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# Boosted optimal weighted least-squares

Abstract : This paper is concerned with the approximation of a function $u$ in a given approximation space $V_m$ of dimension $m$ from evaluations of the function at $n$ suitably chosen points. The aim is to construct an approximation of $u$ in $V_m$ which yields an error close to the best approximation error in $V_m$ and using as few evaluations as possible. Classical least-squares regression, which defines a projection in $V_m$ from $n$ random points, usually requires a large $n$ to guarantee a stable approximation and an error close to the best approximation error. This is a major drawback for applications where $u$ is expensive to evaluate. One remedy is to use a weighted least squares projection using $n$ samples drawn from a properly selected distribution. In this paper, we introduce a boosted weighted least-squares method which allows to ensure almost surely the stability of the weighted least squares projection with a sample size close to the interpolation regime $n=m$. It consists in sampling according to a measure associated with the optimization of a stability criterion over a collection of independent $n$-samples, and resampling according to this measure until a stability condition is satisfied. A greedy method is then proposed to remove points from the obtained sample. Quasi-optimality properties are obtained for the weighted least-squares projection, with or without the greedy procedure. The proposed method is validated on numerical examples and compared to state-of-the-art interpolation and weighted least squares methods.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-02562188
Contributor : Anthony Nouy <>
Submitted on : Monday, May 4, 2020 - 2:42:34 PM
Last modification on : Friday, July 31, 2020 - 12:53:35 PM

### Identifiers

• HAL Id : hal-02562188, version 1
• ARXIV : 1912.07075

### Citation

Cécile Haberstich, Anthony Nouy, Guillaume Perrin. Boosted optimal weighted least-squares. 2020. ⟨hal-02562188⟩

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