Efficient Hyperreduction for Large‐Scale Problems: Exploiting Reducible Constraint Manifolds in Empirical Quadrature Procedure
Bibliographic record
Abstract
ABSTRACT We develop efficient hyperreduction methods for projection‐based model reduction of nonlinear partial differential equations (PDEs) with a large number of parameters and/or large parametric extents. Our formulation is based on the empirical quadrature procedure (EQP), which solves an optimization problem that involves “residual‐matching constraints” over a training parameter set to find a sparse quadrature rule that yields rapid yet accurate approximations of the PDE residual, and solves the constrained optimization via non‐negative least squares (NNLS). Specifically, we extend the EQP and NNLS to provide more efficient offline training for problems that (i) demand tight hyperreduction tolerances, (ii) involve a large number of residual‐matching constraints, and/or (iii) involve a high‐dimensional parameter space. To address (i), we develop second‐order accurate constraints for EQP and a rounding‐error stable NNLS formulation that efficiently provides a solution to the optimization problem with a tight tolerance. To address (ii), we develop NNLS with constraint reduction (NNLS‐CR), which exploits the fact that many constraints are often redundant and systematically constructs a reduced orthogonal set of constraints that still represents all the original constraints. To address (iii), we introduce an EQP method that adaptively constructs the training parameter set and solves the associated constrained optimization problem using a version of NNLS‐CR that admits incremental constraint update. We demonstrate the offline efficiency of the methods, as well as the parametric robustness of the resulting ROMs, using parameterized Navier–Stokes and Reynolds‐averaged Navier–Stokes equations in four different contexts: Shape parameter sweep; flight parameter sweep; ensemble‐based data assimilation; and forward 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 | 0.004 |
| 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 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".