Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
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Bibliographic record
Abstract
Abstract This paper analyses the following question: let A j , j = 1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations ∇⋅ ( A j ∇ u j ) + k 2 n j u j = − f . How small must $\|A_{1} -A_{2}\|_{L^{q}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∥</mml:mo> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>∥</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> </mml:math> and $\|{n_{1}} - {n_{2}}\|_{L^{q}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∥</mml:mo> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>∥</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> </mml:math> be (in terms of k -dependence) for GMRES applied to either $(\mathbf {A}_1)^{-1}\mathbf {A}_2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> or A 2 ( A 1 ) − 1 to converge in a k -independent number of iterations for arbitrarily large k ? (In other words, for A 1 to be a good left or right preconditioner for A 2 ?) We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients A and n . Such a calculation may require the solution of many deterministic Helmholtz problems, each with different A and n , and the answer to the question above dictates to what extent a previously calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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