A Christoffel function weighted least squares algorithm for collocation approximations
Here, we propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation framework. Our investigation is motivated by applications in the collocation approximation of parametric functions, which frequently entails construction of surrogates via orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density defining the orthogonal polynomial family. Our proposed algorithm instead samples with respect to the (weighted) pluripotential equilibrium measure of the domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.
- Research Organization:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sponsoring Organization:
- USDOE National Nuclear Security Administration (NNSA)
- Grant/Contract Number:
- AC04-94AL85000
- OSTI ID:
- 1347352
- Report Number(s):
- SAND-2015-20768J; PII: S002557182016031920
- Journal Information:
- Mathematics of Computation, Vol. 86, Issue 306; ISSN 0025-5718
- Country of Publication:
- United States
- Language:
- English
Web of Science
Pluripotential Numerics
|
journal | June 2018 |
Effectively Subsampled Quadratures for Least Squares Polynomial Approximations
|
journal | January 2017 |
| Compressed sensing approaches for polynomial approximation of high-dimensional functions | preprint | January 2017 |
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