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Sufficient conditions for QMC analysis of finite elements for parametric differential equations

Vesa Kaarnioja, Andreas Rupp, Jay Gopalakrishnan

TL;DR

This work develops a dimension-independent QMC framework for uncertainty quantification in parametric PDEs by analyzing flux-based finite element discretizations under Gevrey-regular randomness. By tying parametric regularity to the flux $\boldsymbol q$ and deriving recursive bounds for parametric derivatives, the authors prove optimal QMC convergence rates for flux-dependent quantities of interest across conforming, mixed, and HDG discretizations of the diffusion equation. Numerical experiments corroborate the theory, showing that accurate flux approximation is crucial for achieving predicted convergence rates and highlighting the effectiveness of flux-centric QoIs in QMC-based UQ. The results provide a versatile blueprint for applying QMC cubature to high-dimensional PDEs with stochastic coefficients, with potential extensions to other equations and adaptive or sparse-grid methods.

Abstract

Parametric regularity of discretizations of flux vector fields satisfying a balance law is studied under some assumptions on a random parameter that links the flux with an unknown primal variable (often through a constitutive law). In the primary example of the stationary diffusion equation, the parameter corresponds to the inverse of the diffusivity. The random parameter is modeled here as a Gevrey-regular random field. Specific focus is on random fields expressible as functions of countably infinite sequences of independent random variables, which may be uniformly or normally distributed. Quasi-Monte Carlo (QMC) error bounds for some quantity of interest that depends on the flux are then derived using the parametric regularity. It is shown that the QMC method converges optimally if the quantity of interest depends continuously on the primal variable, its flux, or its gradient. A series of assumptions are introduced with the goal of encompassing a broad class of discretizations by various finite element methods. The assumptions are verified for the diffusion equation discretized using conforming finite elements, mixed methods, and hybridizable discontinuous Galerkin schemes. Numerical experiments confirm the analytical findings, highlighting the role of accurate flux approximation in QMC methods.

Sufficient conditions for QMC analysis of finite elements for parametric differential equations

TL;DR

This work develops a dimension-independent QMC framework for uncertainty quantification in parametric PDEs by analyzing flux-based finite element discretizations under Gevrey-regular randomness. By tying parametric regularity to the flux and deriving recursive bounds for parametric derivatives, the authors prove optimal QMC convergence rates for flux-dependent quantities of interest across conforming, mixed, and HDG discretizations of the diffusion equation. Numerical experiments corroborate the theory, showing that accurate flux approximation is crucial for achieving predicted convergence rates and highlighting the effectiveness of flux-centric QoIs in QMC-based UQ. The results provide a versatile blueprint for applying QMC cubature to high-dimensional PDEs with stochastic coefficients, with potential extensions to other equations and adaptive or sparse-grid methods.

Abstract

Parametric regularity of discretizations of flux vector fields satisfying a balance law is studied under some assumptions on a random parameter that links the flux with an unknown primal variable (often through a constitutive law). In the primary example of the stationary diffusion equation, the parameter corresponds to the inverse of the diffusivity. The random parameter is modeled here as a Gevrey-regular random field. Specific focus is on random fields expressible as functions of countably infinite sequences of independent random variables, which may be uniformly or normally distributed. Quasi-Monte Carlo (QMC) error bounds for some quantity of interest that depends on the flux are then derived using the parametric regularity. It is shown that the QMC method converges optimally if the quantity of interest depends continuously on the primal variable, its flux, or its gradient. A series of assumptions are introduced with the goal of encompassing a broad class of discretizations by various finite element methods. The assumptions are verified for the diffusion equation discretized using conforming finite elements, mixed methods, and hybridizable discontinuous Galerkin schemes. Numerical experiments confirm the analytical findings, highlighting the role of accurate flux approximation in QMC methods.

Paper Structure

This paper contains 23 sections, 14 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The prominent example of the diffusion equation
  3. Formulations in terms of fluxes
  4. Parametrization of randomness
  5. Notation
  6. Main assumptions
  7. Structure of this document
  8. Quasi-Monte Carlo method
  9. Bounded domain $[-\tfrac{1}{2},\tfrac{1}{2}]^s$
  10. Unbounded domain $\mathbb R^s$
  11. Parametric regularity estimates
  12. The setting
  13. Gevrey regular model on a bounded parameter domain
  14. Unbounded and Gevrey regular model
  15. Finite element methods that fulfill \ref{['IT:norm']}--\ref{['IT:recur']}
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $\varphi$ be defined by eq:phiunif, $s$ be an arbitrary dimension, and let the weighted norm of $F$ in EQ:bounded_norm be bounded for some given weight collection $\gamma$. A randomly shifted QMC rank-1 lattice rule with $n=2^m$ in $s$ dimensions can be constructed by a CBC algorithm with $R$ in where the constant $C_{s,\gamma,\lambda}$ can be given using the Riemann zeta function $\zeta(x)=\s

Figures (1)

  • Figure 1: Root mean square error (ordinates) of the mean value of $u_h(\boldsymbol y)$ in blue, its gradient $-a(\boldsymbol y) \boldsymbol q_h(\boldsymbol y)$ (red), and its flux $\boldsymbol q_h(\boldsymbol y)$ (black). The abscissas illuminate the number of QMC quadrature points per random shift.

Theorems & Definitions (32)

  • Theorem 2.1
  • Theorem 2.2: cf. GrahamKNSSS15
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Lemma 3.4: see GuKa24
  • Lemma 3.5
  • Proof 1
  • Lemma 3.6: see kss12
  • Theorem 3.7
  • ...and 22 more