Boosted optimal weighted least-squares
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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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