Primal and dual-primal iterative substructuring methods of stochastic PDEs
Bibliographic record
Abstract
A novel non-overlapping domain decomposition method is proposed to solve the large-scale linear system arising from the finite element discretization of stochastic partial differential equations (SPDEs). The methodology is based on a Schur complement based geometric decomposition and an orthogonal decomposition and projection of the stochastic processes using Polynomial Chaos expansion. The algorithm offers a direct approach to formulate a two-level scalable preconditioner. The proposed preconditioner strictly enforces the continuity condition on the corner nodes of the interface boundary, while weakly satisfying the continuity condition over the remaining interface nodes. This approach relates to a primal version of an iterative substructuring method. Next, a Lagrange multiplier based dual-primal domain decomposition method is introduced in the context of SPDEs. In the dual-primal method the continuity condition on the corner nodes is strictly satisfied while Lagrange multipliers are used to enforce continuity on the remaining part of the interface boundary. For numerical illustrations, a two dimensional elliptic SPDE with non-Gaussian random coefficients is considered. The numerical results demonstrate the scalability of these algorithms with respect to the mesh size, subdomain size, fixed problem size per subdomain, order of Polynomial Chaos expansion and level of uncertainty in the input parameters. The numerical experiments are performed on a Linux cluster using MPI and PETSc libraries.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 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) | 0.002 | 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 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".