Georgia Institute of Technology College of Sciences

Expansion in a wavelet basis provides useful information ona function in different positions and length-scales. The simplest example of wavelets are the Haar functions, which are just linearcombinations of characteristic functions of cubes, but often moresmoothness is preferred. It is well-known that the notion of Haarfunctions carries over to rather general abstract metric spaces. Whatabout more regular wavelets? It turns out that a neat construction canbe given, starting from averages of the indicator functions over arandom selection of the underlying cubes. This is yet anotherapplication of such probabilistic averaging methods in harmonicanalysis. The talk is based on joint work in progress with P. Auscher.

I will review numerical and analytic results on a system consistingof one or many particles moving in a chaotic billiard under the influence on an electric field and a Gaussian thermostat.

 We will discuss the recent developments of fast image reconstrcution with total variation (TV) regularization whose robustness has been justfied by the theory of compressed sensing. However, the solution of TV based reconstruction encounters two main difficulties on the computational aspect of many applications: the inversion matrix can be large, irregular, and severely ill-conditioned, and the objective is nonsmooth. We introduce two algorithms that tackle the problem using variable splitting and optimized step size selection. The algorithms also provide a general framework for solving large and ill-conditioned linear inversion problem with TV regularization. An important and successful application of TV based image reconstruction in magnetic resonance imaging (MRI) known as paratially parallel imaging (PPI) will be discussed. The numerical results demonstrate significantly improved  efficiency and accuracy over the state-of-the-arts. 

A discussion of the Ding & Lawrence (2003) paper "A statistical sampling algorithm for RNA secondary structure prediction."