Methods in optimization and feasibility for inverse problems and tomography

• Grant number: 13/19504-9
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: February 01, 2014
• End date: January 31, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Alvaro Rodolfo De Pierro
• Grantee: Daniel Reem
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

Many Inverse Problems, especially in emission and X rays computed tomography, are modeled by large scale systems of linear equations that give rise to ill conditioned mathematical problems. In order to find a stable solution it is necessary to add information not contained in the equations. All this information is used through different models where either an optimization or a feasibility problem should be solved. Also, this generates the necessity of studying specific algorithms for this type of mathematical problems as well as the study of the parameters appearing in those algorithms. In this project we propose to study several open problems and theoretical aspects of these algorithms and models. We will study the relationship between the Douglas-Rachford method and relaxed subgradient as well as the acceleration proposed by Nesterov-Beck-Teboulle. Also we will study the possibility of eliminating the underrelaxation parameters analyzing the asymptotic convergence properties. We will analyze the possibility of obtaining a unified theory for the set of divergence measures, like Kullback-Leibler, that are used to measure equations consistency that models the information about noise in data in Inverse Problems. (AU)

Scientific publications

(4)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

REEM, DANIEL; REICH, SIMEON; DE PIERRO, ALVARO. Re-examination of Bregman functions and new properties of their divergences. OPTIMIZATION, v. 68, n. 1, SI, p. 279-348, JAN 2 2019. (13/19504-9)

REEM, DANIEL; REICH, SIMEON; DE PIERRO, ALVARO. A Telescopic Bregmanian Proximal Gradient Method Without the Global Lipschitz Continuity Assumption. JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, v. 182, n. 3, p. 851-884, SEP 2019. (13/19504-9)

REEM, DANIEL; REICH, SIMEON; DE PIERRO, ALVARO. Re-examination of Bregman functions and new properties of their divergences. OPTIMIZATION, v. 68, n. 1, p. 70-pg., 2019-01-02. (13/19504-9)

REEM, DANIEL; DE PIERRO, ALVARO. A new convergence analysis and perturbation resilience of some accelerated proximal forward-backward algorithms with errors. INVERSE PROBLEMS, v. 33, n. 4, APR 2017. (13/19504-9)