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Uncertainty quantification for stationary and time-dependent PDEs subject to Gevrey regular random domain deformations

Ana Djurdjevac, Vesa Kaarnioja, Claudia Schillings, André-Alexander Zepernick

TL;DR

This work tackles uncertainty quantification for PDEs on randomly deformed domains by modeling the domain perturbation in a Gevrey-smooth (nonparametric) framework via a domain-mapping approach. By pulling back the PDEs to a fixed reference domain, the authors derive Gevrey-regular bounds for the pullback coefficients, source terms, and solutions for both the Poisson and heat equations, enabling the design of randomly shifted lattice QMC rules with essentially dimension-independent, linear convergence rates. The analysis also accounts for truncation of the infinite-dimensional input, finite element discretization, and time-stepping errors, yielding explicit overall error bounds that combine these sources. Numerical experiments corroborate the theoretical rates and demonstrate that the Gevrey-regular model captures a broad class of domain uncertainties beyond standard Karhunen–Loève representations, with practical implications for efficient UQ in engineering and biology contexts.

Abstract

We study uncertainty quantification for partial differential equations subject to domain uncertainty. We parameterize the random domain using the model recently considered by Chernov and Le (2024) as well as Harbrecht, Schmidlin, and Schwab (2024) in which the input random field is assumed to belong to a Gevrey smoothness class. This approach has the advantage of being substantially more general than models which assume a particular parametric representation of the input random field such as a Karhunen-Loeve series expansion. We consider both the Poisson equation as well as the heat equation and design randomly shifted lattice quasi-Monte Carlo (QMC) cubature rules for the computation of the expected solution under domain uncertainty. We show that these QMC rules exhibit dimension-independent, essentially linear cubature convergence rates in this framework. In addition, we complete the error analysis by taking into account the approximation errors incurred by dimension truncation of the random input field and finite element discretization. Numerical experiments are presented to confirm the theoretical rates.

Uncertainty quantification for stationary and time-dependent PDEs subject to Gevrey regular random domain deformations

TL;DR

This work tackles uncertainty quantification for PDEs on randomly deformed domains by modeling the domain perturbation in a Gevrey-smooth (nonparametric) framework via a domain-mapping approach. By pulling back the PDEs to a fixed reference domain, the authors derive Gevrey-regular bounds for the pullback coefficients, source terms, and solutions for both the Poisson and heat equations, enabling the design of randomly shifted lattice QMC rules with essentially dimension-independent, linear convergence rates. The analysis also accounts for truncation of the infinite-dimensional input, finite element discretization, and time-stepping errors, yielding explicit overall error bounds that combine these sources. Numerical experiments corroborate the theoretical rates and demonstrate that the Gevrey-regular model captures a broad class of domain uncertainties beyond standard Karhunen–Loève representations, with practical implications for efficient UQ in engineering and biology contexts.

Abstract

We study uncertainty quantification for partial differential equations subject to domain uncertainty. We parameterize the random domain using the model recently considered by Chernov and Le (2024) as well as Harbrecht, Schmidlin, and Schwab (2024) in which the input random field is assumed to belong to a Gevrey smoothness class. This approach has the advantage of being substantially more general than models which assume a particular parametric representation of the input random field such as a Karhunen-Loeve series expansion. We consider both the Poisson equation as well as the heat equation and design randomly shifted lattice quasi-Monte Carlo (QMC) cubature rules for the computation of the expected solution under domain uncertainty. We show that these QMC rules exhibit dimension-independent, essentially linear cubature convergence rates in this framework. In addition, we complete the error analysis by taking into account the approximation errors incurred by dimension truncation of the random input field and finite element discretization. Numerical experiments are presented to confirm the theoretical rates.

Paper Structure

This paper contains 30 sections, 20 theorems, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Outline of the paper.
  4. Notations, preliminaries, and assumptions
  5. Poisson equation.
  6. Heat equation.
  7. Quasi-Monte Carlo integration
  8. Parametric regularity analysis
  9. Parametric regularity of the pullback coefficient
  10. Step 1.
  11. Step 2.
  12. Step 3.
  13. Step 4.
  14. Regularity of the right-hand side
  15. Step 5.
  16. ...and 15 more sections

Key Result

lemma thmcounterlemma

Let $F\in\mathcal{W}_{s,{\boldsymbol{\gamma}}}$ with positive weights $\boldsymbol{\gamma}=(\gamma_{\mathfrak u})_{\mathfrak u\subset\{1,\ldots,s\}}$. An $s$-dimensional randomly shifted lattice rule with $n=2^m$, $m\in\mathbb N$, points can be constructed by a CBC algorithm such that, for $R$ indep where $\mathbb E_{\boldsymbol\Delta}[\cdot]$ denotes the expected value with respect to uniformly d

Figures (3)

  • Figure 1: Left: the triangulation of the reference domain $D_{\rm ref}$ used in all numerical experiments. Middle: a random realization of the random field corresponding to experiment \ref{['eq:ex1']}. Right: a random realization of the random field corresponding to experiment \ref{['eq:ex2']}.
  • Figure 2: The computed $L^2(D_{\rm ref})$ errors for the Poisson equation. Left: the numerical cubature errors corresponding to experiments \ref{['eq:ex1']}--\ref{['eq:ex2']}. Right: the numerical cubature errors corresponding to experiments \ref{['eq:ex3']}--\ref{['eq:ex4']}.
  • Figure 4: The computed $L^2(I;L^2(D_{\rm ref}))$ errors for the heat equation. Left: the numerical cubature errors corresponding to experiments \ref{['eq:ex1']}--\ref{['eq:ex2']}. Right: the numerical cubature errors corresponding to experiments \ref{['eq:ex3']}--\ref{['eq:ex4']}.

Theorems & Definitions (37)

  • definition thmcounterdefinition
  • lemma thmcounterlemma: cf. kuonuyenssurvey
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • ...and 27 more