Arnoldi Preconditioning for Solving Large Linear Biomedical Systems
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Bibliographic record
Abstract
Simulations of biomedical systems often involve solving large, sparse, linear systems of the form Ax = b. In initial value problems, this system is solved at every time step, so a quick solution is essential for tractability. Iterative solvers, especially preconditioned conjugate gradient, are attractive since memory demands are minimized compared to direct methods, albeit at a cost of solution speed. A proper preconditioner can drastically reduce computation and remains an area of active research. In this paper, we propose a novel preconditioner based on system order reduction using the Arnoldi method. Systems of orders up to a million, generated from a finite element method formulation of the elliptic portion of the bidomain equations, are solved with the new preconditioner and performance is compared with that of other preconditioners. Results indicate that the new method converges considerably faster, often within a single iteration. It also uses less memory than an incomplete LU decomposition (ILU). For solving a system repeatedly, the Arnoldi transformation must be continually recomputed, unlike ILU, but this can be done quickly. In conclusion, for solving a system once, the Arnoldi preconditioner offers a greatly reduced solution time, and for repeated solves, will still be faster than an ILU preconditioner.
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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) | 0.000 | 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 | 0.000 |
| 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)
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 it