High order approximation to non-smooth multivariate functions

Common approximation tools return low-order approximations in the vicinities of singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form f=g+r₊ where g,r∈C^M+1(Rⁿ) and the function r₊ is defined by r₊(y)={r(y),r(y)≥00,r(y)<0,∀y∈Rⁿ. Given scattered (or uniform) data points X⊂Rⁿ, we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities.