Hierarchical matrix arithmetic with accumulated updates
Steffen B\"orm

TL;DR
This paper introduces a new hierarchical matrix algorithm that reduces the number of low-rank updates, significantly decreasing preconditioner setup time for large-scale PDE and integral equation problems.
Contribution
A novel algorithm that minimizes low-rank updates in hierarchical matrix computations, improving efficiency for 3D problem preconditioners.
Findings
Setup time reduced by 50% or more
Fewer low-rank updates needed in the algorithm
Enhanced efficiency for large-scale 3D problems
Abstract
Hierarchical matrices can be used to construct efficient preconditioners for partial differential and integral equations by taking advantage of low-rank structures in triangular factorizations and inverses of the corresponding stiffness matrices. The setup phase of these preconditioners relies heavily on low-rank updates that are responsible for a large part of the algorithm's total run-time, particularly for matrices resulting from three-dimensional problems. This article presents a new algorithm that significantly reduces the number of low-rank updates and can reduce the setup time by 50 percent or more.
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Taxonomy
TopicsElectromagnetic Scattering and Analysis · Matrix Theory and Algorithms · Electromagnetic Simulation and Numerical Methods
