On Discrete Least-Squares Projection in Unbounded Domain with Random Evaluations and its Application to Parametric Uncertainty Quantification
Bibliographic record
Abstract
This work is concerned with approximating multivariate functions in an unbounded domain by using a discrete least-squares projection with random point evaluations. Particular attention is given to functions with random Gaussian or gamma parameters. We first demonstrate that the traditional Hermite (Laguerre) polynomials chaos expansion suffers from the instability in the sense that an unfeasible number of points, which is relevant to the dimension of the approximation space, is needed to guarantee the stability in the least-squares framework. We then propose to use the Hermite/Laguerre functions (rather than polynomials) as bases in the expansion. The corresponding design points are obtained by mapping the uniformly distributed random points in bounded intervals to the unbounded domain, which involved a mapping parameter $L$. By using the Hermite/Laguerre functions and a proper mapping parameter, the stability can be significantly improved even if the number of design points scales linearly (up to a logarithmic factor) with the dimension of the approximation space. Apart from the stability, another important issue is the rate of convergence. To speed up the convergence, an effective scaling factor is introduced, and a principle for choosing quasi-optimal scaling factor is discussed. Applications to parametric uncertainty quantification are illustrated by considering a random ODE model together with an elliptic problem with lognormal random input.
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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 | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".