Volume 2, Issue 1, March 2017, Page: 10-16
Wavelet-Based Numerical Homogenization for Scaled Solutions of Linear Matrix Equations
J. B. Allen, Information Technology Laboratory, U.S. Army Engineer Research and Development Center, Vicksburg, USA
Received: Jan. 6, 2017;       Accepted: Jan. 20, 2017;       Published: Feb. 20, 2017
DOI: 10.11648/j.dmath.20170201.13      View  2715      Downloads  80
In this work we investigate the use of wavelet-based numerical homogenization for the solution of various closed form ordinary and partial differential equations, with increasing levels of complexity. In particular, we investigate exact and homogenized (scaled) solutions of the one dimensional Elliptic equation, the two-dimensional Laplace equation, and the two-dimensional Helmholtz equation. For the exact solutions, we utilize a standard Finite Difference approach with Gaussian elimination, while for the homogenized solutions, we applied the wavelet-based numerical homogenization method (incorporating the Haar wavelet basis), and the Schur complement) to arrive at progressive coarse scale solutions. The findings from this work showed that the use of the wavelet-based numerical homogenization with various closed form, linear matrix equations of the type: LU=F affords homogenized scale dependent solutions that can be used to complement multi-resolution analysis, and second, the use of the Schur complement obviates the need to have an a priori exact solution, while the possession of the latter offers the use of simple projection operations.
Numerical Homogenization, Multiscale, Multiresolution, Wavelets
To cite this article
J. B. Allen, Wavelet-Based Numerical Homogenization for Scaled Solutions of Linear Matrix Equations, International Journal of Discrete Mathematics. Vol. 2, No. 1, 2017, pp. 10-16. doi: 10.11648/j.dmath.20170201.13
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
M. Ryvkin, “Employing the Discrete Fourier Transform in the Analysis of Multiscale Problems,” Int. J. Multiscale Computational Engineering, 6, 2008.
P. Wesseling, An Introduction to Multigrid Methods. Wiley, 1992.
A. Brandt, “Multi-level adaptive solutions to boundary value problems,” Math. Comp., 31, 1977.
A. Toselli, and O. Widlund, “Domain Decomposition Methods—Algorithm s and Theory, Springer Series in Computational Mathematics, 34, Springer, New York, 2004.
S. Thirunavukkarasu, M. Guddati, “A domain decomposition method for concurrent coupling of multiscale models,” Int. J. Numerical Methods in Engineering, 92, 2012.
L. Greengard, and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., 73, 1987.
D. F. Martin, P. Colella, M. Anghel, and F. J. Alexander, “Adaptive Mesh Refinement for Multiscale Nonequilibrium Physics,” Computing in Science and Engineering, 2005.
I. Daubechies, Ten Lectures on Wavelets, SIAM, Philidelphia, 1992.
M. Brewster, G. Beylkin, “A multiresolution strategy for numerical homogenization,” Appl. Comput. Harmon. Anal., 2, 1995.
A. Bensoussan, J. L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Pub. Co., Amsterdam, 1978.
G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer-Verlag, New York, 2008.
A. Chertock, and D. Levy, “On Wavelet-based Numerical Homogenization,” Multiscale Model. Simul. 3, 2004.
N. A. Coult, “A Multiresolution Strategy for Homogenization of Partial Differential Equations,” PhD Dissertation, University of Colorado, 1997.
E. B. Tadmor, M. Ortiz and R. Phillips, “Quasicontinuum analyis of defects in solids,” Philos. Mag. A, 73, 1996.
J. Knap and M. Ortiz, “An analysis of the quasicontinuum method,” J. Mech. Phys. Solids, 49, 2001.
Y. Meyer, “Ondelettes sur l’intervalle,” Rev. Mat. Iberoamericana, 7 (2), 1991.
S. Mallat, Multiresolution Approximation and Wavelets, Technical Report, GRASP Lab, Dept. of Computer and Information Science, University of Pennsylvania.
C. Basdevant, V. Perrier, and T. Philipovitch, “Local Spectral Analysis of Turbulent Flows Using Wavelet Transforms,” Vortex Flows and Related Numerical Methods, J. T. Beale et al., (eds.), 1-26, Kluwer Academic Publishers, Netherlands, 1993.
Z. Fuzhen, The Shur Complement and its Applications. Springer Pub., 2005.
Browse journals by subject