Multivariate approximation of functions on irregular domains by weighted least-squares methods
Résumé
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.
Domaines
Mathématiques [math]
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Migliorati - 2021 - Multivariate approximation of functions on irregul.pdf (3.18 Mo)
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