A Reduced Basis Method for Coercive Equations with an Exact Solution Certificate and Spatio-Parameter Adaptivity: Energy-Norm and Output Error Bounds
Bibliographic record
Abstract
We develop a reduced basis method for linear coercive parametrized partial differential equations (PDEs) with two objectives: providing an energy-norm or functional-output a posteriori error bound with respect to the exact weak solution of the PDE as opposed to the typical finite element “truth” solution; providing reliable and efficient construction of a reduced basis model through automatic adaptivity in both physical and parameter spaces. Our error bounds build on two key ingredients. The first is a minimum-residual mixed formulation which provides an approximate solution as well as an upper bound of the dual norm of the residual with respect to the infinite-dimensional function space. The second is an extension of the successive constraint method (SCM) to evaluate a lower bound of the stability constant with respect to the infinite-dimensional function space; the approach builds on a computable lower bound of the minimum eigenvalue associated with the stability constant. Both the minimum-residual mixed formulation and the extended SCM admit offline-online computational decomposition. The offline stage incorporates spatial mesh adaptation and greedy parameter sampling for both the solution approximation and the stability eigenproblem to yield a reliable online system in an efficient manner. The online stage provides an approximate solution and an a posteriori error bound with respect to the exact solution for any parameter value in complexity independent of the size of the finite element spaces. We demonstrate the effectiveness of the approach for a thermal block problem, which exhibits parameter-dependent spatial singularities.
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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 | 0.001 | 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".