High order approximation to non-smooth multivariate functions

2016/04/11 by Amir, Anat, Levin, David
#41A99 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1604.02810

Abstract

Approximations of non-smooth multivariate functions return low-order approximations in the vicinities of the 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), · amp; r(y) ≥ 0
0, · amp; r(y) · lt; 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.

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