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arXiv · 2501.17656

H2-MG: A multigrid method for hierarchical rank structured matrices

Daria Sushnikova·George Turkiyyah·Edmond Chow·David Keyes
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Abstract

This paper presents a new fast iterative solver for large systems involving kernel matrices. Advantageous aspects of H2 matrix approximations and the multigrid method are hybridized to create the H2-MG algorithm. This combination provides the time and memory efficiency of H2 operator representation along with the rapid convergence of a multilevel method. We describe how H2-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 H2 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 H2-MG a valuable addition to the collection of H2 solvers. The algorithm has potential for advancing various fields that require the solution of large, dense, symmetric positive definite matrices.

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Daria Sushnikova, George Turkiyyah, Edmond Chow, David Keyes. 2025-01-29. H2-MG: A multigrid method for hierarchical rank structured matrices. https://arxiv.org/abs/2501.17656

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