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Multivariate approximation of functions on irregular domains by weighted least-squares methods

Abstract : We propose and analyse numerical algorithms based on weighted least squares for the approximation of a bounded real-valued function on a general bounded domain Ω Ă R d. Given any n-dimensional approximation space Vn Ă L 2 pΩq, the analysis in [6] shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations m of the order n ln n. When an L 2 pΩqorthonormal basis of Vn is available in analytic form, such estimators can be constructed using the algorithms described in [6, Section 5]. If the basis also has product form, then these algorithms have computational complexity linear in d and m. In this paper we show that, when Ω is an irregular domain such that the analytic form of an L 2 pΩq-orthonormal basis is not available, stable and quasi-optimally weighted leastsquares estimators can still be constructed from Vn, again with m of the order n ln n, but using a suitable surrogate basis of Vn orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of Ω and Vn. Numerical results validating our analysis are presented.
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Submitted on : Tuesday, October 12, 2021 - 2:25:48 PM
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Giovanni Migliorati. Multivariate approximation of functions on irregular domains by weighted least-squares methods. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2020, 41 (2), pp.1293-1317. ⟨10.1093/imanum/draa023⟩. ⟨hal-03375007⟩

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