Best n term approximation spaces for tensor product wavelet bases

Constructive Approximation

Volumn 24, Issue 1, 2006, Pages 49-70

  Nitsche, Pal Andrej ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

Besov spaces; Best N term approximation; Sparse grids; Tensor product approximation

Indexed keywords

EID: 33645937313     PISSN: 01764276     EISSN: 14320940     Source Type: Journal    
DOI: 10.1007/s00365-005-0609-6     Document Type: Article

References (28)

1. ST. ACHATZ (to appear): Higher order sparse grid methods for elliptic partial differential equations with variable coefficients. Special Issue of Computing on Sparse Grids.
2. TH. APEL (1999): Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics. Stuttgart: Teubner.
3. A. BARINKA, T. BARSCH, P. CHARTON, A. COHEN, S. DAHLKE, W. DAHMEN, K. URBAN (2001): Adaptive wavelet schemes for elliptic problems - Implementations and numerical experiments. SIAM J. Sci. Comput., 23(3):910-939.
4. C. BENNETT, R. SHARPLEY (1988): Interpolation of Operators. New York: Academic Press.
5. P. BINEV, W. DAHMEN, R. A. DEVORE (2004): Adaptive finite element methods with convergence rates. Numer. Math., 97(2):219-268.
6. A. COHEN, W. DAHMEN, R. A. DEVORE (2000): Adaptive wavelet methods for elliptic operator equations: Convergence, rates. Math. Comp., 70(233):27-75.
7. A. COHEN, W. DAHMEN, R. A. DEVORE (2002): Adaptive wavelet methods II - Beyond the elliptic case. Found. Comput. Math., 2(3):203-245.
8. A. COHEN, W. DAHMEN, R. A. DEVORE (2003): Adaptive wavelet schemes for nonlinear variational problems. SIAM J. Numer. Anal., 41(5):1785-1823.
9. A. COHEN, R. A. DEVORE, R. HOCHMUTH (2000): Restricted nonlinear approximation. Constr. Approx., 16:85-113.
10. S. DAHLKE (1999): Besov regularity for elliptic boundary value problems in polygonal domains. Appl. Math. Lett, 12:31-36.
11. S. DAHLKE (2002): Besov regularity of edge singularities for the Poisson equation in polyhedral domains. Numer. Linear Algebra Appl., 9:457-466.
12. W. DAHMEN, A. KUNOTH, K. URBAN (1999): Biorthogonal spline-wavelets on the interval - Stability and moment conditions. Appl. Comput. Harmon. Anal., 6:132-196.
13. W. DAHMEN, R. SCHNEIDER (1998): Wavelets with complementary boundary conditions - Function spaces on the cube. Result. Math., 34:255-293.
14. R. A. DEVORE, G. G. LORENTZ (1993): Constructive Approximation. Grundlehren der Mathematischen Wissenschaften. Berlin: Springer-Verlag.
15. R. A. DEVORE (1998): Nonlinear Approximation. Acta Numerica, Vol. 7. Cambridge, UK: Cambridge University Press, pp. 51-150.
16. M. GRIEBEL, S. KNAPEK (2000): Optimized tensor-pmduct approximation spaces. Constr. Approx., 16(4):525-540.
17. N. KALTON (2003): Quasi-Banach spaces. In: Handbook of the Geometry of Banach Spaces, Vol. 2, pp. 1099-1130. Amsterdam: North-Holland.
18. W. LIGHT, E. CHENEY (1980): Approximation Theory in Tensor Product Spaces. Lecture Notes in Mathematics, Vol. 1169. New York: Springer-Verlag.
19. P. MORIN, R. NOCHETTO, K. SIEBERT (2000): Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal., 38(2).
20. A. NITSCHE (2001): Sparse approximation of singularity functions. Constr. Approx., 21(1):63-81.
21. A. NITSCHE (2004): Sparse Tensor Product Approximation of Elliptic Problems. Dissertation an der Eidgenössischen Technischen Hochschule ETH Zürich, Switzerland.
22. s-norms. In: Metric Function Theory and Related Topics in Analysis (Nicolskij, Kashin, and Izaak, eds.). Moscow: AFC, pp. 137-163.
23. P. OSWALD (1999): Best n-term capacitance approximation on sparse grids. In: Proc. 12th Intern. Conf. on Domain Decomp. Methods in Science and Engineering (T. Chan et al., eds.). Chiba, pp. 437-444.
24. J. PEETRE (1968): A Theory of Interpolation of Normed Spaces. Course Notes. University of Brasilia.
25. S.A. SMOLYAK (1963): Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl., 4:240-243.
26. V. TEMLYAKOV (2002): Nonlinear methods of approximation. Found. Comput. Math., 3:33-107.
27. P. TURPIN (1982): Représentation fonctionelle des espaces vectorielles topologiques. Studia Math., 73:1-10.
28. CH. ZENGER (1991): Sparse grids; in parallel algorithms for PDE. In: Proc. 6th GAMM Seminar, Kiel (W. Hackbusch, ed.). Braunschweig: Vieweg, pp. 241-251.