Learning optimal multigrid smoothers via neural networks Article Swipe
Ru Huang
,
Ruipeng Li
,
Yuanzhe Xi
·
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2102.12071
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2102.12071
Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). The CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories, and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.
Related Topics
Concepts
Multigrid method
Smoothing
Partial differential equation
Computer science
Mathematical optimization
Convolutional neural network
Artificial neural network
Relaxation (psychology)
Convergence (economics)
Laplace operator
Applied mathematics
Algorithm
Mathematics
Artificial intelligence
Mathematical analysis
Social psychology
Computer vision
Economic growth
Psychology
Economics
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2102.12071
- https://arxiv.org/pdf/2102.12071
- OA Status
- green
- Cited By
- 3
- References
- 36
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W3131157216
All OpenAlex metadata
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https://openalex.org/W3131157216Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2102.12071Digital Object Identifier
- Title
-
Learning optimal multigrid smoothers via neural networksWork title
- Type
-
preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2021Year of publication
- Publication date
-
2021-02-24Full publication date if available
- Authors
-
Ru Huang, Ruipeng Li, Yuanzhe XiList of authors in order
- Landing page
-
https://arxiv.org/abs/2102.12071Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2102.12071Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2102.12071Direct OA link when available
- Concepts
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Multigrid method, Smoothing, Partial differential equation, Computer science, Mathematical optimization, Convolutional neural network, Artificial neural network, Relaxation (psychology), Convergence (economics), Laplace operator, Applied mathematics, Algorithm, Mathematics, Artificial intelligence, Mathematical analysis, Social psychology, Computer vision, Economic growth, Psychology, EconomicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
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3Total citation count in OpenAlex
- Citations by year (recent)
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2025: 1, 2021: 2Per-year citation counts (last 5 years)
- References (count)
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36Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
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| publication_date | 2021-02-24 |
| publication_year | 2021 |
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| referenced_works_count | 36 |
| abstract_inverted_index.a | 91, 98 |
| abstract_inverted_index.In | 59 |
| abstract_inverted_index.an | 64 |
| abstract_inverted_index.at | 37 |
| abstract_inverted_index.be | 109 |
| abstract_inverted_index.in | 75 |
| abstract_inverted_index.is | 33, 50 |
| abstract_inverted_index.of | 4, 27, 31, 78, 94, 114, 118 |
| abstract_inverted_index.on | 41, 87, 97, 122 |
| abstract_inverted_index.to | 111 |
| abstract_inverted_index.we | 62 |
| abstract_inverted_index.One | 26 |
| abstract_inverted_index.The | 83 |
| abstract_inverted_index.and | 19, 52, 107, 131 |
| abstract_inverted_index.are | 2, 85 |
| abstract_inverted_index.can | 53, 108 |
| abstract_inverted_index.for | 9, 56, 68 |
| abstract_inverted_index.key | 29 |
| abstract_inverted_index.one | 3 |
| abstract_inverted_index.the | 5, 28, 76, 115 |
| abstract_inverted_index.CNNs | 84 |
| abstract_inverted_index.PDEs | 95 |
| abstract_inverted_index.aims | 36 |
| abstract_inverted_index.each | 42 |
| abstract_inverted_index.form | 77 |
| abstract_inverted_index.from | 14, 22, 72, 90, 103 |
| abstract_inverted_index.grid | 43 |
| abstract_inverted_index.loss | 100 |
| abstract_inverted_index.many | 57 |
| abstract_inverted_index.most | 6 |
| abstract_inverted_index.same | 116 |
| abstract_inverted_index.this | 60 |
| abstract_inverted_index.time | 133 |
| abstract_inverted_index.type | 93 |
| abstract_inverted_index.with | 135 |
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| abstract_inverted_index.based | 96 |
| abstract_inverted_index.class | 117 |
| abstract_inverted_index.given | 92 |
| abstract_inverted_index.graph | 20 |
| abstract_inverted_index.rates | 130 |
| abstract_inverted_index.which | 35 |
| abstract_inverted_index.(PDEs) | 18 |
| abstract_inverted_index.errors | 40 |
| abstract_inverted_index.impose | 54 |
| abstract_inverted_index.level. | 44 |
| abstract_inverted_index.linear | 11 |
| abstract_inverted_index.neural | 80 |
| abstract_inverted_index.paper, | 61 |
| abstract_inverted_index.(CNNs). | 82 |
| abstract_inverted_index.Partial | 15 |
| abstract_inverted_index.applied | 110 |
| abstract_inverted_index.arising | 13 |
| abstract_inverted_index.derived | 102 |
| abstract_inverted_index.finding | 46 |
| abstract_inverted_index.machine | 23 |
| abstract_inverted_index.methods | 1 |
| abstract_inverted_index.optimal | 47 |
| abstract_inverted_index.propose | 63 |
| abstract_inverted_index.results | 121 |
| abstract_inverted_index.rotated | 124 |
| abstract_inverted_index.solving | 10 |
| abstract_inverted_index.systems | 12 |
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| abstract_inverted_index.However, | 45 |
| abstract_inverted_index.adaptive | 66 |
| abstract_inverted_index.compared | 134 |
| abstract_inverted_index.function | 101 |
| abstract_inverted_index.improved | 128 |
| abstract_inverted_index.learning | 24, 69 |
| abstract_inverted_index.methods. | 139 |
| abstract_inverted_index.networks | 81 |
| abstract_inverted_index.operator | 73 |
| abstract_inverted_index.problems | 89, 113, 126 |
| abstract_inverted_index.reducing | 38 |
| abstract_inverted_index.solution | 132 |
| abstract_inverted_index.stencils | 74 |
| abstract_inverted_index.Equations | 17 |
| abstract_inverted_index.Laplacian | 125 |
| abstract_inverted_index.Multigrid | 0 |
| abstract_inverted_index.Numerical | 120 |
| abstract_inverted_index.classical | 136 |
| abstract_inverted_index.efficient | 7, 65 |
| abstract_inverted_index.framework | 67 |
| abstract_inverted_index.multigrid | 32, 104 |
| abstract_inverted_index.optimized | 70 |
| abstract_inverted_index.problems. | 58 |
| abstract_inverted_index.smoothers | 71 |
| abstract_inverted_index.smoothing | 48 |
| abstract_inverted_index.theories, | 106 |
| abstract_inverted_index.Laplacians | 21 |
| abstract_inverted_index.algorithms | 49 |
| abstract_inverted_index.challenges | 55 |
| abstract_inverted_index.components | 30 |
| abstract_inverted_index.relaxation | 138 |
| abstract_inverted_index.smoothing, | 34 |
| abstract_inverted_index.supervised | 99 |
| abstract_inverted_index.techniques | 8 |
| abstract_inverted_index.anisotropic | 123 |
| abstract_inverted_index.convergence | 105, 129 |
| abstract_inverted_index.demonstrate | 127 |
| abstract_inverted_index.large-scale | 112 |
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| abstract_inverted_index.Differential | 16 |
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| abstract_inverted_index.applications. | 25 |
| abstract_inverted_index.convolutional | 79 |
| abstract_inverted_index.high-frequency | 39 |
| abstract_inverted_index.problem-dependent | 51 |
| cited_by_percentile_year | |
| countries_distinct_count | 1 |
| institutions_distinct_count | 3 |
| citation_normalized_percentile |