The solution of multi-scale partial differential equations using wavelets

Wavelets are a powerful new mathematical tool which offers the possibility to treat in a natural way quantities characterized by several length scales. In this article we will show how wavelets can be used to solve partial differential equations which exhibit widely varying length scales and which are therefore hardly accessible by other numerical methods. As a benchmark calculation we solve Poisson's equation for a 3-dimensional Uranium dimer. The length scales of the charge distribution vary by 4 orders of magnitude in this case. Using lifted interpolating wavelets the number of iterations is independent of the maximal resolution and the computational effort therefore scales strictly linearly with respect to the size of the system

Nonlinear Inversion from Partial EIT Data: Computational Experiments

Electrical impedance tomography (EIT) is a non-invasive imaging method in which an unknown physical body is probed with electric currents applied on the boundary, and the internal conductivity distribution is recovered from the measured boundary voltage data. The reconstruction task is a nonlinear and ill-posed inverse problem, whose solution calls for special regularized algorithms, such as D-bar methods which are based on complex geometrical optics solutions (CGOs). In many applications of EIT, such as monitoring the heart and lungs of unconscious intensive care patients or locating the focus of an epileptic seizure, data acquisition on the entire boundary of the body is impractical, restricting the boundary area available for EIT measurements. An extension of the D-bar method to the case when data is collected only on a subset of the boundary is studied by computational simulation. The approach is based on solving a boundary integral equation for the traces of the CGOs using localized basis functions (Haar wavelets). The numerical evidence suggests that the D-bar method can be applied to partial-boundary data in dimension two and that the traces of the partial data CGOs approximate the full data CGO solutions on the available portion of the boundary, for the necessary small $k$ frequencies.Comment: 24 pages, 12 figure

Wavelets and Fast Numerical Algorithms

Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties. For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue. We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents. These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the so-called non-standard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of non-standard forms of several basic operators (e.g. derivatives) will be computed explicitly.Comment: 32 pages, uuencoded tar-compressed LaTeX file. Uses epsf.sty (see `macros'

Computing continuous-time growth models with boundary conditions via wavelets.

This paper presents an algorithm for solving boundary value differential equations, which often arise in economics from the application of Pontryagin’s maximum principle. We propose a wavelet-collocation algorithm, study its convergence properties and illustrate how this approach can be applied to different economic problemsWavelets; Continuous-time growth models; Boundary value problems;

Wavelets and their use

This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to corresponding literature. The multiresolution analysis and fast wavelet transform became a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for achievement of a goal. Analysis of various functions with the help of wavelets allows to reveal fractal structures, singularities etc. Wavelet transform of operator expressions helps solve some equations. In practical applications one deals often with the discretized functions, and the problem of stability of wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves by some examples only. The authors would be grateful for any comments which improve this review paper and move us closer to the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh