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

Kuo, Frances Y.; Nuyens, Dirk

doi:10.1007/978-3-319-91436-7_3

Cited by 7 publications

(8 citation statements)

References 44 publications

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“…where p0 ∈ (0, 1) should be as small as possible while satisfying j≥1 ψj p 0 L∞ < ∞. This part is presented as a step-by-step tutorial in the article [51] from this volume.…”

Section: The Uniform Casementioning

confidence: 99%

“…Also the O(h 2 ) term can be improved by using higher order finite elements. See [50,51] for more details.…”

Section: The Uniform Casementioning

confidence: 99%

“…One crucial step in the analysis of [32] is to choose suitable weight functions j so that the function f has a finite and indeed small norm (8), so that the theoretical setting of Subsection 2.2 can be applied. Again see [50,51] for more details.…”

Section: The Lognormal Case With Karhunen-loève Expansionmentioning

confidence: 99%

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Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

Kuo

Nuyens

2018

Springer Proceedings in Mathematics &Amp; Statistics

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This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube [0, 1] s and in R s , and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension s under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when s is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications. * Frances Y. Kuo ( ):

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“…where p0 ∈ (0, 1) should be as small as possible while satisfying j≥1 ψj p 0 L∞ < ∞. This part is presented as a step-by-step tutorial in the article [51] from this volume.…”

Section: The Uniform Casementioning

confidence: 99%

“…Also the O(h 2 ) term can be improved by using higher order finite elements. See [50,51] for more details.…”

Section: The Uniform Casementioning

confidence: 99%

See 1 more Smart Citation

Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

Kuo

Nuyens

2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…In other words, reasonably estimating these weights is a particular challenge, which might necessitate considerable additional effort. As an example and for additional information, we refer to [23] as a short and general introduction to Quasi-Monte Carlo methods, which is one of the most used approaches as seen above.…”

Section: Introductionmentioning

confidence: 99%

The uniform sparse FFT with application to PDEs with random coefficients

Kämmerer¹,

Potts²,

Taubert³

2021

Preprint

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We develop an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The sparse Fast Fourier Transform (sFFT) detects the most important frequencies in a given search domain and therefore adaptively generates a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function. Our uniform sFFT (usFFT) does this w.r.t. the stochastic domain simultaneously for every node of a finite element mesh in the spatial domain and creates a suitable approximation space for all spatial nodes by joining the detected frequency sets. This strategy allows for a faster and more efficient computation, than just using other algorithms, like for example the sFFT, for each node separately. We then test the usFFT for different examples using periodic, affine and lognormal random coefficients in the PDE problems. The results are significantly better than when using given standard frequency sets and the algorithm does not require any a priori information about the solution.

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“…In this paper we propose the application of a quasi-Monte Carlo method to approximate the expected values with respect to the uncertainty. Quasi-Monte Carlo methods have been shown to perform remarkably well in the application to PDEs with random coefficients [8,9,12,14,21,22,23,24,25,26]. The reason behind their success is that it is possible to design quasi-Monte Carlo rules with error bounds not dependent on the number of uncertain variables, which achieve faster convergence rates compared to Monte Carlo methods in case of smooth integrands.…”

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confidence: 99%

A quasi-Monte Carlo Method for an Optimal Control Problem Under Uncertainty

A.¹,

Kaarnioja²,

Kuo³

et al. 2019

Preprint

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We study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a control function. The robust formulation of the optimization problem is stated as a high-dimensional integration problem over the stochastic variables. It is well known that carrying out a high-dimensional numerical integration of this kind using a Monte Carlo method has a notoriously slow convergence rate; meanwhile, a faster rate of convergence can potentially be obtained by using sparse grid quadratures, but these lead to discretized systems that are non-convex due to the involvement of negative quadrature weights. In this paper, we analyze instead the application of a quasi-Monte Carlo method, which retains the desirable convexity structure of the system and has a faster convergence rate compared to ordinary Monte Carlo methods. In particular, we show that under moderate assumptions on the decay of the input random field, the error rate obtained by using a specially designed, randomly shifted rank-1 lattice quadrature rule is essentially inversely proportional to the number of quadrature nodes. The overall discretization error of the problem, consisting of the dimension truncation error, finite element discretization error and quasi-Monte Carlo quadrature error, is derived in detail. We assess the theoretical findings in numerical experiments.

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

Cited by 7 publications

References 44 publications

Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

The uniform sparse FFT with application to PDEs with random coefficients

A quasi-Monte Carlo Method for an Optimal Control Problem Under Uncertainty

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