Adaptive use of extended systems for the efficient verified solution of nonlinear systems
Elke Just · 2018
This thesis discusses existing methods for the reliable solution of nonlinear systems of equations and presents various approaches to improve these methods. The reliability of all methods is ensured by the application of interval analysis. In particular, the issue of utilizing extended systems is addressed. These systems are obtained from the given system of equations by the introduction of additional variables for suitable subterms. An existing approach to control the usage of extended systems in an overall branch-and-bound scheme is presented. Subsequently, a new, adaptive strategy is developed. This adaptive strategy allows to exploit the advantages of the different extended systems well-aimed and effectively. We further give detailed considerations concerning the usage of the preconditioned interval Newton method on the extended systems. Although the adaptive strategy provides the title for this work, the discussion of techniques for the reliable solution of nonlinear systems is interspersed with further deliberations for improvements. Another significant part of this work is dedicated to the issue of verification tests. These tests can verify the existence and even uniqueness of solutions in a given bounded box. Existing verification methods are discussed and different modifications of the tests are examined. A general scheme to apply verification tests for square systems is given. For non-square systems we discuss the verification of square subsystems and especially possibilities to fix the variables of underdetermined systems. Our theoretical considerations are supported by numerical studies within the framework of the software SONIC, in which the proposed algorithms and strategies have been implemented and tested.