Construction of interlaced polynomial lattice rules for infinitely differentiable functions

Dick, Josef; Goda, Takashi; Suzuki, Kosuke; Yoshiki, Takehito

doi:10.1007/s00211-017-0882-x

Cited by 6 publications

(5 citation statements)

References 33 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%

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%

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

Goda¹,

Suzuki²

2020

Discrepancy Theory

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“…Specific strategies to periodize integrands have been discussed for numerical integration in [28]. Besides single and multiple rank-1 lattice rules [17,21], there are several other sampling strategies for periodic signals such as sparse grids [2,14,15], randomized least square sampling approaches [16,24] and also interlaced scrambled polynomial lattice rules [8,12]. However, we focus on sinlge rank-1 lattice based methods in this paper.…”

Section: Introductionmentioning

confidence: 99%

Efficient multivariate approximation on the cube

Nasdala

Potts

2021

Numer. Math.

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We combine a periodization strategy for weighted $$L_{2}$$ L 2 -integrands with efficient approximation methods in order to approximate multivariate non-periodic functions on the high-dimensional cube $$\left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}$$ - 1 2 , 1 2 d . Our concept allows to determine conditions on the d-variate torus-to-cube transformations $${\psi :\left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}\rightarrow \left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}}$$ ψ : - 1 2 , 1 2 d → - 1 2 , 1 2 d such that a non-periodic function is transformed into a smooth function in the Sobolev space $${\mathcal {H}}^{m}(\mathbb {T}^{d})$$ H m ( T d ) when applying $$\psi $$ ψ . We adapt $$L_{\infty }(\mathbb {T}^{d})$$ L ∞ ( T d ) - and $$L_{2}(\mathbb {T}^{d})$$ L 2 ( T d ) -approximation error estimates for single rank-1 lattice approximation methods and adjust algorithms for the fast evaluation and fast reconstruction of multivariate trigonometric polynomials on the torus in order to apply these methods to the non-periodic setting. We illustrate the theoretical findings by means of numerical tests in up to $$d=5$$ d = 5 dimensions.

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“…Under further conditions on the weights defining the space, a dimension free worst case error C exp(−c log(n) p ) holds for some 1 < p < 2. Dick et al (2017) give a construction of a superpolynomially convergent method. At a cost of O(nd log(n) 2 ) they use a componentby-component construction to get dimension-independent super-polynomial convergence using interlaced polynomial lattice rules.…”

Section: Introductionmentioning

confidence: 99%

“…Under scrambling, Dick (2011) shows that the root mean squared error (RMSE) is Õ(n −α−1/2 ). To obtain super-polynomial convergence Dick et al (2017) let the order of their higher-order digital nets increase with n.…”

Section: Introductionmentioning

confidence: 99%

Super-polynomial accuracy of one dimensional randomized nets using the median-of-means

Pan¹,

Owen²

2021

Preprint

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Let f be analytic on [0, 1] with |f (k) (1/2)| Aα k k! for some constant A and α < 2. We show that the median estimate of µ = 1 0 f (x) dx under random linear scrambling with n = 2 m points converges at the rate O(n −c log(n) ) for any c < 3 log(2)/π 2 ≈ 0.21. We also get a superpolynomial convergence rate for the sample median of 2k − 1 random linearly scrambled estimates, when k = Ω(m). When f has a p'th derivative that satisfies a λ-Hölder condition then the median-of-means has error O(n −(p+λ)+ ) for any > 0, if k → ∞ as m → ∞.

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Construction of interlaced polynomial lattice rules for infinitely differentiable functions

Cited by 6 publications

References 33 publications

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

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

Efficient multivariate approximation on the cube

Super-polynomial accuracy of one dimensional randomized nets using the median-of-means

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