Quasi-Monte Carlo method for solving Fredholm equations
Bibliographic record
Abstract
Abstract A Monte Carlo method used for the estimation of convergent von Neumann series solutions of a Fredholm equation of second kind is considered. The sum z ( d ) ( x ) {z^{(d)}(x)} of d initial terms of the von Neumann series estimating the solution z ( x ) {z(x)} of the equation is represented as a d-dimensional integral over the unit cube H d {H_{d}} . This note presents three examples calculating z ( d ) ( x ) {z^{(d)}(x)} for different kernels with norms ∥ K ∥ < 1 {\lVert K\rVert<1} . We found that z ( d ) ( x ) {z^{(d)}(x)} calculated using a quasi-Monte Carlo (QMC) method converges significantly faster than the corresponding Monte Carlo (MC) estimates in the entire range of ∥ K ∥ {\lVert K\rVert} values. We also found that the average dimension d ^ {\hat{d}} of the integrand in all our examples is small, less than 3. We suggest that the average dimensions d ^ {\hat{d}} of our d-dimensional integrands are bounded as d → ∞ {d\to\infty} .
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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 | 0.004 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.002 | 0.001 |
| 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".