Parallel Domain Decomposition Strategies for Stochastic Elliptic Equations. Part A: Local Karhunen--Loève Representations
Bibliographic record
Abstract
This work presents a method to efficiently determine the dominant Karhunen--Loève (KL) modes of a random process with known covariance function. The truncated KL expansion is one of the most common techniques for the approximation of random processes, primarily because it is an optimal representation, in the mean squared error sense, with respect to the number of random variables in the representation. However, finding the KL expansion involves solving integral problems, which tends to be computationally demanding. This work addresses this issue by means of a work-subdivision strategy based on a domain decomposition approach, enabling the efficient computation of a possibly large number of dominant KL modes. Specifically, the computational domain is partitioned into smaller nonoverlapping subdomains, over which independent local KL decompositions are performed to generate local bases which are subsequently used to discretize the global modes over the entire domain. The latter are determined by means of a Galerkin projection. The procedure leads to the resolution of a reduced Galerkin problem, whose size is not related to the dimension of the underlying discretization space but is actually determined by the desired accuracy and the number of subdomains. It can also be easily implemented in parallel. Extensive numerical tests are used to validate the methodology and assess its serial and parallel performance. The resulting expansion is exploited in Part B to accelerate the solution of the stochastic partial differential equations using a Monte Carlo approach.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".