
Sequential sampling for optimal weighted least squares approximations in hierarchical spaces
We consider the problem of approximating an unknown function u∈ L^2(D,ρ)...
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Gradient Descentbased Doptimal Design for the LeastSquares Polynomial Approximation
In this work, we propose a novel sampling method for Design of Experimen...
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On alternative quantization for doubly weighted approximation and integration over unbounded domains
It is known that for a ρweighted L_qapproximation of single variable f...
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Multivariate approximation of functions on irregular domains by weighted leastsquares methods
We propose and analyse numerical algorithms based on weighted least squa...
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Optimal sampling and Christoffel functions on general domains
We consider the problem of reconstructing an unknown function u∈ L^2(D,μ...
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Optimal pointwise sampling for L^2 approximation
Given a function u∈ L^2=L^2(D,μ), where D⊂ℝ^d and μ is a measure on D, a...
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VoidandCluster Sampling of Large Scattered Data and Trajectories
We propose a data reduction technique for scattered data based on statis...
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Optimal sampling strategies for multivariate function approximation on general domains
In this paper, we address the problem of approximating a multivariate function defined on a general domain in d dimensions from sample points. We consider weighted leastsquares approximation in an arbitrary finitedimensional space P from independent random samples taken according to a suitable measure. In general, leastsquares approximations can be inaccurate and ill conditioned when the number of sample points M is close to N = (P). To counteract this, we introduce a novel method for sampling in general domains which leads to provably accurate and wellconditioned weighted leastsquares approximations. The resulting sampling measure is discrete, and therefore straightforward to sample from. Our main result shows near optimal sample complexity for this procedure; specifically, M = O(N (N)) samples suffice for a well conditioned and accurate approximation. Numerical experiments on polynomial approximation in general domains confirm the benefits of this method over standard sampling.
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