TY - JOUR

T1 - High order approximation to non-smooth multivariate functions

AU - Amir, Anat

AU - Levin, David

N1 - Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2018/7

Y1 - 2018/7

N2 - 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∈CM+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.

AB - 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∈CM+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.

KW - Multivariate functions

KW - Non-smooth functions

KW - Quasi-interpolation

UR - http://www.scopus.com/inward/record.url?scp=85046828931&partnerID=8YFLogxK

U2 - 10.1016/j.cagd.2018.02.004

DO - 10.1016/j.cagd.2018.02.004

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AN - SCOPUS:85046828931

SN - 0167-8396

VL - 63

SP - 31

EP - 65

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

ER -