The power of standard information for multivariate approximation in the randomized setting

Authors:
G. W. Wasilkowski and H. Wozniakowski

Journal:
Math. Comp. **76** (2007), 965-988

MSC (2000):
Primary 41A63, 65C05, 65D15, 65Y20

DOI:
https://doi.org/10.1090/S0025-5718-06-01944-2

Published electronically:
December 13, 2006

MathSciNet review:
2291845

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study approximating multivariate functions from a reproducing kernel Hilbert space with the error between the function and its approximation measured in a weighted -norm. We consider functions with an arbitrarily large number of variables, , and we focus on the *randomized setting* with algorithms using *standard* information consisting of function values at randomly chosen points.

We prove that standard information in the randomized setting is *as powerful* as *linear* information in the worst case setting. Linear information means that algorithms may use arbitrary continuous linear functionals, and by the power of information we mean the speed of convergence of the th minimal errors, i.e., of the minimal errors among all algorithms using function evaluations. Previously, it was only known that standard information in the randomized setting is *no more powerful* than the linear information in the worst case setting.

We also study (*strong*) *tractability* of multivariate approximation in the randomized setting. That is, we study when the minimal number of function evaluations needed to reduce the initial error by a factor is polynomial in (strong tractability), and polynomial in and (tractability). We prove that these notions in the randomized setting for standard information are equivalent to the same notions in the worst case setting for linear information. This result is useful since for a number of important applications only standard information can be used and verifying (strong) tractability for standard information is in general difficult, whereas (strong) tractability in the worst case setting for linear information is known for many spaces and is relatively easy to check.

We illustrate the tractability results for weighted Korobov spaces. In particular, we present necessary and sufficient conditions for strong tractability and tractability. For product weights independent of , we prove that strong tractability is equivalent to tractability.

We stress that all proofs are constructive. That is, we provide randomized algorithms that enjoy the maximal speed of convergence. We also exhibit randomized algorithms which achieve strong tractability and tractability error bounds.

