Adjoint-Based Enforcement of State Constraints in PDE Optimization Problems
Bibliographic record
Abstract
This study demonstrates how the adjoint-based framework traditionally used to compute gradients in PDE optimization problems can be extended to handle general constraints on the state variables. This is accomplished by constructing a projection of the gradient of the objective functional onto a subspace tangent to the manifold defined by the constraint. This projection is realized by solving an adjoint problem defined in terms of the same adjoint operator as used in the system employed to determine the gradient, but with a different forcing. We focus on the "optimize-then-discretize" paradigm in the infinite-dimensional setting where the required regularity of both the gradient and of the projection is ensured. The proposed approach is illustrated with two examples: a simple test problem describing optimization of heat transfer in one direction and a more involved problem where an optimal closure is found for a turbulent flow described by the Navier-Stokes system in two dimensions, both considered subject to different state constraints. The accuracy of the gradients and projections computed by solving suitable adjoint systems is carefully verified and the presented computational results show that the solutions of the optimization problems obtained with the proposed approach satisfy the state constraints with a good accuracy, although not exactly.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| 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 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".