We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.

/pdf/signal-recovery-by-proximal-forward-backward-splitting-41tfo8pc7o.pdf

Signal recovery by proximal forward-backward splitting ∗

Robust stability and control for systems that combine continuous-time and discrete-time dynamics. This article is a tutorial on modeling the dynamics of hybrid systems, on the elements of stability theory for hybrid systems, and on the basics of hybrid control. The presentation and selection of material is oriented toward the analysis of asymptotic stability in hybrid systems and the design of stabilizing hybrid controllers. Our emphasis on the robustness of asymptotic stability to data perturbation, external disturbances, and measurement error distinguishes the approach taken here from other approaches to hybrid systems. While we make some connections to alternative approaches, this article does not aspire to be a survey of the hybrid system literature, which is vast and multifaceted.

Hybrid dynamical systems

Let T be a (possibly nonlinear) continuous operator on Hilbert space . If, for some starting vector x, the orbit sequence {Tkx,k = 0,1,...} converges, then the limit z is a fixed point of T; that is, Tz = z. An operator N on a Hilbert space is nonexpansive?(ne) if, for each x and y in , Even when N has fixed points the orbit sequence {Nkx} need not converge; consider the example N = ?I, where I denotes the identity operator. However, for any the iterative procedure defined by converges (weakly) to a fixed point of N whenever such points exist. This is the Krasnoselskii?Mann (KM) approach to finding fixed points of ne operators. A wide variety of iterative procedures used in signal processing and image reconstruction and elsewhere are special cases of the KM iterative procedure, for particular choices of the ne operator N. These include the Gerchberg?Papoulis method for bandlimited extrapolation, the SART algorithm of Anderson and Kak, the Landweber and projected Landweber algorithms, simultaneous and sequential methods for solving the convex feasibility problem, the ART and Cimmino methods for solving linear systems of equations, the CQ algorithm for solving the split feasibility problem and Dolidze's procedure for the variational inequality problem for monotone operators.

http://iopscience.iop.org/article/10.1088/0266-5611/20/1/006/pdf

A unified treatment of some iterative algorithms in signal processing and image reconstruction

The multiple-sets split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. It generalizes the convex feasibility problem as well as the two-sets split feasibility problem. We propose a projection algorithm that minimizes a proximity function that measures the distance of a point from all sets. The formulation, as well as the algorithm, generalize earlier work on the split feasibility problem. We offer also a generalization to proximity functions with Bregman distances. Application of the method to the inverse problem of intensity-modulated radiation therapy treatment planning is studied in a separate companion paper and is here only described briefly.

/pdf/the-multiple-sets-split-feasibility-problem-and-its-226d4egijg.pdf

The Multiple-Sets Split Feasibility Problem and Its Applications for Inverse Problems

We consider the problem of solving dual monotone inclusions involving sums of composite parallel-sum type operators. A feature of this work is to exploit explicitly the properties of the cocoercive operators appearing in the model. Several splitting algorithms recently proposed in the literature are recovered as special cases.

A splitting algorithm for dual monotone inclusions involving cocoercive operators

Periodic solutions of functional differential inclusions and fixed points of σ-selectionable correspondences

Impulse differential inclusions, and in particular, hybrid control systems, are defined by a differential inclusion (or a control system) and a reset map. A run of an impulse differential inclusion is defined by a sequence of cadences, of reinitialized states and of motives describing the evolution along a given cadence between two distinct consecutive impulse times, the value of a motive at the end of a cadence being reset as the next reinitialized state of the next cadence.



A cadenced run is then defined by constant cadence, initial state and motive, where the value at the end of the cadence is reset at the same reinitialized state. It plays the role of a ‘discontinuous’ periodic solution of a differential inclusion.



We prove that if the sequence of reinitialized states of a run converges to some state, then the run converges to a cadenced run starting from this state, and that, under convexity assumptions, that a cadenced run does exist. Copyright © 2001 John Wiley & Sons, Ltd.

Cadenced runs of impulse and hybrid control systems

This paper is a study of the relations that must exist between a multivalued dynamical system with memory and constraints depending on the past, in order to have the existence of solutions of the dynamical system satisfying the constraints. Such solutions are called viable solutions.

/pdf/functional-viability-theorems-for-differential-inclusions-5dh6g4se44.pdf

Functional viability theorems for differential inclusions with memory

Impulse capture basins of sets under impulse control systems

This article begins by looking at two reports, commissioned in the second half of the 20th century to investigate the impact of social change on education and to propose strategies for tackling new challenges arising from this. The first report, published in 1972, was the so-called Faure report (named after French lawyer and politician Edgar Faure), and the second one, published in 1996, was the so-called Delors report (named after French economist and politician Jacques Delors). Using these two reports as their starting point, the authors of this paper consider the effects of modern information technology on our cultural environment. Today we are experiencing an increasingly virtual world with very real effects. This paper argues that Edgar Faure already foresaw the emergence of a new cultural world in which virtuality would both add to and compete with reality. The authors stress that now, more than ever, access to lifelong education needs to be ensured for everyone, permanently enriched by research and innovation, addressing creativity and entrepreneurship, in addition to continuously evolving skills and competences.

Towards a humanism of knowledge, action and cooperation

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