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Wavelet-Based Numerical Homogenization for Scaled Solutions of Linear Matrix Equations

Received: 6 January 2017     Accepted: 20 January 2017     Published: 20 February 2017
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Abstract

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.

Published in International Journal of Discrete Mathematics (Volume 2, Issue 1)
DOI 10.11648/j.dmath.20170201.13
Page(s) 10-16
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Numerical Homogenization, Multiscale, Multiresolution, Wavelets

References
[1] M. Ryvkin, “Employing the Discrete Fourier Transform in the Analysis of Multiscale Problems,” Int. J. Multiscale Computational Engineering, 6, 2008.
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[3] A. Brandt, “Multi-level adaptive solutions to boundary value problems,” Math. Comp., 31, 1977.
[4] A. Toselli, and O. Widlund, “Domain Decomposition Methods—Algorithm s and Theory, Springer Series in Computational Mathematics, 34, Springer, New York, 2004.
[5] S. Thirunavukkarasu, M. Guddati, “A domain decomposition method for concurrent coupling of multiscale models,” Int. J. Numerical Methods in Engineering, 92, 2012.
[6] L. Greengard, and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., 73, 1987.
[7] D. F. Martin, P. Colella, M. Anghel, and F. J. Alexander, “Adaptive Mesh Refinement for Multiscale Nonequilibrium Physics,” Computing in Science and Engineering, 2005.
[8] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philidelphia, 1992.
[9] M. Brewster, G. Beylkin, “A multiresolution strategy for numerical homogenization,” Appl. Comput. Harmon. Anal., 2, 1995.
[10] A. Bensoussan, J. L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Pub. Co., Amsterdam, 1978.
[11] G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer-Verlag, New York, 2008.
[12] A. Chertock, and D. Levy, “On Wavelet-based Numerical Homogenization,” Multiscale Model. Simul. 3, 2004.
[13] N. A. Coult, “A Multiresolution Strategy for Homogenization of Partial Differential Equations,” PhD Dissertation, University of Colorado, 1997.
[14] E. B. Tadmor, M. Ortiz and R. Phillips, “Quasicontinuum analyis of defects in solids,” Philos. Mag. A, 73, 1996.
[15] J. Knap and M. Ortiz, “An analysis of the quasicontinuum method,” J. Mech. Phys. Solids, 49, 2001.
[16] Y. Meyer, “Ondelettes sur l’intervalle,” Rev. Mat. Iberoamericana, 7 (2), 1991.
[17] S. Mallat, Multiresolution Approximation and Wavelets, Technical Report, GRASP Lab, Dept. of Computer and Information Science, University of Pennsylvania.
[18] 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.
[19] Z. Fuzhen, The Shur Complement and its Applications. Springer Pub., 2005.
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  • APA Style

    J. B. Allen. (2017). Wavelet-Based Numerical Homogenization for Scaled Solutions of Linear Matrix Equations. International Journal of Discrete Mathematics, 2(1), 10-16. https://doi.org/10.11648/j.dmath.20170201.13

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    ACS Style

    J. B. Allen. Wavelet-Based Numerical Homogenization for Scaled Solutions of Linear Matrix Equations. Int. J. Discrete Math. 2017, 2(1), 10-16. doi: 10.11648/j.dmath.20170201.13

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    AMA Style

    J. B. Allen. Wavelet-Based Numerical Homogenization for Scaled Solutions of Linear Matrix Equations. Int J Discrete Math. 2017;2(1):10-16. doi: 10.11648/j.dmath.20170201.13

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  • @article{10.11648/j.dmath.20170201.13,
      author = {J. B. Allen},
      title = {Wavelet-Based Numerical Homogenization for Scaled Solutions of Linear Matrix Equations},
      journal = {International Journal of Discrete Mathematics},
      volume = {2},
      number = {1},
      pages = {10-16},
      doi = {10.11648/j.dmath.20170201.13},
      url = {https://doi.org/10.11648/j.dmath.20170201.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170201.13},
      abstract = {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.},
     year = {2017}
    }
    

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  • TY  - JOUR
    T1  - Wavelet-Based Numerical Homogenization for Scaled Solutions of Linear Matrix Equations
    AU  - J. B. Allen
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    T2  - International Journal of Discrete Mathematics
    JF  - International Journal of Discrete Mathematics
    JO  - International Journal of Discrete Mathematics
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    EP  - 16
    PB  - Science Publishing Group
    SN  - 2578-9252
    UR  - https://doi.org/10.11648/j.dmath.20170201.13
    AB  - 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.
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Author Information
  • Information Technology Laboratory, U.S. Army Engineer Research and Development Center, Vicksburg, USA

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