A study about solving non-linear systems: theoretical perspectives and applications

Author(s)

Tiara Martini dos Santos

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2016-02-22

Examining board members

José Mario Martínez Pérez; Sandra Augusta Santos; Lucio Tunes dos Santos; Juliano de Bem Francisco; Luís Felipe Cesar da Rocha Bueno

• Advisor: José Mario Martínez Pérez

Abstract

The aim of this work is to study and analyse different approaches for solving nonlinear systems. First of all, a sparse version of Newton's method is applied for finding a solution of a horizontal nonlinear complementarity problem (HNCP) associated to a feasible solution of a mathematical programming problem with complementarity constraints (MPCC). The algorithm combines Newton-like and Projected-Gradient directions with a line-search procedure that guarantees global convergence to a stationary point of the merit function associated to this problem. Local quadratic convergence is stated under reasonable hypothesis. Numerical experience on test problems from a well-known collection illustrates the efficiency of the algorithm to find feasible solutions of MPCC in practice. Next, a quasi-Newton strategy for accelerating the convergence of fixed-point iterations is analysed. For that, classical secant updates are considered. Numerical experiments on a training set are developed in order to validate this strategy. After that, the quasi-Newton strategy is applied on the practical problem of represent the kinetic behavior of a PET (Positron Emission Tomography) tracer during cardiac perfusion. The performance of the method when applied to real data problems is illustrated numerically. Finally, a hybrid method combining Newton and Homotopy directions is introduced for solving problems where Newton's method presented difficulties. Initial experiments provide a basis for the presented technic validation (AU)

• FAPESP's process: 12/10444-0 - A new integrated software for nonlinear systems
• Grantee: Tiara Martini dos Santos
• Support Opportunities: Scholarships in Brazil - Doctorate