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Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

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Abstract

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).

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Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, de Cartagena, Spain

    S. Amat

  2. Ecole Centrale Marseille, laboratoire d’analyse topologie et probabilites (LATP), 38 rue Frédéric Joliot Curie, 13451, Marseille cedex 20, France

    K. Dadourian & J. Liandrat

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  1. S. Amat
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  2. K. Dadourian
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  3. J. Liandrat
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Correspondence to J. Liandrat.

Additional information

Communicated by Tomas Sauer.

S. Amat research supported in part by 08662/PI/08 and MTM2007-62945.

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Amat, S., Dadourian, K. & Liandrat, J. Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. Adv Comput Math 34, 253–277 (2011). https://doi.org/10.1007/s10444-010-9151-6

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