Multilevel regularization for image deblurring problems

We consider the de-blurring problem of noisy and blurred signals/images in the case of space invariant point spread functions. The use of appropriate boundary conditions leads to linear systems with structured coefficient matrices related to space invariant operators like Toeplitz, circulants, trigonometric matrix algebras etc. We combine an algebraic multigrid (which is designed ad hoc for structured matrices) with the low-pass projectors typical of the classical geometrical multigrid for elliptic partial differential equations. The smoother is any iterative regularizing method, while the projector is chosen in order to maintain the same algebraic structure at each recursion level and having a low-pass filter property, which is very useful in order to reduce the noise effects. In this way, we obtain a better restored image with a flatter restoration error curve and also in less time than the auxiliary method used as smoother. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

LOW MACH NUMBER LIMIT ON EXTERIOR DOMAINS

This paper is concerned with the low Mach number limit for the compressible Navier-Stokes equations in an exterior domain.We present here an approach based on Strichartz estimate defined on a non trapp...     

A fast alternating minimization algorithm for total variation deblurring without boundary artifacts

Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang (2008) [32]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first one is computationally useful in the case of a symmetric blur, while the second one can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm.     

Vanishing dielectric constant regime for the Navier Stokes Maxwell equations

In this paper we rigorously justify the convergence of smooth solutions of the Navier-Stokes-Maxwell equations towards smooth solutions of the classical 2D parabolic MHD equations in the case of vanishing dielectric constant. The result is achieved by means of higher-order energy estimates.     

LOW MACH NUMBER LIMIT ON EXTERIOR DOMAINS Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

This paper is concerned with the low Mach number limit for the compressible Navier-Stokes equations in an exterior domain.We present here an approach based on Strichartz estimate defined on a non trapping exterior domain and we will be able to show the compactness and strong convergence of the velocity vector field.     

CAMERA: The Center for Advanced Mathematics for Energy Research Applications

Advanced experimental facilities worldwide are probing structure and chemistry, disorder, dynamics and electronic properties, through time, over length scales spanning macroscopic to atomic resolution, in multiple dimensions (e.g., hyperspectral tomography, nano-spectroscopy), under extreme environmental conditions and stimulated reactions. In order to do so, they are collecting more and more data at faster and faster rates. One critical challenge is to build algorithms that can analyze, interpret, and understand the information contained within this experimental data.     

On a Nonlinear Model for Tumor Growth: Global in Time Weak Solutions

We investigate the dynamics of a class of tumor growth models known as mixed models. The key characteristic of these type of tumor growth models is that the different populations of cells are continuously present everywhere in the tumor at all times. In this work we focus on the evolution of tumor growth in the presence of proliferating, quiescent and dead cells as well as a nutrient. The system is given by a multi-phase flow model and the tumor is described as a growing continuum Ω with boundary ?Ω both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation.     

On nondecreasing sequences of regularization parameters for nonstationary iterated Tikhonov

Nonstationary iterated Tikhonov is an iterative regularization method that requires a strategy for defining the Tikhonov regularization parameter at each iteration and an early termination of the iterative process. A classical choice for the regularization parameters is a decreasing geometric sequence which leads to a linear convergence rate. The early iterations compute quickly a good approximation of the true solution, but the main drawback of this choice is a rapid growth of the error for later iterations. This implies that a stopping criteria, e.g. the discrepancy principle, could fail in computing a good approximation. In this paper we show by a filter factor analysis that a nondecreasing sequence of regularization parameters can provide a rapid and stable convergence. Hence, a reliable stopping criteria is no longer necessary. A geometric nondecreasing sequence of the Tikhonov regularization parameters into a fixed interval is proposed and numerically validated for deblurring problems.     

Frame Multiplication Theory and a Vector-Valued DFT and Ambiguity Function

Vector-valued discrete Fourier transforms (DFTs) and ambiguity functions are defined. The motivation for the definitions is to provide realistic modeling of multi-sensor environments in which a useful time–frequency analysis is essential. The definition of the DFT requires associated uncertainty principle inequalities. The definition of the ambiguity function requires a component that leads to formulating a mathematical theory in which two essential algebraic operations can be made compatible in a natural way. The theory is referred to as frame multiplication theory. These definitions, inequalities, and theory are interdependent, and they are the content of the paper with the centerpiece being frame multiplication theory. The technology underlying frame multiplication theory is the theory of frames, short time Fourier transforms, and the representation theory of finite groups. The main results have the following form: frame multiplication exists if and only if the finite frames that arise in the theory are of a certain type, e.g., harmonic frames, or, more generally, group frames. In light of the complexities and the importance of the modeling of time-varying and dynamical systems in the context of effectively analyzing vector-valued multi-sensor environments, the theory of vector-valued DFTs and ambiguity functions must not only be mathematically meaningful, but it must have constructive implementable algorithms, and be computationally viable. This paper presents our vision for resolving these issues, in terms of a significant mathematical theory, and based on the goal of formulating and developing a useful vector-valued theory.