Numerical comparison of regularization algorithms for solving ill-posed problems
type: CreativeWork
description: The problems, where small errors in measured data can lead to large errors in the calculated solution, are called ill-posed. For solving ill-posed problems special methods, regularization methods, are used. These methods contain a free parameter, whose value must be appropriately chosen in order to achieve the balance between the stability of computations and retaining the original form of the problem. This doctoral thesis is devoted to numerical analysis of regularization parameter choice rules with respect to their accuracy. We propose a number of new parameter choice rules that give more accurate results and are more stable than known rules on a wide range of practice-oriented problems, considering different cases about the knowledge of the noise level: fully known, approximately known, or unknown. At the same time, we formulate several strategies to construct new parameter choice rules, confirming the advantages of these strategies by extensive numerical experiments. We also propose extrapolated variants of Tikhonov and Lavrentiev method together with computational evidence that they give better accuracy in case of smooth solution by moderate amount of computations. As conclusion, we make recommendations for selecting the regularization method and the proper parameter choice rule, depending on the problem to be solved. More broadly, this work provides an overview of the accuracy potential of existing a posteriori parameter choice rules.Mittekorrektseteks loetakse ülesandeid, kus väike viga mõõtmisandmetes võib kaasa tuua suure vea arvutatud tulemustes (ülesande lahendis). Selliste ülesannete lahendamiseks kasutatakse spetsiaalseid regulariseerimismeetodeid. Need meetodid sisaldavad vaba parameetrit, mille väärtuse sobiva valimisega on võimalik saavutada tasakaalu arvutuste stabiilsuse ning ülesande originaaliläheduse säilitamise vahel. Doktoritöö on pühendatud regulariseerimismeetodite ja regulariseerimisparameetri valikureeglite numbrilisele analüüsile täpsuse …
abstract: The problems, where small errors in measured data can lead to large errors in the calculated solution, are called ill-posed. For solving ill-posed problems special methods, regularization methods, are used. These methods contain a free parameter, whose value must be appropriately chosen in order to achieve the balance between the stability of computations and retaining the original form of the problem. This doctoral thesis is devoted to numerical analysis of regularization parameter choice rules with respect to their accuracy. We propose a number of new parameter choice rules that give more accurate results and are more stable than known rules on a wide range of practice-oriented problems, considering different cases about the knowledge of the noise level: fully known, approximately known, or unknown. At the same time, we formulate several strategies to construct new parameter choice rules, confirming the advantages of these strategies by extensive numerical experiments. We also propose extrapolated variants of Tikhonov and Lavrentiev method together with computational evidence that they give better accuracy in case of smooth solution by moderate amount of computations. As conclusion, we make recommendations for selecting the regularization method and the proper parameter choice rule, depending on the problem to be solved. More broadly, this work provides an overview of the accuracy potential of existing a posteriori parameter choice rules.Mittekorrektseteks loetakse ülesandeid, kus väike viga mõõtmisandmetes võib kaasa tuua suure vea arvutatud tulemustes (ülesande lahendis). Selliste ülesannete lahendamiseks kasutatakse spetsiaalseid regulariseerimismeetodeid. Need meetodid sisaldavad vaba parameetrit, mille väärtuse sobiva valimisega on võimalik saavutada tasakaalu arvutuste stabiilsuse ning ülesande originaaliläheduse säilitamise vahel. Doktoritöö on pühendatud regulariseerimismeetodite ja regulariseerimisparameetri valikureeglite numbrilisele analüüsile täpsuse …
author: Palm, Reimo
date published: 2010-01-18T06:40:52Z
date modified: 2026-07-24T06:23:42Z
provider: University of Tartu
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Tartu