On decompositions of multivariate functions

Kuo, Frances Y.; Sloan, Ian H.; Wasilkowski, Grzegorz W.; Woźniakowski, Henryk

doi:10.1090/s0025-5718-09-02319-9

Cited by 135 publications

(168 citation statements)

References 27 publications

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“…Several variants of weighted Sobolev spaces have been studied in recent literature, see e.g., [5,6,8,12,21,22,24,25]. Here we consider the unanchored weighted Sobolev space H sob d,r introduced in [5,12], which consists of continuous functions with square-integrable mixed partial derivatives of order r, with the norm in one dimension defined by…”

Section: Weighted Sobolev Spacesmentioning

confidence: 99%

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Periodization strategy may fail in high dimensions

2007

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We discuss periodization of smooth functions f of d variables for approximation of multivariate integrals. The benefit of periodization is that we may use lattice rules, which have recently seen significant progress. In particular, we know how to construct effectively a generator of the rank-1 lattice rule with n points whose worst case error enjoys a nearly optimal bound C d, p n − p . Here C d, p is independent of d or depends at most polynomially on d, and p can be arbitrarily close to the smoothness of functions belonging to a weighted Sobolev space with an appropriate condition on the weights. If F denotes the periodization for f then the error of the lattice rule for a periodized function F is bounded by C d, p n − p F with the norm of F given in the same Sobolev space. For small or moderate d, the norm of F is not much larger than the norm of f . This means that for small or moderate d, periodization is successful and allows us to use optimal properties of lattice rules also for non-periodic functions. The situation is quite different if d is large since the norm of F can be exponentially larger than the norm of f . This can already be seen for f = 1. Hence, the upper bound of the worst case error of 370 Numer Algor (2007) 46:369-391 the lattice rule for periodized functions is quite bad for large d. We conjecture not only that this upper bound is bad, but also that all lattice rules fail for large d. That is, if we fix the number of points n and let d go to infinity then the worst case error of any lattice rule is bounded from below by a positive constant independent of n. We present a number of cases suggesting that this conjecture is indeed true, but the most interesting case, when the sum of the weights of the corresponding Sobolev space is bounded in d, remains open.

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Section: Weighted Sobolev Spacesmentioning

confidence: 99%

“…With α = r, it is clear from comparing just the one-dimensional norms (10) and (12) [12]. This equality in norms allows us to make a sensible comparison between the norms of a non-periodic function f from the weighed Sobolev space and its corresponding periodized function F in the weighted Korobov space.…”

Section: Weighted Sobolev Spacesmentioning

confidence: 99%

Periodization strategy may fail in high dimensions

2007

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“…(this is true because the ANOVA decomposition is known to retain a minimal representation [42]). Therefore the problem has intrinsic low-effective dimension in the superposition sense, and it is expected that (randomized) QMC outperforms MC in this case.…”

Section: Harmonic Oscillatormentioning

confidence: 99%

Quasi-Monte Carlo methods for lattice systems: A first look

Jansen

Leövey

Ammon

et al. 2014

Computer Physics Communications

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a b s t r a c tWe investigate the applicability of quasi-Monte Carlo methods to Euclidean lattice systems for quantum mechanics in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an observable calculated by averaging over random observations generated from an ordinary Markov chain Monte Carlo simulation behaves like N −1/2 , where N is the number of observations. By means of quasi-Monte Carlo methods it is possible to improve this behavior for certain problems to N −1 , or even further if the problems are regular enough. We adapted and applied this approach to simple systems like the quantum harmonic and anharmonic oscillator and verified an improved error scaling. Programming language: C and C++. Program summary Computer: PC.Operating system: Tested on GNU/Linux, should be portable to other operating systems with minimal efforts. Has the code been vectorized or parallelized?: No RAM:The memory usage directly scales with the number of samples and dimensions: Bytes used = ''number of samples'' × ''number of dimensions'' × 8 Bytes (double precision).Classification: 4.13, 11.5, 23.External routines: FFTW 3 library (http://www.fftw.org) Nature of problem:Certain physical models formulated as a quantum field theory through the Feynman path integral, such as quantum chromodynamics, require a non-perturbative treatment of the path integral. The only known approach that achieves this is the lattice regularization. In this formulation the path integral is discretized to a finite, but very high dimensional integral. So far only Monte Carlo, and especially Markov chain-Monte Carlo methods like the Metropolis or the hybrid Monte Carlo algorithm have been used to calculate approximate solutions of the path integral. These algorithms often lead to the undesired effect of autocorrelation in the samples of observables and suffer in any case from the slow asymptotic error behavior proportional to N −1/2 , if N is the number of samples. Solution method:This program applies the quasi-Monte Carlo approach and the reweighting technique (respectively the weighted uniform sampling method) to generate uncorrelated samples of observables of the anharmonic oscillator with an improved asymptotic error behavior. Unusual features:The application of the quasi-Monte Carlo approach is quite revolutionary in the field of lattice field theories. Running time:The running time depends directly on the number of samples N and dimensions d. On modern computers a run with up to N = 2 16 = 65536 (including 9 replica runs) and d = 100 should not take much longer than one minute.

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“…Although we do not know f u , their samples can be obtained from 2 |u| values of the function f , see [12] Then the cost of such sampling of f u is bounded by 2 |u| · $(|u|) ≤ 2 d(ε) · $(d(ε)), where d(ε) is as in (19). Hence it is relatively small even if $ is an exponential function.…”

Section: Remarkmentioning

confidence: 99%