TY - JOUR

T1 - Adaptive multiresolution analysis based on anisotropic triangulations

AU - Cohen, Albert

AU - Dyn, Nira

AU - Hecht, Frédéric

AU - Mirebeau, Jean Marie

PY - 2012

Y1 - 2012

N2 - A simple greedy refinement procedure for the generation of dataadapted triangulations is proposed and studied. Given a function f of two variables, the algorithm produces a hierarchy of triangulations (D j) j≥0 and piecewise polynomial approximations of f on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between f and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the L p norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of f in the case of C 2 functions).

AB - A simple greedy refinement procedure for the generation of dataadapted triangulations is proposed and studied. Given a function f of two variables, the algorithm produces a hierarchy of triangulations (D j) j≥0 and piecewise polynomial approximations of f on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between f and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the L p norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of f in the case of C 2 functions).

UR - http://www.scopus.com/inward/record.url?scp=84858977079&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-2011-02495-6

DO - 10.1090/S0025-5718-2011-02495-6

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AN - SCOPUS:84858977079

SN - 0025-5718

VL - 81

SP - 789

EP - 810

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 278

ER -