Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

Herrmann, Lukas; Schwab, Christoph

doi:10.1051/m2an/2019016

Cited by 25 publications

(31 citation statements)

References 45 publications

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“…Some recent developments in this field can e.g. be found in [32,33,36,38] for the model problem with a lognormal coefficient. Some ideas on a posteriori adaptivity for Monte Carlo methods can e.g.…”

Section: Introductionmentioning

confidence: 99%

Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

et al. 2020

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Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residual-based a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm.

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Section: Introductionmentioning

confidence: 99%

Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

et al. 2020

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“…We note that to handle singularities, due to, e.g., reentrant corners for d = 2, one either has to change the norms to incorporate a weight function as in [17, Proposition 2.3 and Remark 2.4] or use local mesh refinement as, e.g., in [15]. We refer to [17] for the definitions of the norms and omit such details here.…”

Section: Finite Element Approximation Errormentioning

confidence: 99%

“…Section 6 presents the main contribution of this paper where the cost model, the construction and the complexity of the MDFEM algorithm are presented. Finally, a comparison with two existing methods, the QMCFEM [18] and MLQMCFEM [17], shows the benefit of the MDFEM.…”

Section: Introductionmentioning

confidence: 98%

“…The goal is to compute the expected value of some functional of the solution. This method was motivated by the need for new techniques for elliptic PDEs with smooth lognormal diffusion coefficients where the classical QMC approaches can only achieve first-order convergence, see, e.g., [14,15,17,18].…”

Section: Introductionmentioning

confidence: 99%

“…Interlaced polynomial lattice rules have been used before in the PDE context to achieve higher-order convergence but with uniform diffusion in [8]. In contrast, most existing QMC methods for integration with respect to the normal density, and hence most existing QMC based methods for estimating expected values in the PDE context with lognormal diffusion, map the integral to the unit cube by using the inverse of the Gaussian cumulative distribution function and then use randomly shifted lattice rules to approximate this integral, see, e.g., [24,29], and [14,15,17,18,22]. The aim of all these methods is to obtain dimension-independent convergence by making use of weighted function spaces.…”

Section: Introductionmentioning

confidence: 99%

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MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

Nguyen

Nuyens

2021

ESAIM: M2AN

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We introduce the \emph{multivariate decomposition finite element method} (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\bsy) = \sum_{j \ge 1} y_j \, \phi_j$ with $y_j \sim \calN(0,1)$ and a given sequence of functions $\{\phi_j\}_{j \ge 1}$. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the \emph{multivariate decomposition method} (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using \emph{quasi-Monte Carlo} (QMC) methods, and for which we use the \emph{finite element method} (FEM) to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-di\-men\-si\-onal Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of \emph{anchored Gaussian Sobolev spaces} while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon)$ the computational cost is $O(\epsilon^{-1/\lambda-\dd/\lambda}) = O(\epsilon^{-(p^* + \dd/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-\dd/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $\dd = d \, (1+\ddelta)$ for some $\ddelta \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2 + \dd/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j \ge 1}$.

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Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Zhang

2020

WIREs Computational Stats

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Uncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling‐based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time‐consuming if the simulation‐based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and corresponding high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different setting of MLMC and MFMC, aims to address the issue of UQ and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small data sets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. An algorithm summary and the corresponding code implementation are provided for each of the modern MC methods. The extension and application of these methods are discussed in detail. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Monte Carlo Methods Statistical and Graphical Methods of Data Analysis > Sampling

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Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

Cited by 25 publications

References 45 publications

Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

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