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\({\mathcal{H}^{2}}\)-MG: A Multigrid Method for Hierarchical Rank Structured Matrices.

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Title: \({\mathcal{H}^{2}}\)-MG: A Multigrid Method for Hierarchical Rank Structured Matrices.
Authors: Sushnikova, Daria1 (AUTHOR) daria.sushnikova@gmail.com, Turkiyyah, George1 (AUTHOR) George.Turkiyyah@kaust.edu.sa, Chow, Edmond2 (AUTHOR) echow@cc.gatech.edu, Keyes, David1 (AUTHOR) david.keyes@kaust.edu.sa
Source: SIAM Journal on Scientific Computing. 2026, Vol. 48 Issue 1, pA286-A308. 23p.
Subjects: Multigrid methods (Numerical analysis), Iterative methods (Mathematics), Integral operators, Reproducible research, Boundary element methods, Numerical solutions for linear algebra
Abstract: This paper presents a new fast iterative solver for large systems involving kernel matrices. Advantageous aspects of \(\mathcal{H}^{2}\) matrix approximations and the multigrid method are hybridized to create the \(\mathcal{H}^{2}\) -MG algorithm. This combination provides the time and memory efficiency of \(\mathcal{H}^{2}\) operator representation along with the rapid convergence of a multilevel method. We describe how \(\mathcal{H}^{2}\) -MG works, show its linear complexity, and demonstrate its effectiveness on two standard kernels and on a single-layer potential boundary element discretization with complex geometry. The current zoo of \(\mathcal{H}^{2}\) solvers, which includes a wide variety of iterative and direct solvers, so far lacks a method that exploits multiple levels of resolution, commonly referred to in the iterative methods literature as "multigrid" from its origins in a hierarchy of grids used to discretize differential equations. This makes \(\mathcal{H}^{2}\) -MG a valuable addition to the collection of \(\mathcal{H}^{2}\) solvers. The algorithm has the potential for advancing various fields that require the solution of large, dense, symmetric positive definite matrices. Reproducibility of computational results. This paper has been awarded the "SIAM Reproducibility Badge: Code and data available" as recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at and in the supplementary materials ( [269KB]). [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: \({\mathcal{H}^{2}}\)-MG: A Multigrid Method for Hierarchical Rank Structured Matrices.
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  Data: This paper presents a new fast iterative solver for large systems involving kernel matrices. Advantageous aspects of \(\mathcal{H}^{2}\) matrix approximations and the multigrid method are hybridized to create the \(\mathcal{H}^{2}\) -MG algorithm. This combination provides the time and memory efficiency of \(\mathcal{H}^{2}\) operator representation along with the rapid convergence of a multilevel method. We describe how \(\mathcal{H}^{2}\) -MG works, show its linear complexity, and demonstrate its effectiveness on two standard kernels and on a single-layer potential boundary element discretization with complex geometry. The current zoo of \(\mathcal{H}^{2}\) solvers, which includes a wide variety of iterative and direct solvers, so far lacks a method that exploits multiple levels of resolution, commonly referred to in the iterative methods literature as "multigrid" from its origins in a hierarchy of grids used to discretize differential equations. This makes \(\mathcal{H}^{2}\) -MG a valuable addition to the collection of \(\mathcal{H}^{2}\) solvers. The algorithm has the potential for advancing various fields that require the solution of large, dense, symmetric positive definite matrices. Reproducibility of computational results. This paper has been awarded the "SIAM Reproducibility Badge: Code and data available" as recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at and in the supplementary materials ( [269KB]). [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1137/24M171944X
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      – SubjectFull: Iterative methods (Mathematics)
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      – SubjectFull: Integral operators
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      – SubjectFull: Reproducible research
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      – SubjectFull: Boundary element methods
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      – SubjectFull: Numerical solutions for linear algebra
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      – TitleFull: \({\mathcal{H}^{2}}\)-MG: A Multigrid Method for Hierarchical Rank Structured Matrices.
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              Text: 2026
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