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Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

Alexey Chernov, Tung Le

TL;DR

This work analyzes parametric elliptic semilinear PDEs with parameter-dependent coefficients and forcing, establishing that analytic or Gevrey-$\delta$ data in the parameter induce the same level of parametric regularity in the solution. Using the novel alternative-to-factorial technique, it derives Gevrey-$\delta$ bounds for the solution in $H^{1}_{0}(D)$ and, under stronger data, in $H^{2}(D)$ with pointwise consequences. These regularity results yield rigorous convergence estimates for numerical integration in uncertainty quantification, demonstrated for Gauss-Legendre quadrature and randomly shifted lattice QMC, and validated by numerical experiments. The findings quantify how smoothness of inputs translates into fast convergence for high-dimensional parameter integration, enabling efficient UQ workflows for nonlinear reaction-diffusion models with random coefficients.

Abstract

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

TL;DR

This work analyzes parametric elliptic semilinear PDEs with parameter-dependent coefficients and forcing, establishing that analytic or Gevrey- data in the parameter induce the same level of parametric regularity in the solution. Using the novel alternative-to-factorial technique, it derives Gevrey- bounds for the solution in and, under stronger data, in with pointwise consequences. These regularity results yield rigorous convergence estimates for numerical integration in uncertainty quantification, demonstrated for Gauss-Legendre quadrature and randomly shifted lattice QMC, and validated by numerical experiments. The findings quantify how smoothness of inputs translates into fast convergence for high-dimensional parameter integration, enabling efficient UQ workflows for nonlinear reaction-diffusion models with random coefficients.

Abstract

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.
Paper Structure (14 sections, 11 theorems, 111 equations, 2 figures)

This paper contains 14 sections, 11 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The falling factorial estimates
  4. Multi-index notation
  5. Gevrey-class and analytic functions
  6. Elliptic semilinear PDEs with countably many parameters
  7. $L^{p}$ spaces and the Hölder inequality
  8. Well-posedness of the variational formulation
  9. Parametric regularity
  10. Parametric regularity in $H^{1}(D)$
  11. Parametric regularity in $H^{2}(D)$ and pointwise estimates
  12. Applications and numerical experiments
  13. Gauss-Legendre quadrature
  14. Quasi-Monte Carlo method for Gevrey functions

Key Result

Lemma 2.1

For all integers $n\geq 1$ and $k\geq 2$ the following identities hold

Figures (2)

  • Figure 1: Quadrature error $\varepsilon ^{(1)}_{n}$ (left) with respect to the number $n$ of quadrature points and Quadrature error $\varepsilon ^{(2)}_{n}$ (right) with respect to $N = n^{1/2}$.
  • Figure 2: Convergence of the quadrature error with respect to the number of samples $n$ for four methods: QMC analytic ($\varepsilon ^{{\mathrm{QMC}},(1)}_{n}$), QMC Gevrey ($\varepsilon ^{{\mathrm{QMC}},(2)}_{n}$), MC analytic ($\varepsilon ^{{\mathrm{MC}},(1)}_{n}$), MC Gevrey ($\varepsilon ^{{\mathrm{MC}},(2)}_{n}$).

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • ...and 14 more