pdf for uncertainty quantification (UQ) problems involving elliptic PDEs with random inputs. It is a follow up to Gilbert, Kuo, Srikumar, SIAM J. Numer. Anal. 63 (2025), pp. 1025-1054, which introduced a method of density estimation for a class of UQ problems, based on computing the integral formulations of the cdf and pdf by performing an initial smoothing preintegration step and then applying a quasi-Monte Carlo quadrature rule to approximate the remaining high-dimensional integral. That paper focussed on the quadrature aspect of the method, whereas this paper studies the spatial discretisation of the PDE using finite element methods. First, it is shown that the finite element approximation satisfies the required assumptions for the preintegration theory, including the important monotonicity condition. Then the finite element error is analysed and finally, the combined finite element and quasi-Monte Carlo error is bounded. It is shown that under similar assumptions, the cdf and pdf can be approximated with the same rate of convergence as the much simpler problem of computing expected values.

Finite element analysis of density estimation using preintegration for elliptic PDE with random input

TL;DR

This work analyzes the finite element (FE) component of density estimation for elliptic PDEs with random input, focusing on the FE error in the preintegration-based method that yields smooth cdf and pdf for uncertainty quantification. By proving FE monotonicity, deriving two robust lower bounds, and establishing parametric regularity for the FE approximation, the authors obtain FE error bounds for the cdf and pdf that match the rates of the simpler expected-value problem. They then integrate FE with quasi-Monte Carlo (QMC) quadrature to bound the overall mean-square error of the density estimates, obtaining a convergence rate of order in space and in the number of QMC points, for any small , up to constants depending on the problem’s stochastic dimension. The results provide a complete error framework for density estimation in random elliptic PDEs, enabling accurate and efficient UQ via preintegration, FE discretization, and QMC integration.

Abstract

This paper analyses the finite element component of the error when using preintegration to approximate the cdf and pdf for uncertainty quantification (UQ) problems involving elliptic PDEs with random inputs. It is a follow up to Gilbert, Kuo, Srikumar, SIAM J. Numer. Anal. 63 (2025), pp. 1025-1054, which introduced a method of density estimation for a class of UQ problems, based on computing the integral formulations of the cdf and pdf by performing an initial smoothing preintegration step and then applying a quasi-Monte Carlo quadrature rule to approximate the remaining high-dimensional integral. That paper focussed on the quadrature aspect of the method, whereas this paper studies the spatial discretisation of the PDE using finite element methods. First, it is shown that the finite element approximation satisfies the required assumptions for the preintegration theory, including the important monotonicity condition. Then the finite element error is analysed and finally, the combined finite element and quasi-Monte Carlo error is bounded. It is shown that under similar assumptions, the cdf and pdf can be approximated with the same rate of convergence as the much simpler problem of computing expected values.
Paper Structure (15 sections, 11 theorems, 59 equations)

This paper contains 15 sections, 11 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Background on random PDEs and smoothing by preintegration
  3. PDEs with random input
  4. Finite element methods
  5. Density estimation using smoothing by preintegration
  6. Density estimation for the discrete problem
  7. Preintegration theory for the discrete problem
  8. Monotonicity of the FE approximation
  9. Monotonicity of the FE approximation
  10. Two lower bounds for the FE approximation
  11. Parametric regularity of the FE approximation
  12. Error analysis
  13. FE error
  14. Combined QMC and FE error analysis
  15. Conclusion

Key Result

Theorem 2.1

Suppose that Assumption asm:pde holds and for ${\boldsymbol{y}} = ({\boldsymbol{w}}, {\boldsymbol{z}}) \in \mathbb{R}^{2s + 1}$ let $\ell(\cdot, {\boldsymbol{w}}) \in H^{-1 + \tau'}(D)$ for some $\tau' \in (0, 1]$. Then, the solution of eq:varpde satisfies $u(\cdot, {\boldsymbol{y}}) \in V \cap H^{1 where $C_{\tau'} < \infty$ is independent of ${\boldsymbol{y}}$.

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Corollary 4.4
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • ...and 2 more