Parametric Regularity Analysis and Quasi-Monte Carlo Cubature for Discontinuous Galerkin Approximations of Elliptic PDEs with Random Coefficients

Core Concepts

The core message of this paper is to develop a tailored quasi-Monte Carlo (QMC) cubature framework for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations with random coefficients. The authors prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen, and provide a detailed parametric regularity analysis for DG solutions.

Abstract

This paper considers the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. The authors investigate both the affine and uniform and the lognormal models for the input random field, and focus on approximating the expected value of the PDE response subject to input uncertainty.
The key highlights and insights are:

The authors prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. This is an important result, as DG methods are non-conforming and the existing QMC theory cannot be directly applied.

The parametric regularity bounds for DG solutions, which are developed in this work, are also useful for other methods such as sparse grids.

Numerical results are provided that confirm the theoretical findings.

The authors first introduce the necessary notations and preliminaries, including the two models for the random diffusion coefficient - the affine and uniform model, and the lognormal model. They then provide an overview of quasi-Monte Carlo cubature and the analysis for conforming finite element methods.
The core of the paper focuses on the DG framework. The authors derive the DG variational formulation and discuss the necessary stability estimates. They then prove the key parametric regularity result for the DG solution, showing that the same type of bounds as for conforming finite elements can be obtained. This allows them to develop the tailored QMC cubature theory for DG approximations.
Finally, numerical experiments are presented that validate the theoretical findings and demonstrate the effectiveness of the proposed QMC approach for DG methods.

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Key Insights Distilled From

Quasi-Monte Carlo and discontinuous Galerkin

by Vesa Kaarnio... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2207.07698.pdf

Deeper Inquiries

How can the proposed QMC framework be extended to other types of non-conforming discretization schemes beyond the interior penalty DG method considered in this work

To extend the proposed Quasi-Monte Carlo (QMC) framework to other types of non-conforming discretization schemes beyond the interior penalty DG method, one would need to adapt the regularity analysis and error estimation techniques to suit the specific characteristics of the new discretization methods.
For instance, methods like Hybrid High-Order (HHO) or Hybridizable Discontinuous Galerkin (HDG) could be considered. These methods involve additional degrees of freedom at the element interfaces, which may require a different treatment in the QMC analysis compared to the IPDG method.
The key steps in extending the QMC framework would involve:

Defining the Function Spaces: Characterizing the function spaces used in the new discretization scheme and understanding their properties with respect to the random coefficients.
Deriving Regularity Bounds: Developing parametric regularity bounds for the new method, considering the impact of the non-conforming nature of the discretization on the QMC convergence rates.
Error Estimation: Adapting the error estimation techniques to account for the specific features of the new method, such as the presence of additional degrees of freedom or different stabilization parameters.

By carefully adapting the QMC framework to suit the characteristics of the new non-conforming discretization schemes, it is possible to extend the applicability of QMC methods to a broader range of PDE problems.

What are the potential challenges in applying the QMC cubature approach to more complex PDE problems, such as those involving nonlinear or time-dependent random coefficients

Applying the QMC cubature approach to more complex PDE problems involving nonlinear or time-dependent random coefficients presents several challenges that need to be addressed:

Increased Dimensionality: Nonlinear or time-dependent coefficients can lead to higher-dimensional parameter spaces, which can significantly impact the convergence rates of QMC methods. Special attention needs to be paid to the choice of QMC rules and the design of the cubature points to ensure efficient convergence.

Nonlinearity: Nonlinear coefficients can introduce additional complexities in the parametric regularity analysis, requiring more sophisticated techniques to estimate the regularity of the solution with respect to the random parameters.

Temporal Dependencies: Time-dependent coefficients introduce a temporal dimension to the problem, necessitating the development of QMC strategies that can handle time discretization alongside the parametric uncertainty.

Adaptive Strategies: Adapting QMC methods to handle the dynamic nature of time-dependent coefficients or the nonlinearity of the problem requires the development of adaptive strategies that can adjust the sampling and cubature points accordingly.

Addressing these challenges would involve a combination of advanced mathematical analysis, algorithmic developments, and computational strategies to effectively apply QMC cubature to complex PDE problems with nonlinear or time-dependent random coefficients.

Can the parametric regularity analysis techniques developed in this paper be leveraged to improve the efficiency of other uncertainty quantification methods, such as sparse grid quadratures or multilevel Monte Carlo

The parametric regularity analysis techniques developed in this paper can indeed be leveraged to enhance the efficiency of other uncertainty quantification methods, such as sparse grid quadratures or multilevel Monte Carlo (MLMC).

Sparse Grid Quadratures: By incorporating the parametric regularity bounds derived for the DG method, one can optimize the selection of sparse grid points and levels based on the smoothness of the solution with respect to the random parameters. This can lead to more efficient sparse grid quadratures with improved convergence rates.

Multilevel Monte Carlo (MLMC): The regularity analysis can guide the design of the MLMC hierarchy, helping to determine the optimal level of refinement at each level based on the parametric regularity of the solution. This can result in more effective variance reduction and improved computational efficiency in estimating the expected value of the PDE response subject to input uncertainty.

By integrating the insights from the parametric regularity analysis into these methods, it is possible to enhance their performance and applicability to a wider range of PDE problems with random coefficients.