Scattered Data Approximation

Optimum experimental designs

We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditioned $\ell^1$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. We present theoretical analysis to motivate the algorithm and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. Numerical examples are also provided to demonstrate that our proposed algor...

A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

Full length article: On optimal polynomial meshes

Possibility of approximation, analytic functions Possibility of approximation, continued Interpolation and lemniscates Degree of convergence of polynomials. Overconvergence Best Approximation by polynomials Orthogonality and least squares Interpolation by polynomials Interpolation by rational functions Approximation by rational functions Interpolation and functions analytic in the unit circle Approximation with auxiliary conditions and to non-analytic functions Existence and uniqueness of rational functions of best approximation Appendix Bibliography Index.

Interpolation and approximation

We show that the property of being a (weakly) admissible mesh for multivariate polynomials is preserved by small perturbations on real and complex Markov compacts. Applications are given to smooth transformations of polynomial meshes and to polynomial interpolation.

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Small perturbations of polynomial meshes

Optimal polynomial admissible meshes on some classes of compact subsets of R d

We obtain good discrete sets for real or complex multivariate polynomial approximation (admissible meshes) on compact sets satysfying a Markov polynomial inequality, by analytic transformations. Then we apply the result to the construction of near optimal admissible meshes, and we discuss two examples concerning complex analytic curves and real analytic cylindroids.

/pdf/analytic-transformations-of-admisible-meshes-149161tdse.pdf

Analytic transformations of admisible meshes

We present a brief survey on the compression of discrete measures by Caratheodory-Tchakaloff Subsampling, its implementation by Linear or Quadratic Programming and the application to multivariate polynomial Least Squares. We also give an algorithm that computes the corresponding Caratheodory-Tchakaloff (CATCH) points and weights for polynomial spaces on compact sets and manifolds in 2D and 3D.

/pdf/caratheodory-tchakaloff-subsampling-1wii8z6vza.pdf

Caratheodory-Tchakaloff Subsampling

Using the approximation theory notions of polynomial mesh and Dubiner distance in a compact set, we derive error estimates for total degree polynomial optimization on Chebyshev grids of the hypercube.

/pdf/a-note-on-total-degree-polynomial-optimization-by-chebyshev-2rwtf2q464.pdf

Federico Piazzon

Papers

Small perturbations of polynomial meshes

Optimal polynomial admissible meshes on some classes of compact subsets of R d

Analytic transformations of admisible meshes

Caratheodory-Tchakaloff Subsampling

A note on total degree polynomial optimization by Chebyshev grids