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IMA Journal of Numerical Analysis

Complexity of deterministic and randomized methods for multivariate integration problems for the class HpΛ(Id)

Fang Gensun, Peixin Ye

Published Feb 7, 2005• 6 citations
Publisher

Abstract

We introduce a new class of multivariate functions HpΛ(Id), which includes some classical sets of functions such as isotropic and anisotropic Sobolev and Hölder–Nikolskii as special cases or as a subclass in the sense of continuous imbedding. Then we prove that when p > 1 the ε-complexity of the integration problem in the randomized setting as well as in the average case setting is significantly smaller than that in the worst case deterministic setting.

DOI
10.1093/imanum/dri006

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We introduce a new class of multivariate functions HpΛ(Id), which includes some classical sets of functions such as isotropic and anisotropic Sobolev and Hölder–Nikolskii as special cases or as a subclass in the sense of continuous imbedding. Then we prove that when p > 1 the ε-complexity of the integration problem in the randomized setting as well as in the average case setting is significantly smaller than that in the worst case deterministic setting.
This paper reformulates the problems of approximation of imbedding and integration from anisotropic Sobolev classes B(Wrp([0, 1]d)) in the deterministic, randomized and average case settings, and obtains the exact orders of the n-th minimal error of these problems in all three settings. The results show that in the case of B(Wrp([0, 1]d)) being not imbedded into the space of continuous functions C([0, 1]d), the randomized and average case error are essential smaller than the deterministic ones. Quantitatively, this maximal gain for imbedding problem amounts to the factor n-1+ε for any ε > 0, and for integration problem the gain is up to factor n-1 which is also the maximal speedup over the deterministic algorithms observed so far in natural numerical problems.
We obtain the optimal order of high-dimensional integration complexity in the quantumcomputation model in anisotropic Sobolev classes W∞^r ([0, 1]^d) and Hǒlder Nikolskii classes H∞^r([0, 1]^d). It is proved that for these classes of functions there is a speed-up of quantum algorithms over deterministic classical algorithms due to factor n^-1 and over randomized classical methods due to factor n^-1/2. Moreover, we give an estimation for optimal query complexity in the class H∞^∧ (D) whose smoothness index is the boundary of some complete set in Z+^d.
We obtain the optimal order of high-dimensional integration complexity in the quantum computation model in anisotropic Sobolev classes W∞r([0,1]d)W_{\infty}^{\bf r}([0,1]^d)W∞r​([0,1]d) and Hölder Nikolskii classes H∞r([0,1]d)H_{\infty}^{\bf r}([0,1]^d)H∞r​([0,1]d). It is proved that for these classes of functions there is a speed-up of quantum algorithms over deterministic classical algorithms due to factor n−1n^{-1}n−1 and over randomized classical methods due to factor n−1/2n^{-1/2}n−1/2. Moreover, we give an estimation for optimal query complexity in the class H∞Λ(D)H_{\infty}^{\Lambda}(D)H∞Λ​(D) whose smoothness index is the boundary of some complete set in Z+d\mathbb{Z}_+^dZ+d​.
We present an algorithm for multivariate integration over cubes that is unbiased and has optimal order of convergence (in the randomized sense as well as in the worst case setting) for all Sobolev spaces Hr,mix([0,1]d)H^{r, mix}([0,1]^d)Hr,mix([0,1]d) and Hs([0,1]d)H^s([0,1]^d)Hs([0,1]d) for s>d/2s>d/2s>d/2.
We propose isotropic probability measures defined on classes of smooth multivariate functions. These provide a natural extension of the classical isotropic Wiener measure to multivariate functions from C 2r . We show that in the corresponding average case setting the minimal errors of algorithms that use n function values are Θ (n -(d+4r+1)/2d ) and Θ (n -(4r+1)/2d ) for the integration and L 2 -approximation problems respectively. Here d is the number of variables of the corresponding class of functions. This means that the minimal average errors depend essentially on the number d of variables. In particular, for d large relative to r, the L 2 -approximation problem is intractable. The integration and L 2 -approximation problems have been recently studied with measures whose covariance kernels are tensor products. The results for these measures and for isotropic measures differ significantly.