Numeričko rješavanje problema sedlaste točke

master's thesis

• Author: Mihovil Telišman
• Mentor: Nela Bosner (mentor)
• Committee member: Nela Bosner (predsjednik povjerenstva)
• Committee member: Vjeran Hari (član povjerenstva)
• Committee member: Luka Grubišić (član povjerenstva)
• Committee member: Mirko Primc (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-11-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu opisani su neki problemi koji se svode na sustave sa sedlastom točkom među kojima su najvažniji problemi inkompresibilnog toka i problemi iz područja optimizacije. Da bi se mogli konstruirati efikasni algoritmi za rješavanje, bilo je potrebno iznijeti teorijsku podlogu matrica koji se javljaju prilikom diskretizacija promatranih problema. Iz teorijskih razmatranja zaključili smo da promatrane matrice imaju općenito jako loša spektralna svojstva zbog svoje blizine singularnim matricama. Ipak, koristeći se prije svega Schurovim komplementom, neke zapreke je moguće zaobići i u kombinaciji sa prekondiciniranjem konstruirati efikasne algoritme za rješavanje takvih sustava. Uzawa metoda je povijesno prva metoda koja je efikasno rješavala određenu klasu ovih problema dok su se kasnije pojavile modernije metode u potpunosti zasnovane na iteracijama iz Krilovljevih potprostora čiji je najznačajniji predstavnik GMRES metoda. Međutim, zbog loše uvjetovanosti matrica gotovo je uvijek potrebno provoditi prekondicioniranje. Teorija odabira matrice prekondicioniranja je iznimno složena sa mnogo otvorenih pitanja. Zbog toga smo mi obradili numerički jednostavnije primjere.

Abstract (english)

In this paper we described some of the problems that lead to saddle point systems with special emphasis on incompressible flow problems and optimization problems. In order to construct efficient solution algorithms, it was necessary to understand the properties of matrices that occur as a result of discretization of a continuous problems. The most important conclusion of this theory is that matrices from saddle point systems have very poor spectral properties due to their vicinity to singular matrices. Nevertheless, using Schur complement decomposition and other techniques, it is possible to overcome certain obstacles and in combination with preconditioning construct a viable solution algorithms for this ill-conditioned systems. The first effective method developed for solving a certain class of saddle point systems is Uzawa method. Later, many methods were developed that are primary based on Krylov subspace iterations with the most prominent of them GMRES method. Unfortunately, due to ill-conditioned nature of saddle point systems, some sort of preconditioning is almost always required. Since the theory of determining a precondition matrix is very complex with many unanswered questions, the numerical examples in this paper are a bit simpler.

• Keywords: sustavi sa sedlastom točkom; problemi inkompresibilnog toka; problemi optimizacije; Schurov komplement; Uzawa metoda; GMRES metoda
• Keywords (english): saddle point systems; incompressible flow problems; optimization problems; Schur complement; Uzawa method; GMRES method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:089984
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-03-29 11:50:04