Scalable Domain Decomposition Methods for Nonlinear and Time-Dependent Stochastic Systems
Bibliographic record
Abstract
Computational modelling is one of the most important tools to understand and predict real-life physical processes. However, the accuracy of their predictions becomes questionable when the uncertainties associated with model parameters, assumptions to the mathematical models and the noise in experimental data are not properly accounted for. Sampling-based approaches to handle these uncertainties become overwhelmingly expensive for large-scale models with high resolution discretizations in space/time. This thesis proposes a sampling-free intrusive stochastic Galerkin-based approach to handle the uncertainties associated with model parameters for time-dependent and nonlinear problems. The increased cost of solving high resolution models using this sampling-free approach is handled using domain decomposition (DD)-based solvers by efficiently distributing the workload to many processes. Developing parallel scalable iterative solvers for uncertainty quantification of these high-resolution models in high performance computing (HPC) environments is the main objective of this thesis. An acoustic wave propagation model with a random field representation of wave speed is handled using a non-overlapping DD method. The symmetric and positive-definite coefficient matrix of the system can be solved using a conjugate-gradient iterative method and associated Neumann-Neumann vertex-based preconditioner in two dimensions. However, the complex spatial coupling and the coupling among the stochastic expansion coefficients can affect the scalabilities of the solver in three dimensions. Hence, a wirebasket-based preconditioner is utilized to enrich the coarse grid allowing better global error propagation and improved scalability for the elastic wave propagation model. For nonlinear stochastic partial differential equations (PDEs), the coefficient matrix of the associated linearized algebraic system is non-symmetric which requires the use of generalized minimum residual (GMRES) method-based iterative solvers. A multilevel Schwarz preconditioner combining DD and algebraic multigrid method is proposed for efficient error reduction for large-scale models. The scalabilities of the proposed solver are demonstrated for the PDE-based compartmental model of the geospatial spread of COVID-19 considering uncertain population movement in a large geographical domain of Southern Ontario.
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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) | 0.000 | 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 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".