A Multi-Index Quasi--Monte Carlo Algorithm for Lognormal Diffusion Problems

Robbe, Pieterjan; Nuyens, Dirk; Vandewalle, Stefan

doi:10.1137/16m1082561

Cited by 13 publications

(13 citation statements)

References 31 publications

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“…The discussion pertaining to MLMC is applicable to MLQMC, with the exception of the part related to the computation of the number of samples. Unlike MLMC, the number of samples for MLQMC is not based on a formula as in Equation (10). In order to satisfy the statistical constraint, i.e., Equation (9), an adaptive algorithm is used, see [9].…”

Section: E[q Mlmcmentioning

confidence: 99%

“…In this work, we use a shifted rank-1 lattice rule in order to generate the MLQMC sample points x (r,n) , as is commonly done in other work regarding MLQMC, see for example [9,10,22]. Rank-1 lattice rules belong to one of two major classes of QMC points, i.e.…”

Section: E[q Mlmcmentioning

confidence: 99%

“…For example, an alternative non-random selection of sampling points can be used, as in Quasi-Monte Carlo [5,6] and Latin Hypercube [7] sampling methods. Other techniques, such as Multilevel Monte Carlo (MLMC) [8], Multilevel Quasi-Monte Carlo (MLQMC) [9] and its generalizations, see, e.g., [10,11], can speed up the method. These improved Monte Carlo methods are based on a hierarchy of meshes with increasing resolution, where samples on coarser meshes are computationally less expensive than samples on finer meshes.…”

Section: Introductionmentioning

confidence: 99%

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p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

et al. 2020

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Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.

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Section: E[q Mlmcmentioning

confidence: 99%

Section: E[q Mlmcmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

et al. 2020

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“…A different QMC theory for the lognormal case is offered in [26]. Further PDE computations with higher order QMC are reported in [15], and with multi-level and multi-index QMC in [40]. QMC has also been applied to PDEs on the sphere [35], holomorphic equations [8], Bayesian inversion [3,41], stochastic wave propagation [12,13], and eigenproblems [16].…”

Section: Beyond the Surveymentioning

confidence: 99%

Application of Quasi-Monte Carlo Methods to PDEs with Random Coefficients – An Overview and Tutorial

Kuo

Nuyens

2018

Springer Proceedings in Mathematics &Amp; Statistics

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This article provides a high-level overview of some recent works on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. It is based on an indepth survey of a similar title by the same authors, with an accompanying software package which is also briefly discussed here. Embedded in this article is a step-by-step tutorial of the required analysis for the setting known as the uniform case with first order QMC rules. The aim of this article is to provide an easy entry point for QMC experts wanting to start research in this direction and for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods. * Frances Y. Kuo ( ):

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“…For example, alternative non-random selections of sampling points can be used, as in Quasi-Monte Carlo [5,6] and Latin Hypercube [7] sampling methods. Also, variance reduction techniques, such as Multilevel Monte Carlo (MLMC) [8], Multilevel Quasi-Monte Carlo (MLQMC) [9] and its generalizations, see, e.g., [10,11], can speed up the method. These improved Monte Carlo methods are based on a hierarchy of increasing resolution meshes where samples on coarser meshes are computationally less expensive than on finer meshes.…”

Section: Introductionmentioning

confidence: 99%

h- and p-refined Multilevel Monte Carlo Methods for Uncertainty Quantification in Structural Engineering

Blondeel,

Robbe,

van hoorickx

et al. 2019

Preprint

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Practical structural engineering problems are often characterized by significant uncertainties. Historically, one of the prevalent methods to account for this uncertainty has been the standard Monte Carlo (MC) method. Recently, improved sampling methods have been proposed, based on the idea of variance reduction by employing a hierarchy of mesh refinements. We combine an h-and p-refinement hierarchy with the Multilevel Monte Carlo (MLMC) and Multilevel Quasi-Monte Carlo (MLQMC) method. We investigate the applicability of these novel combination methods on three structural engineering problems, for which the uncertainty resides in the Young's modulus: the static response of a cantilever beam with elastic material behavior, its static response with elastoplastic behavior, and its dynamic response with elastic behavior.The uncertainty is either modeled by means of one random variable sampled from a univariate Gamma distribution or with multiple random variables sampled from a gamma random field. This random field results from a truncated Karhunen-Loève (KL) expansion. In this paper, we compare the computational costs of these Monte Carlo methods. We demonstrate that MQLMC and MLMC have a significant speedup with respect to MC, regardless of the mesh refinement hierarchy used. We empirically demonstrate that the MLQMC cost is optimally proportional to −1 under certain conditions, where is the tolerance on the root-mean-square error (RMSE). In addition, we show that, when the uncertainty is modeled as a random field, the multilevel methods combined with p-refinement have a significant lower computation cost than their counterparts based on h-refinement. We also illustrate the effect the uncertainty models have on the uncertainty bounds in the solutions. An uncertain Young's modulus modeled as a single random variable has much larger uncertainty bounds on its solution than an uncertain Young's modulus modeled as a random field.

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A Multi-Index Quasi--Monte Carlo Algorithm for Lognormal Diffusion Problems

Cited by 13 publications

References 31 publications

p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

Application of Quasi-Monte Carlo Methods to PDEs with Random Coefficients – An Overview and Tutorial

h- and p-refined Multilevel Monte Carlo Methods for Uncertainty Quantification in Structural Engineering

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