Quasi-Monte Carlo method for solving Fredholm equations
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>d</m:mi> <m:mo>)</m:mo> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {z^{(d)}(x)} of d initial terms of the von Neumann series estimating the solution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>z</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {z(x)} of the equation is represented as a d -dimensional integral over the unit cube <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mi>d</m:mi> </m:msub> </m:math> {H_{d}} . This note presents three examples calculating <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>d</m:mi> <m:mo>)</m:mo> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {z^{(d)}(x)} for different kernels with norms <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>∥</m:mo> <m:mi>K</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {\lVert K\rVert<1} . We found that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>d</m:mi> <m:mo>)</m:mo> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∥</m:mo> <m:mi>K</m:mi> <m:mo>∥</m:mo> </m:mrow> </m:math> {\lVert K\rVert} values. We also found that the average dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover> <m:mi>d</m:mi> <m:mo>^</m:mo> </m:mover> </m:math> {\hat{d}} of the integrand in all our examples is small, less than 3. We suggest that the average dimensions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover> <m:mi>d</m:mi> <m:mo>^</m:mo> </m:mover> </m:math> {\hat{d}} of our d -dimensional integrands are bounded as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>→</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:math> {d\to\infty} .
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 it