Rosa Donat

Nonlinear multiscale decompositions: The approach of A. Harten

Data‐dependent interpolatory techniques can be used in the reconstruction step of a multiresolution scheme designed “à la Harten”. In this paper we carefully analyze the class of Essentially Non‐Oscillatory (ENO) interpolatory techniques described in [11] and their potential to improve the compression capabilities of multiresolution schemes. When dealing with nonlinear multiresolution schemes the issue of stability also needs to be carefully considered.

Tensor product multiresolution analysis with error control for compact image representation

A class of multiresolution representations based on nonlinear prediction is studied in the multivariate context based on tensor product strategies. In contrast to standard linear wavelet transforms, these representations cannot be thought of as a change of basis, and the error induced by thresholding or quantizing the coefficients requires a different analysis. We propose specific error control algorithms which ensure a prescribed accuracy in various norms when performing such operations on the coefficients. These algorithms are compared with standard thresholding, for synthetic and real images.

Shock-Vortex Interactions at High Mach Numbers

We perform a computational study of the interaction of a planar shock wave with a cylindrical vortex. We use a particularly robust High Resolution Shock Capturing scheme, Marquinas scheme, to obtain high quality, high resolution numerical simulations of the interaction. In the case of a very-strong shock/vortex encounter, we observe a severe reorganization of the flow field in the downstream region, which seems to be due mainly to the strength of the shock. The numerical data is analyzed to study the driving mechanisms for the production of vorticity in the interaction.

Point Value Multiscale Algorithms for 2D Compressible Flows

The numerical simulation of physical problems modeled by systems of conservation laws is difficult due to the presence of discontinuities in the solution. High-order shock capturing schemes combine sharp numerical profiles at discontinuities with a highly accurate approximation in smooth regions, but usually their computational cost is quite large. Following the idea of A. Harten [Comm. Pure Appl. Math., 48 (1995), pp. 1305--1342] and Bihari and Harten [SIAM J. Sci. Comput., 18 (1997), pp. 315--354], we present in this paper a method to reduce the execution time of such simulations. It is based on a point value multiresolution transform that is used to detect regions with singularities. In these regions, an expensive high-resolution shock capturing scheme is applied to compute the numerical flux at cell interfaces. In smooth regions a cheap polynomial interpolation is used to deduce the value of the numerical divergence from values previously obtained on lower resolution scales. This method is applied to solve the two-dimensional compressible Euler equations for two classical configurations. The results are analyzed in terms of quality and efficiency.

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

A nonlinear multiresolution scheme within Hartens framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.

Interpolation and Approximation of Piecewise Smooth Functions

This paper provides approximation orders for a class of nonlinear interpolation procedures for uniformly sampled univariate data. The interpolation is based on essentially nonoscillatory (ENO) and subcell resolution (SR) reconstruction techniques. These nonlinear techniques aim at reducing significantly the approximation error for functions with isolated singularities and are therefore attractive for applications such as shock computations or image compression. We prove that in the presence of isolated singularities, the approximation order provided by the interpolation procedure is improved by a factor of

Multiresolution Based on Weighted Averages of the Hat Function I: Linear Reconstruction Techniques

h

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

relative to the linear methods, where h is the sampling rate. Moreover, for h below a critical value, we recover the optimal approximation order as for uniformly smooth functions.

Adaptive interpolation of images

In this paper we analyze a particular example of the general framework developed in [A. Harten, {\it SIAM J. Numer. Anal}., 33 (1996) pp. 1205--1256], the case in which the discretization operator is obtained by taking local averages with respect to the hat function. We consider a class of reconstruction procedures which are appropriate for this multiresolution setting and describe the associated prediction operators that allow us to climb up the ladder from coarse to finer levels of resolution. In Part I we use data-independent (linear) reconstruction techniques as our approximation tool. We show how to obtain multiresolution transforms in bounded domains and analyze their stability with respect to perturbations.

Stability Through Synchronization in Nonlinear Multiscale Transformations

In this paper we describe and analyze a class of nonlinear \mr 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.