Integral operators on sparse grids

SIAM Journal on Numerical Analysis

Volumn 39, Issue 5, 2001, Pages 1794-1809

  Knapek, S ^a   Koster, F ^a  

^a NONE

Author keywords

Biorthogonal wavelets; Boolean blending; Compression; Hyperbolic cross approximation; Integral equations; Optimized approximation spaces; Sparse grids

Indexed keywords

EID: 0036412880     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142900375426     Document Type: Article

References (34)

1. K. I. BABENKO, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables, Dokl. Akad. Nauk SSSR, 132 (1960), pp. 672-075.
2. G. BASZENSKI, nth order polynomial spline blending, in Multivariate Approximation Theory III, W. Schemp and K. Zeller, eds., Birkhäuser, Basel, 1985, pp. 35-46.
3. S. BERTOLUZZA, C. CANUTO, AND K. URBAN, On the adaptive computation of integrals of wavelets, Appl. Numer. Math., 34 (2000), pp. 13-38.
4. G. BEYLKIN, R. COIFMAN, AND V. ROKHLIN, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math., 44 (1991), pp. 141-183.
5. H.-J. BUNGARTZ, M. GRIEBEL, D. RÖSCHKE, AND C. ZENGER, Pointwise convergence of the combination technique for Laplace's equation, East-West J. Numer. Math., 1 (1994), pp. 21-45.
6. J. C. CAVENDISH, W. J. GORDON, AND C. H. HALL, Ritz-Galerkin approximations in blending function spaces, Numer. Math., 26 (1976), pp. 155-178.
7. A. COHEN, I. DAUBECHIES, AND P. VIAL, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal., 1 (1993), pp. 54-81.
8. W. DAHMEN, Stability of multiscale transformations, J. Fourier Anal. Appl., 2 (1996), pp. 341-361.
9. W. DAHMEN, Wavelet and multiscale methods for operator equations, Acta Numer., 6 (1997), pp. 55-228.
10. W. DAHMEN, S. PRÖSSDORF, AND R. SCHNEIDER, Wavelet approximation methods for pseudo-differential equations II: Matrix compression and fast solution, Adv. Comput. Math., 1 (1993), pp. 259-335.
11. F.-J. DELVOS, d-variate Boolean interpolation, J. Approx. Theory, 34 (1982), pp. 99-114.
12. R. A. DEVORE, B. JAWERTH, AND V. POPOV, Compression of wavelet decompositions, Amer. J. Math., 114 (1992), pp. 737-785.
13. R. A. DEVORE, S. V. KONYAGIN, AND V. N. TEMLYAKOV, Hyperbolic wavelet approximation, Constr. Approx., 14 (1998), pp. 1-26.
14. DIN' ZUNG, Number of integral points in a certain set and the approximation of functions of several variables, Math. Notes, 36 (1984), pp. 736-744.
15. K. FRANK, Complexity of local solution of multivariate integral equations, J. Complexity, 11 (1995), pp. 416-434.
16. K. FRANK, S. HEINRICH, AND S. PEREVERZEV, Information complexity of multivariate Fredholm equations in Sobolev classes, J. Complexity, 12 (1996), pp. 17-34.
17. T. GERSTNER AND M. GRIEBEL, Numerical integration using sparse grids, Numer. Algorithms, 18 (1998), pp. 209-232.
18. M. GRIEBEL AND S. KNAPEK, Optimized tensor product approximation spaces, Constr. Approx., 16 (2000), pp. 525-540.
19. M. GRIEBEL AND F. KOSTER, Orthogonal Wavelets on the Interval, Tech. report SFB-256 Bericht Nr. 576, Universität Bonn, Germany, 1998.
20. M. GRIEBEL, P. OSWALD, AND T. SCHIEKOFER, Sparse grids for boundary integral equations, Numer. Math., 83 (1999), pp. 279-312.
21. R. HOCHMUTH, S. KNAPEK, AND G. ZUMBUSCH, Tensor Products of Sobolev Spaces and Applications, Tech. report SFB-256 Bericht Nr. 685, Universität Bonn, Germany, 2000.
22. S. KNAPEK, Hyperbolic Cross Approximation of Integral Operators with Smooth Kernel, Tech. report SFB-256 Bericht Nr. 665, Universität Bonn, Germany, 2000.
23. S. KNAPEK, Approximation und Kompression mit Tensorprodukt-Multiskalen-Räumen, Ph.D. thesis, Institut für Angewandte Mathematik, Universität Bonn, Germany, 2000.
24. Y. MEYER AND R. COIFMAN, Wavelets: Calderon-Zymund and Multilinear Operators, Cambridge Stud. Adv. Math. 48, Cambridge University Press, Cambridge, 1997.
25. P. OSWALD, On discrete norm estimates related to multilevel preconditioners in the finite element method, in Constructive Theory of Functions, K.G. Ivanov. P. Petrushev, and B. Sendov, eds., Publ. House Bulgar Acad. Sci., Sofia, 1992, pp. 203-214.
26. S. V. PEREVERZEV, On the complexity of finding solutions of Fredholm equations of the second kind with differentiate kernels, I, Ukrain. Mat. Zh., 40 (1988), pp. 84-91.
27. S. V. PEREVERZEV, On the complexity of finding solutions of Fredholm equations of the second kind with differentiate kernels, II, Ukrain. Mat. Zh., 41 (1989), pp. 169-173.
28. C. PFLAUM AND A. ZHOU, Error analysis of the combination technique, Numer. Math., 84 (1999), pp. 327-350.
29. R. SCHNEIDER, Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten Lösung großer vollbesetzter Gleichungssysteme, Advances in Numerical Analysis, Teubner, Stuttgart, 1998.
30. S. A. SMOLYAK, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR, 4 (1963), pp. 240-243.
31. W. SWELDENS AND R. PIESSENS, Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions, SIAM J. Numer. Anal., 31 (1994), pp. 1240-1264.
32. V. N. TEMLYAKOV, Approximation of Periodic Functions, Nova Science, New York, 1993.
33. J. F. TRAUB, G. W. WASILKOWSKI, AND H. WOZNIAKOWSKI, Information-Based Complexity, Academic Press, New York, 1988.
34. C. ZENGER, Sparse grids, in Parallel Algorithms for Partial Differential Equations, Notes Numer. Fluid Mech. 31, W. Hackbusch, ed., Vieweg, Braunschweig, 1991, pp. 241-251.