A Christoffel function weighted least squares algorithm for collocation approximations
Abstract
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.
- Authors:
- Publication Date:
- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1347352
- Report Number(s):
- SAND-2015-20768J
Journal ID: ISSN 0025-5718; PII: S002557182016031920
- Grant/Contract Number:
- AC04-94AL85000
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Mathematics of Computation
- Additional Journal Information:
- Journal Volume: 86; Journal Issue: 306; Journal ID: ISSN 0025-5718
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING
Citation Formats
Narayan, Akil, Jakeman, John D., and Zhou, Tao. A Christoffel function weighted least squares algorithm for collocation approximations. United States: N. p., 2016.
Web. doi:10.1090/mcom/3192.
Narayan, Akil, Jakeman, John D., & Zhou, Tao. A Christoffel function weighted least squares algorithm for collocation approximations. United States. https://doi.org/10.1090/mcom/3192
Narayan, Akil, Jakeman, John D., and Zhou, Tao. Mon .
"A Christoffel function weighted least squares algorithm for collocation approximations". United States. https://doi.org/10.1090/mcom/3192. https://www.osti.gov/servlets/purl/1347352.
@article{osti_1347352,
title = {A Christoffel function weighted least squares algorithm for collocation approximations},
author = {Narayan, Akil and Jakeman, John D. and Zhou, Tao},
abstractNote = {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.},
doi = {10.1090/mcom/3192},
journal = {Mathematics of Computation},
number = 306,
volume = 86,
place = {United States},
year = {Mon Nov 28 00:00:00 EST 2016},
month = {Mon Nov 28 00:00:00 EST 2016}
}
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Works referenced in this record:
Asymptotics for Christoffel Functions with Varying Weights
journal, November 2000
- Totik, Vilmos
- Advances in Applied Mathematics, Vol. 25, Issue 4
Asymptotics for Christoffel functions for general measures on the real line
journal, December 2000
- Totik, Vilmos
- Journal d'Analyse Mathématique, Vol. 81, Issue 1
Géza Freud, orthogonal polynomials and Christoffel functions. A case study
journal, September 1986
- Nevai, Paul
- Journal of Approximation Theory, Vol. 48, Issue 1
High-Order Collocation Methods for Differential Equations with Random Inputs
journal, January 2005
- Xiu, Dongbin; Hesthaven, Jan S.
- SIAM Journal on Scientific Computing, Vol. 27, Issue 3
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials
journal, June 2013
- Kroó, A.; Lubinsky, D. S.
- Canadian Journal of Mathematics, Vol. 65, Issue 3
Fekete points and convergence towards equilibrium measures on complex manifolds
preprint, January 2009
- Berman, Robert J.; Boucksom, Sebastien; Nystrom, David Witt
- arXiv
A non-adapted sparse approximation of PDEs with stochastic inputs
text, January 2010
- Doostan, Alireza; Owhadi, Houman
- arXiv
A Survey of Weighted Approximation for Exponential Weights
text, January 2007
- Lubinsky, Doron S.
- arXiv
Works referencing / citing this record:
Pluripotential Numerics
journal, June 2018
- Piazzon, Federico
- Constructive Approximation, Vol. 49, Issue 2
Effectively Subsampled Quadratures for Least Squares Polynomial Approximations
journal, January 2017
- Seshadri, Pranay; Narayan, Akil; Mahadevan, Sankaran
- SIAM/ASA Journal on Uncertainty Quantification, Vol. 5, Issue 1
Compressed sensing approaches for polynomial approximation of high-dimensional functions
preprint, January 2017
- Adcock, Ben; Brugiapaglia, Simone; Webster, Clayton G.
- arXiv