Polynomial-time algorithms for multivariate linear problems with finite-order weights: Average case setting

Foundations of Computational Mathematics

Volumn 9, Issue 1, 2009, Pages 105-132

  Wasilkowski, G W ^a   Wozniakowski H ^b,c  

^a UNIVERSITY OF KENTUCKY   (United States)

^b Columbia University ^*  (United States)

^c UNIVERSITY OF WARSAW   (Poland)

Author keywords

Average case setting; Finite order weights; Multivariate linear problems; Polynomial time algorithms; Small effective dimension; Tractability

Indexed keywords

AVERAGE CASE; EFFECTIVE DIMENSIONS; FINITE-ORDER WEIGHTS; LINEAR PROBLEMS; POLYNOMIAL-TIME ALGORITHMS; TRACTABILITY;

BANACH SPACES; POLYNOMIALS;

ALGORITHMS;

EID: 58849089245     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-007-9003-z     Document Type: Article

References (23)

1. R. E. Caflisch W. Morokoff A. Owen 1997 Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension J. Comput. Finance 1 27 46
2. J. Dick I. H. Sloan X. Wang H. Woźniakowski 2006 Good lattice rules in weighted Korobov spaces with general weights Numer. Math. 103 63 97
3. B. Efron C. Stein 1981 The jackknife estimate of variance Ann. Stat. 9 586 596
4. F. J. Hickernell, G. W. Wasilkowski, and H. Woźniakowski, Tractability of linear multivariate problems in the average case setting, in Monte Carlo and Quasi-Monte Carlo Methods (H. Niederreiter, ed.), 2006, in press.
5. F. Y. Kuo I. H. Sloan 2005 Quasi-Monte Carlo methods can be efficient for integration over product of spheres J. Complex. 21 196 210
6. H.-H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, Vol. 463, Springer, Berlin, 1975.
7. M. Kwas Y. Li 2003 Worst case complexity of multivariate Feynman-Kac path integration J. Complex. 19 730 743
8. E. Novak and H. Woźniakowski, When are integration and discrepancy tractable?, in Foundation of Computational Mathematics (Oxford, 1999) (R. A. DeVore, A. Iserles, and E. Süli, eds.), pp. 211-266, Cambridge University Press, Cambridge, 2001.
9. A. Papageorgiou G. W. Wasilkowski 1990 On the average complexity of multivariate problems J. Complex. 6 1 23
10. 1-approximation of multivariate functions with bounded mixed derivatives of second order Numer. Algorithms 36 229 246
11. I. H. Sloan X. Wang H. Woźniakowski 2004 Finite-order weights imply tractability of multivariate integration J. Complex. 20 46 74
12. S. A. Smolyak 1963 Quadrature and interpolation formulas for tensor products of certain classes of functions Dokl. Akad. Nauk SSSR 4 240 243
13. I. M. Sobol 2001 Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates Math. Comput. Simul. 55 271 280
14. J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski, Information-Based Complexity, Academic Press, New York, 1988.
15. J. F. Traub and A. G. Werschulz, Complexity and Information, Cambridge University Press, Cambridge, 1988.
16. X. Wang K. T. Fang 2003 Effective dimensions and quasi-Monte Carlo integration J. Complex. 19 101 124
17. X. Wang and I. H. Sloan, Why are high-dimensional finance problems often of low effective dimension? 2003, submitted.
18. N. N. Vakhania, Probability Distributions on Linear Spaces, North-Holland, New York, 1981.
19. G. W. Wasilkowski 1986 Information of varying cardinality J. Complex. 2 204 228
20. G. W. Wasilkowski H. Woźniakowski 1995 Explicit cost bounds of algorithms for multivariate tensor product problems J. Complex. 11 1 56
21. G. W. Wasilkowski H. Woźniakowski 1999 Weighted tensor-product algorithms for linear multivariate problems J. Complex. 15 402 447
22. G. W. Wasilkowski H. Woźniakowski 2004 Finite-order weights imply tractability of linear multivariate problems J. Approx. Theory 130 57 77
23. G. W. Wasilkowski H. Woźniakowski 2005 Polynomial-time algorithms for multivariate problems with finite-order weights: Worst case setting Found. Comput. Math. 5 451 491