Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration

Yoshiki, Takehito

doi:10.32917/hmj/1499392824

Cited by 20 publications

(31 citation statements)

References 25 publications

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“…One straightforward idea is to construct infinite order digital nets and sequences and then study their propagation rule. In this line of research, we refer to [62,67] for the Walsh analysis of infinitely many times differentiable functions, and furthermore, to [44,59,60,61,13,43] for the relevant literature.…”

Section: Discussionmentioning

confidence: 99%

“…The first assertion on the decay of the Walsh coefficients was shown in [3], while the second assertion of the sparsity of the Walsh coefficients was shown in [38]. Regarding the first assertion, we also refer to more recent works [62,67] which introduce different approaches from the one by Dick [7,8,9] for evaluating the Walsh coefficients. In many cases, one may obtain smaller constants C α,b .…”

Section: Optimal Order L 2 -Discrepancy Boundsmentioning

confidence: 97%

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4. Recent advances in higher order quasi-Monte Carlo methods

Goda¹,

Suzuki²

2020

Discrepancy Theory

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In this article we review some of recent results on higher order quasi-Monte Carlo (HoQMC) methods. After a seminal work by Dick (2007Dick ( , 2008 who originally introduced the concept of HoQMC, there have been significant theoretical progresses on HoQMC in terms of discrepancy as well as multivariate numerical integration. Moreover, several successful and promising applications of HoQMC to partial differential equations with random coefficients and Bayesian estimation/inversion problems have been reported recently. In this article we start with standard quasi-Monte Carlo methods based on digital nets and sequences in the sense of Niederreiter, and then move onto their higher order version due to Dick. The Walsh analysis of smooth functions plays a crucial role in developing the theory of HoQMC, and the aim of this article is to provide a unified picture on how the Walsh analysis enables recent developments of HoQMC both for discrepancy and numerical integration. P ⊂ [0, 1] s be a finite multiset, that is, if an element occurs multiple times, it is counted according to its multiplicity. Then I(f ) is approximated bywhere |P | denotes the cardinality of P . It is obvious that this algorithm is exact for any choice of P if f is a constant function, but except for such a trivial case, a careful design of P and the accompanying theoretical analysis are required to show that the algorithm works well for various functions f . One fairly easy but sensible approach is to choose each point x independently and uniformly from [0, 1] s . This is widely known under the name of Monte Carlo methods [39]. Many fundamental results in probability theory, including the law of large numbers and the central limit theorem, apply to this approach. Looking at I(f ; P ) as a random variable (with P being the underlying stochastic variable), we have E[I(f ; P )] = I(f ) and V[I(f ; P )] = V[f ] |P | ,

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Section: Discussionmentioning

confidence: 99%

Section: Optimal Order L 2 -Discrepancy Boundsmentioning

confidence: 97%

4. Recent advances in higher order quasi-Monte Carlo methods

Goda¹,

Suzuki²

2020

Discrepancy Theory

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“…Therefore, for simplicity, here we take q = r = ∞ and denote the corresponding norm by The following theorem is adjusted from [23, Theorem 3.10] for b = 2. We made use of a new result in [107] that a constant C α,b which usually appears is exactly 1 when b = 2. Furthermore, we followed the proof of [84, Theorem 5.1] to obtain the factor 2/n instead of 2/(n − 1).…”

Section: Weighted Space Of Smooth Functions Over [0 1] S and Interlamentioning

confidence: 99%

Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

2016

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This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers, and contrasts, the uniform case versus the lognormal case, singlelevel algorithms versus multi-level algorithms, first order QMC rules versus higher order QMC rules, and deterministic QMC methods versus randomized QMC methods. It gives a summary of the error analysis and proof techniques in a unified view, and provides a practical guide to the software for constructing and generating QMC points tailored to the PDE problems. The analysis for the uniform case can be generalized to cover a range of affine parametric operator equations.

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“…(computations with C = 1 yielded different generating vectors, and led to slight artefacts on high levels L ≥ 7 in this example). For the presently used base b = 2, the choice C = 1.0 holds [29].…”

Section: Affine-parametric Linear Test Problemmentioning

confidence: 99%

Multilevel higher-order quasi-Monte Carlo Bayesian estimation

Dick

Gantner

Gia

et al. 2017

Math. Models Methods Appl. Sci.

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We propose and analyze deterministic multilevel approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive gaussian measurement data. The algorithms use a multilevel (ML) approach based on deterministic, higher order quasi-Monte Carlo (HoQMC) quadrature for approximating the high-dimensional expectations, which arise in the Bayesian estimators, and a Petrov-Galerkin (PG) method for approximating the solution to the underlying partial differential equation (PDE). This extends the previous single-level approach from [J. Dick, R. N. Gantner, Q. T. Le Gia and Ch. Schwab, Higher order Quasi-Monte Carlo integration for Bayesian Estimation. Report 2016-13, Seminar for Applied Mathematics, ETH Zürich (in review)].Compared to the single-level approach, the present convergence analysis of the multilevel method requires stronger assumptions on holomorphy and regularity of the countably-parametric uncertaintyto-observation maps of the forward problem. As in the single-level case and in the affine-parametric case analyzed in [ J. Dick, F.Y. Kuo, Q. T. Le Gia and Ch. Schwab, Multi-level higher order QMC Galerkin discretization for affine parametric operator equations. Accepted for publication in SIAM J. Numer. Anal., 2016], we obtain sufficient conditions which allow us to achieve arbitrarily high, algebraic convergence rates in terms of work, which are independent of the dimension of the parameter space. The convergence rates are limited only by the spatial regularity of the forward problem, the discretization order achieved by the Petrov Galerkin discretization, and by the sparsity of the uncertainty parametrization.We provide detailed numerical experiments for linear elliptic problems in two space dimensions, with s = 1024 parameters characterizing the uncertain input, confirming the theory and showing that the ML HoQMC algorithms outperform, in terms of error vs. computational work, both multilevel Monte Carlo (MLMC) methods and single-level (SL) HoQMC methods.

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Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration

Cited by 20 publications

References 25 publications

4. Recent advances in higher order quasi-Monte Carlo methods

4. Recent advances in higher order quasi-Monte Carlo methods

Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

Multilevel higher-order quasi-Monte Carlo Bayesian estimation

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