Abstract
We review recent developments in techniques for representing data in terms of its local scale components. These techniques allow data compression through elimination of scale-coefficients that are sufficiently small in the transformed representation. This capability for data compression can be used to reduce the cost of many numerical solution algorithms either by applying it to the numerical solution operator in order to get an approximate sparse representation, or by applying it to the numerical solution itself in order to reduce the number of quantities that need to be computed.
This work was partially supported by the National Aeronautics and Space Administration under NASA Contract Nos. NAS1-18605 and NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), M/S 132C, NASA Langley Research Center, Hampton, VA, 23681-0001, and partially supported at UCLA under ONR-N00014-92-J-1890 and NSF-DMS91-03104.
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References
Abgrall, R. and Harten, A., 1994. “Multiresolution analysis on unstructured meshes: preliminary report,” UCLA CAM Report 94-26.
Arandiga, F. and Candela, V., 1992. “Multiresolution standard form of a matrix,” UCLA CAM Report 92-37.
Arandiga, F., Candela V., and Donat, R., 1992. “Fast multiresolution algorithms for solving linear equations: A comparative study,” UCLA CAM Report 92-52.
Arandiga, F., Donat, R., and Harten, A., 1993. “Multiresolution based on weighted averages of the hat function,” UCLA CAM Report 93-34.
Bacry, E., Mallat, S., and Papanicolau, G., 1992. “A wavelet based space-time adaptive numerical methods for partial differential equations,” Mathematical Modelling and Numerical Analysis 26, pp. 703–834.
Beylkin, G., Coifman, R., and Rokhlin, V., 1991. “Fast wavelet transform and numerical algorithms. I,” Comm. Pure Appl. Math. 44, pp. 141–183.
Brandt, A. and Lubrecht, A.A., 1990. “Multilevel matrix multiplication and fast solution of integral equations,” J. Comp. Phys. 90, pp. 348–370.
Cohen, A., Daubechies, I., and Feauveau, J.-C,1992. “Biorthogonal bases of compactly suported wavelets,” Comm. Pure Appl. Math. 45, pp. 485–560.
Daubechies, I., 1988. “Orthonormal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, pp. 909–996.
Engquist, B., Osher, S., and Zhong, S., 1992. “Fast wavelet algorithms for linear evolution equations,” ICASE Report 92-14.
Harten, A., 1993a. “Discrete multiresolution analysis and generalized wavelets,” J. Appl. Num. Math. 12, pp. 153–193; also UCLA CAM Report 92-08.
Harten, A., 1993b. “Multiresolution representation of data. I. Preliminary Report,” UCLA CAM Report No. 93-13.
Harten, A., 1994a. “Adaptive multiresolution schemes for shock computations,” J. Comp. Phys. 115, pp. 319–338; also UCLA CAM Report 93-06.
Harten, A., 1994b. “Multiresolution representation of data. II. General Framework,” UCLA CAM Report No. 94-10.
Harten, A., 1995. “Multiresolution algorithms for the numerical solution of hyperbolic conservation laws,” Comm. Pure Appl. Math. 48, pp. 1305–1342; also UCLA CAM Report 93-03.
Harten, A., Engquist, B., Osher, S., and Chakravarthy, S., 1987. “Uniformly high order accurate essentially non-oscillatory schemes, III,” J. Comp. Phys. 71, pp. 231–303.
Harten, A. and Yad-Shalom, I., 1994. “Fast multiresolution algorithms for matrix-vector multiplication,” SIAM J. Numer. Anal. 31, pp. 1191–1218; also ICASE Report 92-55.
Liandrat, J. and Tchamitchian, Ph., 1990. “Resolution of the 1D regularized Burgers’ equation using a spatial wavelet approximation,” ICASE Report 90-83.
Mallat, S., 1989. “Multiresolution approximation and wavelets orthonormal bases of L2(R),” Trans. Amer. Math. Soc. 315, pp. 69–87.
Meyer, Y., 1990. Ondelettes et Opérateurs, Hermann.
Maday, Y. and Ravel, J.C., 1992. “Adaptivité par Ondelettes: condition aux limites et dimensions supérieures,” Tech. Report, Université Pierre et Marie Curie, Lab. D’Analyse Numérique.
Strang, G., 1989. “Wavelets and dilation equations: A brief introduction,” SIAM Review 31, pp. 614–627.
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Harten, A. (1997). Multiresolution Representation and Numerical Algorithms: A Brief Review. In: Keyes, D.E., Sameh, A., Venkatakrishnan, V. (eds) Parallel Numerical Algorithms. ICASE/LaRC Interdisciplinary Series in Science and Engineering, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5412-3_11
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DOI: https://doi.org/10.1007/978-94-011-5412-3_11
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