The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

Authors:
Fred J. Hickernell and Xiaoqun Wang

Journal:
Math. Comp. **71** (2002), 1641-1661

MSC (2000):
Primary 65C05, 65D30

Published electronically:
August 2, 2001

MathSciNet review:
1933048

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Abstract: Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation.

Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor is bounded by for some exponent and some positive constant . The -exponent of tractability is defined as the smallest power of in these bounds. It is shown by using Monte Carlo quadrature that the -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.

**[Aro50]**N. Aronszajn,*Theory of reproducing kernels*, Trans. Amer. Math. Soc.**68**(1950), 337–404. MR**0051437**, 10.1090/S0002-9947-1950-0051437-7**[CMO97]**R. E. Caflisch, W. Morokoff, and A. Owen,*Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension*, J. Comput. Finance**1**(1997), 27-46.**[DR84]**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR**760629****[Duf96]**D. Duffie,*Dynamic asset pricing theory*, Princeton University Press, Princeton, New Jersey, 1996.**[Fau82]**Henri Faure,*Discrépance de suites associées à un système de numération (en dimension 𝑠)*, Acta Arith.**41**(1982), no. 4, 337–351 (French). MR**677547****[FW94]**K.-T. Fang and Y. Wang,*Number-theoretic methods in statistics*, Monographs on Statistics and Applied Probability, vol. 51, Chapman & Hall, London, 1994. MR**1284470****[Hal60]**J. H. Halton,*On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals*, Numer. Math.**2**(1960), 84–90. MR**0121961****[Hic98]**Fred J. Hickernell,*A generalized discrepancy and quadrature error bound*, Math. Comp.**67**(1998), no. 221, 299–322. MR**1433265**, 10.1090/S0025-5718-98-00894-1**[HW00]**F. J. Hickernell and H. Woźniakowski,*Integration and approximation in arbitrary dimensions*, Adv. Comput. Math.**12**(2000), no. 1, 25–58. High dimensional integration. MR**1758946**, 10.1023/A:1018948631251**[MC94]**William J. Morokoff and Russel E. Caflisch,*Quasi-random sequences and their discrepancies*, SIAM J. Sci. Comput.**15**(1994), no. 6, 1251–1279. MR**1298614**, 10.1137/0915077**[Nie92]**Harald Niederreiter,*Random number generation and quasi-Monte Carlo methods*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1172997****[NW00]**E. Novak and H. Wozniakowski,*When are integration and discrepancy tractable?*, Foundations of Computational Mathematics (R. De Vore, A. Iserles, E. Süli, eds.), Chap. 8, Cambridge University Press, Cambridge, 2001.**[NX96]**Harald Niederreiter and Chaoping Xing,*Quasirandom points and global function fields*, Finite fields and applications (Glasgow, 1995) London Math. Soc. Lecture Note Ser., vol. 233, Cambridge Univ. Press, Cambridge, 1996, pp. 269–296. MR**1433154**, 10.1017/CBO9780511525988.022**[Sai88]**Saburou Saitoh,*Theory of reproducing kernels and its applications*, Pitman Research Notes in Mathematics Series, vol. 189, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. MR**983117****[SJ94]**I. H. Sloan and S. Joe,*Lattice methods for multiple integration*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR**1442955****[Sob69]**I. M. \cyr{T}sobol′,*Mnogomernye kvadraturnye formuly i funktsii Khaara*, Izdat. “Nauka”, Moscow, 1969 (Russian). MR**0422968****[Sob98]**I. M. Sobol,*On quasi-Monte Carlo integrations*, Math. Comput. Simulation**47**(1998), no. 2-5, 103–112. IMACS Seminar on Monte Carlo Methods (Brussels, 1997). MR**1641362**, 10.1016/S0378-4754(98)00096-2**[SW98]**Ian H. Sloan and Henryk Woźniakowski,*When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?*, J. Complexity**14**(1998), no. 1, 1–33. MR**1617765**, 10.1006/jcom.1997.0463**[TW98]**J. F. Traub and A. G. Werschulz,*Complexity and information*, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1998. MR**1692462****[Wah90]**Grace Wahba,*Spline models for observational data*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR**1045442****[Woz91]**H. Woźniakowski,*Average case complexity of multivariate integration*, Bull. Amer. Math. Soc. (N.S.)**24**(1991), no. 1, 185–194. MR**1072015**, 10.1090/S0273-0979-1991-15985-9**[WW96]**Grzegorz W. Wasilkowski and Henryk Woźniakowski,*On tractability of path integration*, J. Math. Phys.**37**(1996), no. 4, 2071–2086. MR**1380891**, 10.1063/1.531493

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Additional Information

**Fred J. Hickernell**

Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

Email:
fred@hkbu.edu.hk

**Xiaoqun Wang**

Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Email:
xwang@math.tsinghua.edu.cn

DOI:
https://doi.org/10.1090/S0025-5718-01-01377-1

Keywords:
Quasi-Monte Carlo methods,
Monte Carlo methods,
tractability,
infinite dimensional integration

Received by editor(s):
May 24, 2000

Received by editor(s) in revised form:
October 18, 2000

Published electronically:
August 2, 2001

Additional Notes:
This work was supported by a Hong Kong Research Grants Council grant RGC/97-98/47 and by the NSF of China Grants 79970120 and 10001021.

Article copyright:
© Copyright 2001
American Mathematical Society