Optimal Preconditioners for Hybrid Direct-Iterative H-Matrix Solvers in Boundary Element Methods
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 H-matrix solvers with a fast direct H-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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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".