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A Christoffel function weighted least squares algorithm for collocation approximations

Journal Article · · Mathematics of Computation
DOI:https://doi.org/10.1090/mcom/3192· OSTI ID:1347352
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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 Laboratories (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, Journal Name: Mathematics of Computation Journal Issue: 306 Vol. 86; ISSN 0025-5718
Country of Publication:
United States
Language:
English

References (9)

Asymptotics for Christoffel Functions with Varying Weights journal November 2000
Asymptotics for Christoffel functions for general measures on the real line journal December 2000
Géza Freud, orthogonal polynomials and Christoffel functions. A case study journal September 1986
High-Order Collocation Methods for Differential Equations with Random Inputs journal January 2005
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials journal June 2013
Fekete points and convergence towards equilibrium measures on complex manifolds preprint January 2009
User-friendly tail bounds for sums of random matrices text January 2010
A non-adapted sparse approximation of PDEs with stochastic inputs text January 2010
A Survey of Weighted Approximation for Exponential Weights text January 2007

Cited By (3)

Effectively Subsampled Quadratures for Least Squares Polynomial Approximations journal January 2017
Compressed sensing approaches for polynomial approximation of high-dimensional functions preprint January 2017
Pluripotential Numerics journal June 2018

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Related Subjects

97 MATHEMATICS AND COMPUTING