A Model Reduction Framework with the Empirical Quadrature Procedure for High-dimensional Shape-parameterized Partial Differential Equations
Bibliographic record
Abstract
This thesis details a goal-oriented model reduction framework for parameterized nonlinear partial differential equations (PDEs), with an emphasis on shape-deformation problems with high-dimensional and non-affine parameter dependence. The framework builds on five technical ingredients: free-form deformation, which provides implicit geometry-independent shape deformations; reduced basis spaces, which provide rapidly convergent approximations of the parameterized solutions; the empirical quadrature procedure (EQP), which provides efficient hyper-reduction for nonlinear PDEs with non-affine parameter dependence; the dual-weighted residual (DWR) method, which provides a posteriori error estimation; and a greedy sampling method based on saturated parameter space assumptions, which provides efficient adaptive training in high-dimensional parameter spaces. This generalized framework enables automatic training of reduced order models that meets a user-prescribed error tolerance for a wide range of continuum mechanics problems. The versatility of this framework is demonstrated for many-query shape deformation problems in acoustics, aerodynamics, and hyperelasticity. The eventual application of this framework is to accelerate many-query problems such as shape optimization and uncertainty quantification.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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".