Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator

In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators $A=\partial\Phi+B$, where $\partial\Phi$ is the subdifferential of a convex lower semicontinuous function $\Phi$, and $B$ is a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by $A$, and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems. The time discretization of these dynamics gives various forward-backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical splitting methods for optimization, and multi-criteria decision processes.Comment: 25 page

Optimization Methods for Image Regularization from Poisson Data

This work regards optimization techniques for image restoration problems in presence of Poisson noise. In several imaging applications (e.g. Astronomy, Microscopy, Medical Imaging) such noise is predominant; hence regularization techniques are needed in order to obtain satisfying restored images. In a variational framework, the image restoration problem consists in finding a minimum of a functional, which is the sum of two terms,: the fit–to–data and the regularization one. The trade–off between these two terms is measured by a regularization parameter. The estimation of such a parameter is very difficult due to the presence of Poisson noise. In this thesis we investigate three models regarding this parameter: a Discrepancy Model, Constrained Model and the Bregman procedure. The former two provide an estimation for the regularization parameter, but in some cases, such as low counts images, they do not allow to obtain satisfactory results. On the other hand, in presence of such images the Bregman procedure provides reliable results and, moreover, it allows to use an overestimation of the regularization parameter, giving satisfying restored images; furthermore, this procedure permits to gain a contrast enhancement on the final result. In the first part of the work, the basics on image restoration problems are recalled, and a survey on the state–of–the–art methods is given, with an original contribution regarding scaling techniques in ε–subgradient methods. Then, the Discrepancy and the Constrained Models are analyzed from both theoretical and practical point of view, developing suitable numerical techniques for their solution; furthermore, an inexact version of the Bregman procedure is introduced: such a version allows to have a minor computational cost and maintains the same theoretical features of the exact version. Finally, in the last part, a wide experimentation shows the computational efficiency of the inexact Bregman procedure; furthermore, the three models are compared, showing that in high counts images they provide similar results, while in case of low counts images the Bregman procedure provides reliable restored images. This last consideration is evident not only on test problems, but also in problems coming from Astronomy imaging, particularly in case of High Dynamic Range images, as shown in the final part of the experimental section

Large Scale Inverse Problems

This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation &amp Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

The 2nd International Conference on Mathematical Modelling in Applied Sciences, ICMMAS’19, Belgorod, Russia, August 20-24, 2019 : book of abstracts

The proposed Scientific Program of the conference is including plenary lectures, contributed oral talks, poster sessions and listeners. Five suggested special sessions / mini-symposium are also considered by the scientific committe