QMC Rules of Arbitrary High Order: Reproducing Kernel Hilbert Space Approach

Baldeaux, Jan; Dick, Josef

doi:10.1007/s00365-009-9074-y

Cited by 26 publications

(49 citation statements)

References 25 publications

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“…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.…”

Section: Optimal Order L 2 -Discrepancy Boundsmentioning

confidence: 98%

“…Another major step was made in a series of papers [36,37,38], where the authors refined the integration error analysis for smooth functions due to Dick [7,8] and proved that order (2α + 1) digital nets and sequences achieve the best possible order of the worst-case error for a reproducing kernel Hilbert space with dominating mixed smoothness α, which is (log N ) (s−1)/2 /N α . Note that the original work by Dick [8] proves the worst-case error of order (log N ) sα /N α for order α digital nets and sequences, see also [3]. Other than higher order digital nets and sequences, only the Frolov lattice rule in conjunction with periodization of integrands has been proven to achieve the same, best possible order of the worst-case error so far [29,64,45].…”

Section: Introductionmentioning

confidence: 95%

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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: Optimal Order L 2 -Discrepancy Boundsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 95%

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

Goda¹,

Suzuki²

2020

Discrepancy Theory

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“…To be more precise, there it was shown (cf. [26]) that the worst case integration error of a digital (t, α, 1, αm × m, s)-net over F q satisfies…”

Section: Lemmamentioning

confidence: 98%

“…In [26], the implementation of digital (t, α, min(1, α/d), dm × m, s)-nets over F q was discussed and consequently digital (t, α, min(1, α/d), dm × m, s)-nets were used for numerical experiments.…”

Section: Definitionmentioning

confidence: 99%

On the approximation of smooth functions using generalized digital nets

Baldeaux

Dick

Kritzer

2009

Journal of Complexity

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a b s t r a c tIn this paper, we study an approximation algorithm which firstly approximates certain Walsh coefficients of the function under consideration and consequently uses a Walsh polynomial to approximate the function. A similar approach has previously been used for approximating periodic functions, using lattice rules (and Fourier polynomials), and for approximating functions in Walsh Korobov spaces, using digital nets. Here, the key ingredient is the use of generalized digital nets (which have recently been shown to achieve higher order convergence rates for the integration of smooth functions). This allows us to approximate functions with square integrable mixed partial derivatives of order α > 1 in each variable. The approximation error is studied in the worst case setting in the L 2 norm. We also discuss tractability of our proposed approximation algorithm, investigate its computational complexity, and present numerical examples.

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“…To simplify the situation, instead of an infinite sequence, let us consider a chain of α reals (I (1) n ) m−α+1≤n≤m with each given by…”

Section: Richardson Extrapolationmentioning

confidence: 99%

Richardson Extrapolation of Polynomial Lattice Rules

Dick¹,

Goda²,

Yoshiki³

2019

SIAM J. Numer. Anal.

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We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness α ≥ 2 defined over the s-dimensional unit cube. We propose a new quasi-Monte Carlo (QMC)-based quadrature rule, named extrapolated polynomial lattice rule, which achieves the almost optimal rate of convergence. Extrapolated polynomial lattice rules are constructed in two steps: i) construction of classical polynomial lattice rules over F b with α consecutive sizes of nodes, b m−α+1 , . . . , b m , and ii) recursive application of Richardson extrapolation to a chain of α approximate values of the integral obtained by consecutive polynomial lattice rules.We prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence of the worst-case error in Sobolev spaces with general weights. Then, by restricting to product weights, we show that such good extrapolated polynomial lattice rules can be constructed by the fast component-by-component algorithm under a computable quality criterion. The required total construction cost is of order (s + α)N log N , which improves the currently known result for interlaced polynomial lattice rule, that is of order sαN log N . We also study the dependence of the worst-case error bound on the dimension.A big advantage of our method compared to interlaced polynomial lattice rules is that the fast QMC matrix vector method can be used in this setting, while still achieving the same rate of convergence. Such a method was previously not known.Numerical experiments for test integrands support our theoretical result.

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QMC Rules of Arbitrary High Order: Reproducing Kernel Hilbert Space Approach

Cited by 26 publications

References 25 publications

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

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

On the approximation of smooth functions using generalized digital nets

Richardson Extrapolation of Polynomial Lattice Rules

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