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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A Christoffel function weighted least squares algorithm for collocation approximations
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by Akil Narayan, John D. Jakeman and Tao Zhou PDF
Math. Comp. 86 (2017), 1913-1947 Request permission

Abstract:

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.
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Additional Information
  • Akil Narayan
  • Affiliation: Department of Mathematics and Scientific Computing and Imaging (SCI) Institute, University of Utah, University of Utah, Salt Lake City, Utah 84112
  • MR Author ID: 932761
  • Email: akil@sci.utah.edu
  • John D. Jakeman
  • Affiliation: Computer Science Research Institute, Sandia National Laboratories, 1450 Innovation Parkway, SE, Albuquerque, New Mexico 87123
  • MR Author ID: 862962
  • Email: jdjakem@sandia.gov
  • Tao Zhou
  • Affiliation: LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
  • MR Author ID: 908982
  • Email: tzhou@lsec.cc.ac.cn
  • Received by editor(s): December 14, 2014
  • Received by editor(s) in revised form: June 8, 2015, September 19, 2015, and January 29, 2016
  • Published electronically: November 28, 2016
  • Additional Notes: The first author was partially supported by AFOSR FA9550-15-1-0467 and DARPA N660011524053
    The third author was supported the National Natural Science Foundation of China (Award Nos. 91530118 and 11571351)
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1913-1947
  • MSC (2010): Primary 65D15, 41A10
  • DOI: https://doi.org/10.1090/mcom/3192
  • MathSciNet review: 3626543