Computing discrepancies of Smolyak quadrature rules

Journal of Complexity

Volumn 12, Issue 4, 1996, Pages 287-314

  Frank, Karin ^a   Heinrich, Stefan ^a  

^a University of Koblenz Landau   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030511609     PISSN: 0885064X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcom.1996.0020     Document Type: Article

References (31)

1. ABRAMOWITZ, M., AND STEGUN, I. A. (1972), "Handbook of Mathematical Functions," National Bureau of Standards, Washington, D.C.
2. ARONSZAJN, N. (1950), Theory of reproducing kernels, Trans. Amer. Math. Soc. 68, 337-404.
3. BABENKO, K. I. (1960), Approximation of a certain class of periodic functions of several variables by trigonometric polynomials, Dokl. Akad. Nauk SSSR 132, 982-985 [in Russian].
4. BASZENSKI G., AND DELVOS, F.-J. (1993), Multivariate boolean midpoint rules, in "Numerical Integration IV," ISNM, Vol. 112 (H. Brass and G. Hämmerlin, Eds.), pp. 1-11, Birkhäuser, Basel.
5. BASZENSKI, G., DELVOS, F.-J., AND JESTER, S. (1992), Blending approximation with sine functions, in "Numerical Methods of Approximation, 9," ISNM, Vol. 105 (D. Braess and L. L. Schumaker, Eds.), pp. 1-19, Birkhäuser, Basel.
6. BONK, TH. (1994), A new algorithm for multi-dimensional adaptive numerical quadrature, in "Adaptive Methods - Algorithms, Theory and Applications," (W. Hackbusch and G. Wittum, Eds.), pp. 54-68, Vieweg, Braunschweig.
7. BRASS, H. (1977), "Quadraturverfahren," Vandenhoeck & Rupprecht, Göttingen.
8. DELVOS, F.-J. (1982), d-Variate boolean interpolation, J. Approx. Theory 34, 99-114.
9. GRIEBEL, M. SCHNEIDER, M., AND ZENGER CH. (1992), A combination technique for the solution of sparse grid problems, in "Iterative Methods in Linear Algebra," (R. Beauwens and P. de Groen, Eds.), Elsevier, North-Holland, pp. 263-281.
10. 2 Discrepancy," Tech. Rep. 267/95, Universität Kaiserslautern; Math. Comp., to appear.
11. 2 Discrepancy," Tech. Rep. 267/95, Universität Kaiserslautern; Math. Comp., to appear.
12. MÜLLER-GRONBACH, T. (1996), Hyperbolic cross designs for approximation of random fields, preprint.
13. NIEDERREITER, H. (1978), Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84, 957-1041.
14. NIEDERREITER, H. (1992), "Random Number Generation and Quasi-Monte Carlo Methods," CBMS-NSF Rep. 63, SIAM.
15. NOVAK, E., AND RITTER, K. (1996), High dimensional integration of smooth functions over cubes, Numer. Math., to appear.
16. PASKOV, S. H. (1993), Average case complexity of multivariate integration for smooth functions, J. Complexity 9, 291-312.
17. RITTER, K., WASILKOWSKI, G. W., AND WOŹNIAKOWSKI, H. (1995), Multivariate integration and approximation for random fields satisfying Sacks-Ylvisaker conditions, Ann. Appl. Prob. 5, 518-540.
18. SMOLYAK, S. A. (1963), Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 148, 1042-1045 [in Russian]; Soviet. Math. Dokl. 4, 240-243 [English transl.].
19. SMOLYAK, S. A. (1963), Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 148, 1042-1045 [in Russian]; Soviet. Math. Dokl. 4, 240-243 [English transl.].
20. SACKS, J., AND YLVISAKER, D. (1970), Statistical design and integral approximation, in "Proc. 12th. Biennial Seminar of the Canadian Math. Congress" (R. Pyke, Ed.), pp. 115-136, Can. Math. Soc., Montreal.
21. TEMLYAKOV, V. N. (1990), On a way of obtaining lower estimates for the errors of quadrature formulas, Mat. Sb. 181, 1403-1413; (1992), Math. USSR Sb. 71, 247-257 [English transl.].
22. TEMLYAKOV, V. N. (1990), On a way of obtaining lower estimates for the errors of quadrature formulas, Mat. Sb. 181, 1403-1413; (1992), Math. USSR Sb. 71, 247-257 [English transl.].
23. TEMLYAKOV, V. N. (1993), On approximate recovery of functions with bounded mixed derivative, J. Complexity 9, 41-59.
24. TEZUKA, S. (1995), "Uniform Random Numbers: Theory and Practice," Kluwer Academic, Dordrecht.
25. WARNOCK, T. T. (1972), Computational investigations of low discrepancy point sets, in Applications of Number Theory to Numerical Analysis, (S. K. Zaremba, Ed.). Academic Press, New York.
26. WERNER, D. (1995), "Funktionalanalysis," Springer-Verlag, Berlin.
27. WOŹNIAKOWSKI, H. (1991), Average case complexity of multivariate integration, Bull. Amer. Math. Soc. 24, 185-194.
28. WASILKOWSKI, G. W., AND WOŹNIAKOWSKI, H. (1995), Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity 11, 1-56.
29. ZAREMBA, S. K. (1968), Some applications of multidimensional integration by parts, Ann. Polon. Math. 21, 85-96.
30. ZENGER, CH. (1991), Sparse grids, in "Parallel Algorithms for Partial Differential Equations," (W. Hackbusch, Ed.), Vieweg, Braunschweig, pp. 241-251.
31. ZIELINSKI, J. (1994), Multidimensional Integration mittels Tensorprodukt-Technik, Tech. Rep., Universität Kaiserslautern.