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On the optimal convergence rate of universal and nonuniversal algorithms for multivariate integration and approximation

Authors: Michael Griebel and Henryk Wozniakowski
Journal: Math. Comp. 75 (2006), 1259-1286
MSC (2000): Primary 65D30, 65D15, 41A45
Published electronically: May 1, 2006
MathSciNet review: 2219028
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Abstract: We study the maximal rate of convergence (mrc) of algorithms for (multivariate) integration and approximation of $ d$-variate functions from reproducing kernel Hilbert spaces $ H(K_{d})$. Here $ K_{d}$ is an arbitrary kernel all of whose partial derivatives up to order $ r$ satisfy a Hölder-type condition with exponent $ 2\beta$. Algorithms use $ n$ function values and we analyze their rate of convergence as $ n$ tends to infinity. We focus on universal algorithms which depend on $ d$, $ r$, and $ \beta$ but not on the specific kernel $ K_d$, and nonuniversal algorithms which may depend additionally on $ K_d$.

For universal algorithms the mrc is $ (r+\beta)/d$ for both integration and approximation, and for nonuniversal algorithms it is $ 1/2+ (r+\beta)/d$ for integration and $ a+(r+\beta)/d$ with $ a\in[1/(4+4(r+\beta)/d),1/2]$ for approximation. Hence, the mrc for universal algorithms suffers from the curse of dimensionality if $ d$ is large relative to $ r+\beta$, whereas the mrc for nonuniversal algorithms does not since it is always at least $ 1/2$ for integration, and $ 1/4$ for approximation. This is the price we have to pay for using universal algorithms. On the other hand, if $ r+\beta$ is large relative to $ d$, then the mrc for universal and nonuniversal algorithms is approximately the same.

We also consider the case when we have the additional knowledge that the kernel $ K_d$ has product structure, $ K_{d,r,\beta}=\bigotimes_{j=1}^dK_{r_j,\beta_j}$. Here $ K_{r_j,\beta_j}$ are some univariate kernels whose all derivatives up to order $ r_j$ satisfy a Hölder-type condition with exponent $ 2\beta_j$. Then the mrc for universal algorithms is $ q:=\min_{j=1,2,\dots,d }(r_j+\beta_j)$ for both integration and approximation, and for nonuniversal algorithms it is $ 1/2 +q$ for integration and $ a+q$ with $ a\in[1/(4+4q),1/2]$ for approximation. If $ r_j\ge1$ or $ \beta_j\ge\beta>0$ for all $ j$, then the mrc is at least $ \min(1,\beta)$, and the curse of dimensionality is not present. Hence, the product form of reproducing kernels breaks the curse of dimensionality even for universal algorithms.

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Additional Information

Michael Griebel
Affiliation: Institut für Numerische Simulation, Universität Bonn, Wegelerstrasse 6, D-53113 Bonn, Germany

Henryk Wozniakowski
Affiliation: Department of Computer Science, Columbia University, New York, NY 10027, USA; and Institute of Applied Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

Received by editor(s): July 25, 2005
Received by editor(s) in revised form: August 15, 2005
Published electronically: May 1, 2006
Additional Notes: The research of the first author was supported in part by the Sonderforschungsbereich 611 Singuläre Phänomene und Skalierung in Mathematischen Modellen sponsored by the Deutsche Forschungsgemeinschaft.
The research of the second author was supported in part by the National Science Foundation.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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