Boundary control for inverse Cauchy problems of the Laplace equations
Bibliographic record
Abstract
Abstract: The method of fundamental solu-tions is coupled with the boundary control tech-nique to solve the Cauchy problems of theLaplaceEquations. Themainideaoftheproposedmethod is to solve a sequence of direct problemsinsteadofsolvingtheinverse problemdirectly. Inparticular,weusea boundarycontroltechniquetoobtain an approximation of the missing Dirichletboundary data; the Tikhonov regularization tech-nique and the L-curve method are employed toachieve such goal stably. Once the boundarydataon the whole boundary are known, the numericalsolution to the Cauchy problem can be obtainedby solving a direct problem. Numerical exam-plesare providedfor verificationsoftheproposedmethod on the steady-state heat conductionprob-lems. Keyword: Method of fundamental solution,methodofparticularsolution,collocationmethod,Tikhonovregularization,L-curve. 1 Introduction The Cauchy problem for an elliptic equation isa typical ill-posed problem whose solution doesnot depend continuously on the boundary data.That is, a small error in the specified data mayresult in an enormous error in the numerical so-lution. This problem appears in many applica-tions for example in the cardiography, the non-destructive testing, and etc. Stable and efficientnumerical methods are of highimportance. How-ever, it is well-known that the Cauchy problemfor an elliptic equation is ill-posed without any
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".