High dimensional numerical problems

Nonlinear Analysis, Theory, Methods and Applications

Volumn 30, Issue 3, 1997, Pages 1439-1446

  Novak, Erich ^a  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

Author keywords

Boolean methods; Complexity of numerical problems; Curse of dimension; Global optimization; High dimensional problems; Information based complexity; Integration; Optimal error bounds; Partially separable problems; Smolyak's construction; Sparse grids

Indexed keywords

EID: 0043133443     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0362-546X(96)00134-4     Document Type: Article

References (30)

1. BAKHVALOV N. S., On approximate computation of integrals, Vestnik MGU, Ser. Math. Mech. Astron. Phys. Chem. 4, 3-18 (1959). [In Russian]
2. CALVIN J. M., Average performance of a class of adaptive algorithms for global optimization, preprint, Georgia Institute of Technology (1995).
3. DELVOS F.-J. & SCHEMPP W., Boolean methods in interpolation and approximation, Pitman Research Notes in Mathematics Series 230. Longman, Essex (1989).
4. GENZ A. C., A package for testing multiple integration subroutines, in: Keast, P., Fairweather, G., eds., Numerical Integration, 337-340, Kluwer, Dordrecht (1987).
5. GENZ A. C. & MALIK A. A., Algorithm 019: an adaptive algorithm for numerical integration over an n-dimensional rectangular region, J. Comp. Appl. Math. 6, 295-302 (1980).
6. GRIEBEL M., SCHNEIDER M. & ZENGER Ch., A combination technique for the solution of sparse grid problems, in: Beauwens, R., de Groen, P., eds., Iterative Methods in Linear Algebra, 263-281, Elsevier, North-Holland (1992).
7. HEINRICH S., Random approximation in numerical analysis, in Functional Analysis, K. D. Bierstedt et al., eds., Lecture Notes in Pure and Applied Mathematics 150, Marcel Dekker, New York, 123-171 (1993).
8. KACEWICZ B. Z., How to increase the order to get minimal-error algorithms for systems of ODE, Numer. Math. 45, 93-104 (1984).
9. KACEWICZ B. Z., On sequential and parallel solution of initial value problems, J. Complexity 6, 136-148 (1990).
10. KAHANER D. K., A survey of existing multidimensional quadrature routines, Contemporary Mathematics 115, 9-22 (1991).
11. NEMIROVSKI A. S., & YUDIN D. B., Problem Complexity and Method Efficiency on Optimization, John Wiley and Sons, Chicester (1983).
12. NOVAK E., Deterministic and stochastic error bounds in numerical analysis. Lecture Notes in Mathematics 1349, Springer-Verlag, Berlin (1988).
13. NOVAK E., On the power of adaption, J. Complexity, to appear.
14. NOVAK E. & RITTER K., Global optimization using hyperbolic cross points, in: State of the art in global optimization: computational methods and applications, C. A. Floudas, P. M. Pardalos (eds.), Kluwer, Dordrecht, 19-33 (1996).
15. NOVAK E. & RITTER K., High dimensional integration of smooth functions over cubes, Numer. Math., to appear.
16. PARDALOS P. M. & SHALLOWAY D. & XUE G. (eds.), Global minimization of nonconvex energy functions: molecular conformation and protein folding, DIMACS Series Vol. 23, American Math. Soc., Providence (1996).
17. PASKOV S. H. & TRAUB J. F., Faster valuation of financial derivatives, J. Portfolio Management 22, 113-120 (1995).
18. SLOAN I. H. & JOE S., Lattice Methods for Multiple Integration, Clarendon Press, Oxford (1994).
19. SMOLYAK S. A., Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl. 4, 240-243 (1963).
20. TEMLYAKOV V. N., Approximation of Periodic Functions, Nova Science Publishers, New York (1994).
21. TRAUB J. F. & WASILKOWSKI G. W. & WOŹNIAKOWSKI H., Information-based complexity, Academic Press, San Diego (1988).
22. VAN DOOREN P. & DE RIDDER L., An adaptive algorithm for numerical integration over an n-dimensional cube, J. Comp. Appl. Math. 2, 207-217 (1976).
23. VESELY F. J., Computational physics, an introduction, Plenum Press, New York (1994).
24. WASILKOWSKI G. W. & WOŹNIAKOWSKI H., Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity 11, 1-56 (1995).
25. WASILKOWSKI G. W. & WOŹNIAKOWSKI H., On tractability of path integration, J. Math. Phys. 37, 2071-2088 (1996).
26. WERSCHULZ A. G., The computational complexity of differential and integral equations, Oxford Science Publications, Oxford (1991).
27. WOŹNIAKOWSKI H., Average case complexity of linear multivariate problems, Part 1 and 2, J. Complexity 8, 337-392 (1992).
28. WOŹNIAKOWSKI H., Tractability and strong tractability of linear multivariate problems, J. Complexity 10, 96-128 (1994).
29. WOŹNIAKOWSKI H., Tractability and strong tractability of multivariate tensor product problems, in: Proc. of 2nd Gauss Symp., München 1993, J. Computing and Information 4, 1-19 (1994).
30. ZENGER Ch., Sparse grids, in: Hackbusch, W., ed., Parallel Algorithms for Partial Differential Equations, 241-251, Vieweg, Braunschweig (1991).