On the Spectral Clustering of a Class of Multigrid Preconditioners Article Swipe
Lorca, José Pablo Lucero
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2511.12298
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2511.12298
This paper studies a common two-level multigrid construction for block-structured linear systems and identifies a simple way to describe how its smoothing and coarse-grid components interact. By examining the method through a collection of small coupling modes, we show that its behavior can be captured by a single scalar quantity for each mode. The main result is an explicit choice of smoothing parameters that makes all modes respond in the same way, causing the nontrivial eigenvalues of the preconditioned operator to collapse to a single value. This gives a clear and self-contained description of the ideal version of the method and provides a concrete target for designing related schemes. Although the exact spectral collapse requires ideal components, we also show that the same construction naturally produces operators that resemble those used in practical discretizations. Examples from finite-difference and discontinuous Galerkin settings illustrate how the ideal parameters can be used in practice.
Related Topics
Concepts
Smoothing
Mathematics
Ideal (ethics)
Applied mathematics
Eigenvalues and eigenvectors
Operator (biology)
Multigrid method
Scalar (mathematics)
Linear system
Simple (philosophy)
Cluster analysis
Class (philosophy)
Mathematical optimization
Coupling (piping)
Algorithm
Spectrum (functional analysis)
Spectral clustering
Computer science
Discontinuous Galerkin method
Logarithm
Metadata
- Type
- preprint
- Landing Page
- https://doi.org/10.48550/arxiv.2511.12298
- OA Status
- green
- OpenAlex ID
- https://openalex.org/W7106014405
All OpenAlex metadata
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https://openalex.org/W7106014405Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2511.12298Digital Object Identifier
- Title
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On the Spectral Clustering of a Class of Multigrid PreconditionersWork title
- Type
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preprintOpenAlex work type
- Publication year
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2025Year of publication
- Publication date
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2025-11-15Full publication date if available
- Authors
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Lorca, José Pablo LuceroList of authors in order
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https://doi.org/10.48550/arxiv.2511.12298Publisher landing page
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://doi.org/10.48550/arxiv.2511.12298Direct OA link when available
- Concepts
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Smoothing, Mathematics, Ideal (ethics), Applied mathematics, Eigenvalues and eigenvectors, Operator (biology), Multigrid method, Scalar (mathematics), Linear system, Simple (philosophy), Cluster analysis, Class (philosophy), Mathematical optimization, Coupling (piping), Algorithm, Spectrum (functional analysis), Spectral clustering, Computer science, Discontinuous Galerkin method, LogarithmTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
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