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Semi-algebraic Approximation Using Christoffel–Darboux Kernel

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Constructive Approximation Aims and scope

Abstract

We provide a new method to approximate a (possibly discontinuous) function using Christoffel–Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy solutions to nonlinear hyperbolic PDEs, or using numerical integration from finitely many evaluations of the function. While most of the existing methods construct a piecewise polynomial approximation, we construct a semi-algebraic approximation whose estimation and evaluation can be performed efficiently. An appealing feature of this method is that it deals with nonsmoothness implicitly so that a single scheme can be used to treat smooth or nonsmooth functions without any prior knowledge. On the theoretical side, we prove pointwise convergence almost everywhere as well as convergence in the Lebesgue one norm under broad assumptions. Using more restrictive assumptions, we obtain explicit convergence rates. We illustrate our approach on various examples from control and approximation. In particular, we observe empirically that our method does not suffer from the Gibbs phenomenon when approximating discontinuous functions.

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Notes

  1. A semi-algebraic function is a function whose graph is semi-algebraic, i.e., defined with finitely many polynomial inequalities.

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Acknowledgements

We are grateful to Quentin Vila for his technical input on discontinuous solutions of PDEs and to Milan Korda, Victor Magron and Matteo Tacchi for interesting discussions. This work was partly funded by the ERC Advanced Grant Taming and was also conducted in the framework of the regional program “Atlanstic 2020, Research, Education and Innovation in Pays de la Loire”, supported by the French Region Pays de la Loire and the European Regional Development Fund. E. Pauwels and J.B. Lasserre are also partially supported by the AI Interdisciplinary Institute ANITI funding through the French program “Investing for the Future PIA3”, under the Grant agreement number ANR-19-PI3A-0004.

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Authors and Affiliations

  1. LS2N, École Centrale de Nantes and CNRS UMR 6004, 44000, Nantes, France

    Swann Marx

  2. IRIT-UPS, Université de Toulouse, 118 route de Narbonne, 31400, Toulouse, France

    Edouard Pauwels

  3. Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Tillmann Weisser

  4. LAAS-CNRS, Université de Toulouse, 7 Avenue du Colonel Roche, 31400, Toulouse, France

    Didier Henrion & Jean Bernard Lasserre

  5. Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 4, 16206, Prague, Czechia

    Didier Henrion

  6. IMT-UPS, Université de Toulouse, 118 Route de Narbonne, 31400, Toulouse, France

    Jean Bernard Lasserre

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  1. Swann Marx
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  2. Edouard Pauwels
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  3. Tillmann Weisser
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  4. Didier Henrion
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  5. Jean Bernard Lasserre
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Correspondence to Swann Marx.

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Communicated by Remi Gribonval.

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Marx, S., Pauwels, E., Weisser, T. et al. Semi-algebraic Approximation Using Christoffel–Darboux Kernel. Constr Approx 54, 391–429 (2021). https://doi.org/10.1007/s00365-021-09535-4

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  • DOI: https://doi.org/10.1007/s00365-021-09535-4

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