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Quadrature Strategies for Constructing Polynomial Approximations

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Uncertainty Modeling for Engineering Applications

Part of the book series: PoliTO Springer Series ((PTSS))

Abstract

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach, and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton’s method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies.

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Notes

  1. 1.

    Known colloquially as the Jacobi matrix.

  2. 2.

    In the case of data-driven distributions, kernel density estimation or even a maximum likelihood estimation may be required to obtain a probability distribution that can be used by the discretized Stieltjes procedure.

  3. 3.

    The codes to replicate the figures in this paper can be found at the website: www.effective-quadratures.org/publications.

  4. 4.

    By solving the basis pursuit (and de-noising) problem.

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Acknowledgements

This work was carried out while PS was visiting the Department of Mechanical, Chemical and Materials Engineering at Universitá di Cagliari in Cagliari, Sardinia; the financial support of the University’s Visiting Professor Program is gratefully acknowledged. The authors are also grateful to Akil Narayan for numerous discussions on polynomial approximations and quadratures.

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Authors and Affiliations

  1. Department of Engineering, University of Cambridge, Cambridge, UK

    Pranay Seshadri

  2. The Alan Turing Institute, London, UK

    Pranay Seshadri

  3. Department of Mechanical Engineering, Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA

    Gianluca Iaccarino

  4. Department of Mechanical, Chemical and Materials Engineering, Universitá di Cagliari, Cagliari, Sardinia, Italy

    Tiziano Ghisu

Authors
  1. Pranay Seshadri
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  2. Gianluca Iaccarino
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  3. Tiziano Ghisu
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Corresponding author

Correspondence to Gianluca Iaccarino .

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Editors and Affiliations

  1. Department of Electronics and Telecommunications (DET), Politecnico di Torino, Turin, Italy

    Flavio Canavero

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Seshadri, P., Iaccarino, G., Ghisu, T. (2019). Quadrature Strategies for Constructing Polynomial Approximations. In: Canavero, F. (eds) Uncertainty Modeling for Engineering Applications. PoliTO Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-030-04870-9_1

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