
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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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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Optimal sampling strategies for multivariate function approximation on general domains
In this paper, we address the problem of approximating a multivariate fu...
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Boosted optimal weighted leastsquares
This paper is concerned with the approximation of a function u in a give...
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Approximation and quadrature by weighted least squares polynomials on the sphere
Given a sequence of MarcinkiewiczZygmund inequalities in L_2 on a usual...
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Learning Functions of Few Arbitrary Linear Parameters in High Dimensions
Let us assume that f is a continuous function defined on the unit ball o...
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Optimal sampling and assay for soil organic carbon estimation
The world needs around 150 Pg of negative carbon emissions to mitigate c...
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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) from its evaluations at given sampling points x^1,…,x^m∈ D, where D⊂ℝ^d is a general domain and μ a probability measure. The approximation is picked from a linear space V_n of interest where n=(V_n). Recent results have revealed that certain weighted leastsquares methods achieve near best approximation with a sampling budget m that is proportional to n, up to a logarithmic factor ln(2n/ε), where ε>0 is a probability of failure. The sampling points should be picked at random according to a wellchosen probability measure σ whose density is given by the inverse Christoffel function that depends both on V_n and μ. While this approach is greatly facilitated when D and μ have tensor product structure, it becomes problematic for domains D with arbitrary geometry since the optimal measure depends on an orthonormal basis of V_n in L^2(D,μ) which is not explicitly given, even for simple polynomial spaces. Therefore sampling according to this measure is not practically feasible. In this paper, we discuss practical sampling strategies, which amount to using a perturbed measure σ that can be computed in an offline stage, not involving the measurement of u. We show that near best approximation is attained by the resulting weighted leastsquares method at nearoptimal sampling budget and we discuss multilevel approaches that preserve optimality of the cumulated sampling budget when the spaces V_n are iteratively enriched. These strategies rely on the knowledge of apriori upper bounds on the inverse Christoffel function. We establish such bounds for spaces V_n of multivariate algebraic polynomials, and for general domains D.
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