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), 2024. Published by Science Publishing Group |
Numerical Homogenization, Multiscale, Multiresolution, Wavelets
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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
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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Download@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}
}
TY - JOUR T1 - Wavelet-Based Numerical Homogenization for Scaled Solutions of Linear Matrix Equations AU - J. B. Allen Y1 - 2017/02/20 PY - 2017 N1 - https://doi.org/10.11648/j.dmath.20170201.13 DO - 10.11648/j.dmath.20170201.13 T2 - International Journal of Discrete Mathematics JF - International Journal of Discrete Mathematics JO - International Journal of Discrete Mathematics SP - 10 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. VL - 2 IS - 1 ER -
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