Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm
Abstract
Algebraic multigrid (AMG) coarse spaces are commonly constructed so that they exhibit the so-called weak approximation property (WAP) which is a necessary and sufficient condition for uniform two-grid convergence. Here, this paper studies a modification of such coarse spaces so that the modified ones provide approximation in energy norm. Our modification is based on the projection in energy norm onto an orthogonal complement of original coarse space. This generally leads to dense modified coarse space matrices, which is hence computationally infeasible. To remedy this, based on the fact that the projection involves inverse of a well-conditioned matrix, we use polynomials to approximate the projection and, therefore, obtain a practical, sparse modified coarse matrix and prove that the modified coarse space maintains computationally feasible approximation in energy norm. We present some numerical results for both PDE discretization matrices as well as graph Laplacian ones, which are in accordance with our theoretical results.
- Authors:
-
- Tufts Univ., Medford, MA (United States). Dept. of Mathematics
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing; Portland State Univ., OR (United States). Dept. of Mathematics
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
- OSTI Identifier:
- 1669216
- Report Number(s):
- LLNL-JRNL-744422
Journal ID: ISSN 0895-4798; 899470
- Grant/Contract Number:
- AC52-07NA27344; DMS-1619640; DMS-1620063
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Matrix Analysis and Applications
- Additional Journal Information:
- Journal Volume: 40; Journal Issue: 3; Journal ID: ISSN 0895-4798
- Publisher:
- SIAM
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; AMG; strong approximation property; weak approximation property
Citation Formats
Hu, Xiaozhe, and Vassilevski, Panayot S. Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm. United States: N. p., 2019.
Web. doi:10.1137/18m1165190.
Hu, Xiaozhe, & Vassilevski, Panayot S. Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm. United States. https://doi.org/10.1137/18m1165190
Hu, Xiaozhe, and Vassilevski, Panayot S. Tue .
"Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm". United States. https://doi.org/10.1137/18m1165190. https://www.osti.gov/servlets/purl/1669216.
@article{osti_1669216,
title = {Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm},
author = {Hu, Xiaozhe and Vassilevski, Panayot S.},
abstractNote = {Algebraic multigrid (AMG) coarse spaces are commonly constructed so that they exhibit the so-called weak approximation property (WAP) which is a necessary and sufficient condition for uniform two-grid convergence. Here, this paper studies a modification of such coarse spaces so that the modified ones provide approximation in energy norm. Our modification is based on the projection in energy norm onto an orthogonal complement of original coarse space. This generally leads to dense modified coarse space matrices, which is hence computationally infeasible. To remedy this, based on the fact that the projection involves inverse of a well-conditioned matrix, we use polynomials to approximate the projection and, therefore, obtain a practical, sparse modified coarse matrix and prove that the modified coarse space maintains computationally feasible approximation in energy norm. We present some numerical results for both PDE discretization matrices as well as graph Laplacian ones, which are in accordance with our theoretical results.},
doi = {10.1137/18m1165190},
journal = {SIAM Journal on Matrix Analysis and Applications},
number = 3,
volume = 40,
place = {United States},
year = {Tue Sep 24 00:00:00 EDT 2019},
month = {Tue Sep 24 00:00:00 EDT 2019}
}
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