A Multigrid Method for Adaptive Sparse Grids
Sparse grids have become an important tool to reduce the number of degrees of freedom of discretizations of moderately high-dimensional partial differential equations; however, the reduction in degrees of freedom comes at the cost of an almost dense and unconventionally structured system of linear e...
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| Format: | Article |
| Language: | en_US |
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Society for Industrial and Applied Mathematics
2016
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| Online Access: | http://hdl.handle.net/1721.1/100938 https://orcid.org/0000-0002-5045-046X |
| _version_ | 1826215596129255424 |
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| author | Peherstorfer, Benjamin Zimmer, Stefan Zenger, Christoph Bungartz, Hans-Joachim |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Peherstorfer, Benjamin Zimmer, Stefan Zenger, Christoph Bungartz, Hans-Joachim |
| author_sort | Peherstorfer, Benjamin |
| collection | MIT |
| description | Sparse grids have become an important tool to reduce the number of degrees of freedom of discretizations of moderately high-dimensional partial differential equations; however, the reduction in degrees of freedom comes at the cost of an almost dense and unconventionally structured system of linear equations. To guarantee overall efficiency of the sparse grid approach, special linear solvers are required. We present a multigrid method that exploits the sparse grid structure to achieve an optimal runtime that scales linearly with the number of sparse grid points. Our approach is based on a novel decomposition of the right-hand sides of the coarse grid equations that leads to a reformulation in so-called auxiliary coefficients. With these auxiliary coefficients, the right-hand sides can be represented in a nodal point basis on low-dimensional full grids. Our proposed multigrid method directly operates in this auxiliary coefficient representation, circumventing most of the computationally cumbersome sparse grid structure. Numerical results on nonadaptive and spatially adaptive sparse grids confirm that the runtime of our method scales linearly with the number of sparse grid points and they indicate that the obtained convergence factors are bounded independently of the mesh width. |
| first_indexed | 2024-09-23T16:36:24Z |
| format | Article |
| id | mit-1721.1/100938 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:36:24Z |
| publishDate | 2016 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/1009382022-09-29T20:17:32Z A Multigrid Method for Adaptive Sparse Grids Peherstorfer, Benjamin Zimmer, Stefan Zenger, Christoph Bungartz, Hans-Joachim Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Peherstorfer, Benjamin Sparse grids have become an important tool to reduce the number of degrees of freedom of discretizations of moderately high-dimensional partial differential equations; however, the reduction in degrees of freedom comes at the cost of an almost dense and unconventionally structured system of linear equations. To guarantee overall efficiency of the sparse grid approach, special linear solvers are required. We present a multigrid method that exploits the sparse grid structure to achieve an optimal runtime that scales linearly with the number of sparse grid points. Our approach is based on a novel decomposition of the right-hand sides of the coarse grid equations that leads to a reformulation in so-called auxiliary coefficients. With these auxiliary coefficients, the right-hand sides can be represented in a nodal point basis on low-dimensional full grids. Our proposed multigrid method directly operates in this auxiliary coefficient representation, circumventing most of the computationally cumbersome sparse grid structure. Numerical results on nonadaptive and spatially adaptive sparse grids confirm that the runtime of our method scales linearly with the number of sparse grid points and they indicate that the obtained convergence factors are bounded independently of the mesh width. 2016-01-20T01:44:58Z 2016-01-20T01:44:58Z 2015-10 2014-12 Article http://purl.org/eprint/type/JournalArticle 1064-8275 1095-7197 http://hdl.handle.net/1721.1/100938 Peherstorfer, Benjamin, Stefan Zimmer, Christoph Zenger, and Hans-Joachim Bungartz. “A Multigrid Method for Adaptive Sparse Grids.” SIAM Journal on Scientific Computing 37, no. 5 (January 2015): S51–S70. © 2015 Society for Industrial and Applied Mathematics https://orcid.org/0000-0002-5045-046X en_US http://dx.doi.org/10.1137/140974985 SIAM Journal on Scientific Computing Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society for Industrial and Applied Mathematics Society for Industrial and Applied Mathematics |
| spellingShingle | Peherstorfer, Benjamin Zimmer, Stefan Zenger, Christoph Bungartz, Hans-Joachim A Multigrid Method for Adaptive Sparse Grids |
| title | A Multigrid Method for Adaptive Sparse Grids |
| title_full | A Multigrid Method for Adaptive Sparse Grids |
| title_fullStr | A Multigrid Method for Adaptive Sparse Grids |
| title_full_unstemmed | A Multigrid Method for Adaptive Sparse Grids |
| title_short | A Multigrid Method for Adaptive Sparse Grids |
| title_sort | multigrid method for adaptive sparse grids |
| url | http://hdl.handle.net/1721.1/100938 https://orcid.org/0000-0002-5045-046X |
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