DECAY CHARACTERISTICS OF THE HAT INTERPOLATION WAVELET COEFFICIENTS IN THE TWO-DIMENSIONAL MULTIRESOLUTION REPRESENTATION

Title & Authors
DECAY CHARACTERISTICS OF THE HAT INTERPOLATION WAVELET COEFFICIENTS IN THE TWO-DIMENSIONAL MULTIRESOLUTION REPRESENTATION
KWON KIWOON; KIM YOON YOUNG;

Abstract
The objective of this study is to analyze the decay characteristics of the hat interpolation wavelet coefficients of some smooth functions defined in a two-dimensional space. The motivation of this research is to establish some fundamental mathematical foundations needed in justifying the adaptive multiresolution analysis of the hat-interpolation wavelet-Galerkin method. Though the hat-interpolation wavelet-Galerkin method has been successful in some classes of problems, no complete error analysis has been given yet. As an effort towards this direction, we give estimates on the decaying ratios of the wavelet coefficients at children interpolation points to the wavelet coefficient at the parent interpolation point. We also give an estimate for the difference between non-adaptively and adaptively interpolated representations.
Keywords
hat interpolation wavelet;decay characteristics;adaptive wavelet-Galerkin method;
Language
English
Cited by
References
1.
L. Andersson, N. Hall, B. Jawerth, and G. Peters, Wavelets on closed subsets of the real line, Topics in the Theory and Applications of Wavelets, Academic press, 1993, 1-14

2.
I. Babuska and A. Miller, A feedback finite element method with a-posteriori error estimation, I. The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg. 61 (1987), 1-40

3.
I. Babuska and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), 736-754

4.
R. Balder and C. Zenger, The solution of multidimensional real Helmholtz equations on sparse grids, SIAM J. Sci. Comp. 17 (1996), 631-646

5.
R. E. Bank, T. F. Dupont, and H. Yserentant, The hierarchical basis multigrid method, Numer. Math. 52 (1988), 427-458

6.
R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), 283-301

7.
A. Barinka, T. Barsch, P. Charton, A. Cohen, S. Dahlke, W. Dahmen, and U. Karsten, Adaptive wavelet schemes for elliptic problems - implementation and numerical experiments, SIAM J. Sci. Comput. 23 (2001), 910-939

8.
F. Bornemann, B. Erdmann, and R. Kornhuber, A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. Numer. Anal. 33 (1996), 1188-1204

9.
A. Brandt, Multi-level adaptive solutions to boundary value problems, Math. Comput. 31 (1977), 333-390

10.
A. Brandt, Multi-level adaptive technique for fast numerical solution to boundary value problems, Proc. 3rd Int. Conf. on Numerical Methods in Fluid Mechanics, Lecture Notes in Physics, 18, 82-89, Springer-Verlag, 1973

11.
W. L. Briggs, A Multigrid Tutorial, SIAM, 1987

12.
H.-J. Bungatz, Dunne Gitter und deren Anwendung bei der adaptiven Losung der dreidimensionalen Poisson-Gleichung, Technischen Universitat Munchen, 1992

13.
M. A. Christon and D. W. Roach, The numerical performance of wavelets for PDEs: the multi-scale finite element, Comput. Mech. 25 (2000), 230-244

14.
A. Cohen, W. Dahmen, and R. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates, Math. Comp. 70 (2001), 27-75

15.
A. Cohen, I. Daubechies, and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), 485-560

16.
A. Cohen and R. Masson, Wavelet methods for second-order elliptic problems, preconditioning, and adaptivity, SIAM J. Sci. Comp. 21 (1999), 1006-1026

17.
S. Dahlke, W. Dahmen, R. Hochmuth, and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, Appl. Numer. Math. 23 (1997), 21-47

18.
W. Dahmen, Wavelet methods for PDEs - some recent developments, J. Comput. Appl. Math. 128 (2001), 133-185

19.
J. M. de Villiers, K. M. Goosen, and B. M. Herbst, Dubuc-Deslauriers subdivision for finite sequences and interpolation wavelets on an interval, SIAM J. Math. Anal. 35 (2003) 423452

20.
D. Donoho, Interpolating wavelet transform, Stanford University, 1992

21.
J. Douglas Jr., T. Dupont, and M. F. Wheeler, An $L^{\infty}$estimate and a super-convergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials, Rev. Francaise Automat. Informat. Recherche Operationnelle Ser Rouge 8 (1974), 61-66

22.
K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numer. 4 (1995), 105-158

23.
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1983

24.
M. Griebel, Adaptive sparse grid multilevel methods for elliptic PDEs based on finite diffrences, Computing 61 (1998), 151-179

25.
M. Griebel and S. Knapek, Optimized tensor-product approximation spaces, Constr. Approx. 16 (2000), 525-540

26.
W. Hackbusch, On the convergence of multigrid iterations, Beit. Numer. Math. 9 (1981), 231-329

27.
W. Hackbusch, On the multi-grid method applied to difference equations, Computing, 20 (1978), 291-306

28.
W. Hackbusch, Survey of convergence proofs for multigrid iterations, Special topics of applied mathematics, Proceedings, Bonn, Oct. 1979, 151-164, Elsevier, 1980

29.
G.-W. Jang, J. E. Kim, and Y. Y. Kim, Multiscale Galerkin method using inter- polation wavelets for two dimensional elliptic problems in general domains, Int. J. Numer. Methods Engrg. 59 (2004), 225-253

30.
Y. Y. Kim and G.-W. Jang, Hat interpolation wavelet-based multi-scale Galerkin method for thin-walled box beam analysis, Int. J. Numer. Methods Engrg. 53 (2002), 1575-1592

31.
P. Krysl, E. Grinspun and P. Schroder, Natural hierarchical refinement for finite element methods, Int. J. Numer. Methods Engrg. 56 (2003), 1109-1124

32.
S. Mallat, A wavelet tour of signal processing, Academic Press, 1998

33.
E. J. Stollnitz, T. D. DeRose, and D. H. Salesin, Wavelets for computer graphics: theory and applications, 21-31, Morgan Kaufmann publishers, 1996

34.
P. Wesseling, An introduction to multigrid methods, John Wiley & Sons, 1992

35.
H. Yserentant, On the multi-level splitting of finite element spaces, Numer. Math. 49 (1986), 379-412

36.
H. Yserentant, Two preconditioners based on the multi-level splitting of finite element spaces, Numer. Math. 58 (1990), 163-184

37.
C. Zenger, Sparse grids, Parallel algorithms for partial differential equations, Notes Numer. Fluid Mech., Vieweg 31 (1991)

38.
O. C. Zienkiewicz and A. Craig, Adaptive refinement, error estimates, multigrid solution, and hierarchic finite element method concepts, Accuracy Estimates and Adaptive Refinements in Finite Element Computations, 25-59, John Wiley & Sons, 1986

39.
O. C. Zienkiewicz, D. W. Kelly, J. Gago, and I. Babuska, Hierarchical finite element approaches, error estimates and adaptive refinement, The mathematics of finite elements and applications IV, 313 346, Academic Press, 1982