Statistical inverse inference for solving the first kind of nonlinear Fredholm integral equation
Bibliographic record
Abstract
Inverse problems, widely applied in geology, oceanography, and geophysics, are often ill-posed, presenting significant challenges. This study addresses the first kind of nonlinear Fredholm integral equation derived from gravity measurement and magnetic relief problems, which are highly sensitive to data perturbations. Conventional methods, such as Newton iteration with regularization, provide stable solutions but fail to account for data randomness, yielding only single approximations. We propose a Bayesian statistical inversion framework to overcome these limitations, transforming the problem into posterior distribution characterization. We estimate solutions and quantify their uncertainty using Markov chain Monte Carlo (MCMC) methods, specifically the preconditioned Crank–Nicolson (pCN) algorithm. Numerical simulations demonstrate the method's effectiveness in gravity and magnetic problems, offering a robust tool for uncertainty analysis in stochastic inverse problems.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.009 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".