A meshless P1-Lagrange Interpolation operator in Rd

Abstract Generally, meshless interpolations are not polynomial, in this paper, we propose a meshless polynomial local interpolator. This interpolator uses Lagrange polynomials P 1 and a nearest neighbors strategy, and have yet been introduced in a previous work [1] investigating the relation between an artificial neural network and the transport of the identity towards a given function to be interpolated. Unfortunately, it has been reported in [1] that the order of convergence of the interpolation deteri- orates when the size of the d -simplexe becomes too small, because the built d -simplexe does not necessary contain the point at which inter- polation is computed. The aim of this note is to remedy this difficulty. Here, we study the problem of interpolating d-dimensional data with P 1 polynomials by minimizing a data-fidelity cost function, involving the barycentric coordinates of the d + 1 -nearest neighbors. Data points dis- tributions are not required to be structured, and the nearest neighbors are computed through a penalized minimization process in order to get a no degenerate d -simplex containing the point at which the interpolation is computed. The proposed interpolation operator is local, and a global one can be built which will be no more polynomial. Some numerical results are provided showing the efficiency of the proposed method.