**1.**N. Aronszajn, Theory of reproducing kernels,*Trans. AMS***68**(1950), 337-404. MR**0051437 (14:479c)****2.**N. S. Bakhvalov, On approximate calculation of integrals,*Vestnik MGV, Ser. Mat. Mekh. Astron. Fiz. Him.***4**(1959), 3-18. MR**0115275 (22:6077)****3.**J. Dick, I. H. Sloan, X. Wang and H. Wozniakowski, Good lattice rules in weighted Korobov spaces with general weights, to appear in*Numer. Math.***4.**B. L. Granovskii and S. M. Ermakov, The Monte Carlo method,*J. Soviet Math.***7**(1977), 161-192.**5.**M. Griebel and H. Wozniakowski, On the optimal convergence rate of universal and non-universal algorithms for multivariate integration and approximation, to appear in*Math. Computation*.**6.**S. Heinrich, Lower bounds for the complexity of Monte Carlo function approximation,*J. Complexity***8**(1992), 277-300. MR**1187419 (93h:65007)****7.**S. Heinrich, Monte Carlo complexity of global solution of integral equations,*J. Complexity***14**(1998), 151-175. MR**1629093 (99g:65135)****8.**F. J. Hickernell and H. Wozniakowski, Integration and approximation in arbitrary dimensions,*Advances in Comput. Math.***12**(2000), 25-58. MR**1758946 (2001d:65017)****9.**F. Kuo, I. H. Sloan and H. Wozniakowski, Lattice rules for multivariate approximation in the worst case setting, in*Monte Carlo and Quasi-Monte Carlo Methods 2004*, eds. H. Niederreiter and D. Talay, Springer-Verlag, Berlin, pp. 289-330, 2006. MR**2208715 (2006j:41044)****10.**F. Kuo, I. H. Sloan and H. Wozniakowski, Lattice rules for multivariate approximation in the average case setting, in progress.**11.**P. Mathé, Random approximation of Sobolev embeddings,*J. Complexity***7**(1991), 261-281. MR**1128021 (93d:65008)****12.**A. S. Nemirovsky and D. B. Yudin,*Problem Complexity and Method Efficiency in Optimization*, Wiley, New York, 1983. MR**0702836 (84g:90079)****13.**E. Novak,*Deterministic and Stochastic Error Bounds in Numerical Analysis,*Lect. Notes in Math.**1349**, Springer-Verlag, Berlin, 1988. MR**0971255 (90a:65004)****14.**E. Novak, Stochastic properties of quadrature formulas,*Numer. Math.***53**(1988), 609-620. MR**0954773 (89i:65026)****15.**E. Novak, Optimal linear randomized methods for linear operators in Hilbert spaces,*J. Complexity***8**(1992), 22-36. MR**1153612 (93f:65048)****16.**E. Novak, I. H. Sloan and H. Wozniakowski, Tractability of approximation for weighted Korobov spaces on classical and quantum computers,*Found. Comput. Math.***4**(2004), 121-156. MR**2049868 (2005h:81097)****17.**A. Papageorgiou and G. W. Wasilkowski, On the average complexity of multivariate problems,*J. Complexity***6**(1990), 1-23. MR**1048027 (91b:94020)****18.**H. Pfeiffer, Monte Carlo methods with few random bits,*Ph.D. Thesis*, University of Jena, 2005.**19.**L. Plaskota,*Noisy Information and Computational Complexity,*Cambridge Univ. Press, Cambridge, 1996. MR**1446005 (99b:65189)****20.**K. Ritter,*Average-Case Analysis of Numerical Problems,*Lect. Notes in Math.**1733**, Springer-Verlag, Berlin, 2000. MR**1763973 (2001i:65001)****21.**J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski,*Information-Based Complexity*, Academic Press, New York, 1988. MR**0958691 (90f:68085)****22.**J. F. Traub and A. G. Werschulz,*Complexity and Information*, Cambridge Univ. Press, Cambridge, 1998. MR**1692462 (2000m:65170)****23.**J. F. Traub and H. Wozniakowski,*A General Theory of Optimal Algorithms*, Academic Press, New York, 1980. MR**0584446 (84m:68041)****24.**G. Wahba,*Spline Models for Observational Data,*CBMS-NSF**59**, Philadelphia, 1990. MR**1045442 (91g:62028)****25.**G. W. Wasilkowski, Some nonlinear problems are as easy as the approximation problem,*Comput. Math. Appl.***10**(1984), 351-363. MR**0777390 (86j:41023)****26.**G. W. Wasilkowski, Randomization for continuous problems,*J. Complexity***5**(1989), 195-218. MR**1006106 (91a:65006)****27.**G. W. Wasilkowski and H. Wozniakowski, Weighted tensor-product algorithms for linear multivariate problems,*J. of Complexity***15**, (1999), 402-447. MR**1716741 (2000h:65200)****28.**G. W. Wasilkowski and H. Wozniakowski, On the power of standard information for weighted approximation,*Found. Comput. Math.***1**(2001), 417-434. MR**1857723 (2002g:41052)****29.**A. G. Werschulz,*The Computational Complexity of Differential and Integral Equations,*Oxford Univ. Press, Oxford, 1991. MR**1144521 (93a:68061)****30.**A. G. Werschulz and H. Wozniakowski, Tractability of quasilinear problems I: general results, II: second-order elliptic problems, submitted for publication, 2005.**31.**H. Wozniakowski, Tractability and strong tractability of linear multivariate problems,*J. Complexity*,**10**(1994), 96-128. MR**1263379 (95d:65117)**

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Additional Information

**G. W. Wasilkowski**

Affiliation:
Department of Computer Science, University of Kentucky, 773 Anderson Hall, Lexington, Kentucky 40506-0046

Email:
greg@cs.uky.edu

**H. Wozniakowski**

Affiliation:
Department of Computer Science, Columbia University, New York, NY 10027, USA, and Institute of Applied Mathematics, University of Warsaw, Warsaw, Poland

Email:
henryk@cs.columbia.edu

DOI:
https://doi.org/10.1090/S0025-5718-06-01944-2

Keywords:
Weighted multivariate approximation,
randomized algorithms,
Monte Carlo methods,
tractability

Received by editor(s):
December 8, 2005

Published electronically:
December 13, 2006

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.