Optimal Preconditioners for Hybrid Direct-Iterative $\mathcal {H}$-Matrix Solvers in Boundary Element Methods
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Bibliographic record
Abstract
The paper proposes a new approach to the fast solution of matrix equations resulting from boundary element discretization of integral equations. By hybridizing fast iterative <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}$</tex-math></inline-formula>-matrix solvers with a fast direct <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}$</tex-math></inline-formula>-matrix preconditioner factorization, we create a framework that can be tuned between the extremes of a direct solver and a unpreconditioned iterative solver. This tuning is largely achieved using a single numerical parameter representing the preconditioner tolerance. A more complicated scheme involving two different tolerances is also briefly considered. The proposed framework is demonstrated on a high-order accurate Locally Corrected Nyström solution of surface integral equations for PEC targets. Examples consider various scattering problems including those featuring strong physical resonances. We show that appropriately choosing the preconditioner tolerance achieves the prescribed solution accuracy with minimal CPU time. Expanding from one to two tolerance parameters further enhances the framework by providing the flexibility to dynamically adjust tolerance, enabling higher compression while maintaining accuracy and fast convergence. This adaptive strategy offers significant potential for optimizing the balance between memory usage and CPU time in the future.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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