Multiresolution Based on Weighted Averages of the Hat Function II: Nonlinear Reconstruction Techniques

In this paper we describe and analyze a class of nonlinear multiresolution schemes for the multiresolution setting which corresponds to discretization by local averages with respect to the hat function. These schemes are based on the essentially-non-oscillatory (ENO) interpolatory procedure described in [A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, J. Comput. Phys., 71 (1987), pp. 231–302]. We show that by allowing the approximation to fit the local nature of the data, one can improve the compression capabilities of the multiresolution algorithms. The question of stability for nonlinear (data-dependent) reconstruction techniques is also addressed.