0000000000297103
AUTHOR
Jacques Liandrat
A fully adaptive wavelet algorithm for parabolic partial differential equations
We present a fully adaptive numerical scheme for the resolution of parabolic equations. It is based on wavelet approximations of functions and operators. Following the numerical analysis in the case of linear equations, we derive a numerical algorithm essentially based on convolution operators that can be efficiently implemented as soon as a natural condition on the space of approximation is satisfied. The algorithm is extended to semi-linear equations with time dependent (adapted) spaces of approximation. Numerical experiments deal with the heat equation as well as the Burgers equation.
A fully adaptive multiresolution scheme for image processing
A nonlinear multiresolution scheme within Harten's framework [A. Harten, Discrete multiresolution analysis and generalized wavelets, J. Appl. Numer. Math. 12 (1993) 153-192; A. Harten, Multiresolution representation of data II, SIAM J. Numer. Anal. 33 (3) (1996) 1205-1256] is presented. It is based on a centered piecewise polynomial interpolation fully adapted to discontinuities. Compression properties of the multiresolution scheme are studied on various numerical experiments on images.
Image compression based on a multi-directional map-dependent algorithm
Abstract This work is devoted to the construction of a new multi-directional edge-adapted compression algorithm for images. It is based on a multi-scale transform that is performed in two steps: a detection step producing a map of edges and a prediction/multi-resolution step which takes into account the information given by the map. A short analysis of the multi-scale transform is performed and an estimate of the error associated to the largest coefficients for a piecewise regular function with Lipschitz edges is provided. Comparisons between this map-dependent algorithm and different classical algorithms are given.
Error bounds for a convexity-preserving interpolation and its limit function
AbstractError bounds between a nonlinear interpolation and the limit function of its associated subdivision scheme are estimated. The bounds can be evaluated without recursive subdivision. We show that this interpolation is convexity preserving, as its associated subdivision scheme. Finally, some numerical experiments are presented.