Numerical solutions to some ill-posed problems

dc.contributor.authorHoang, Nguyen Si
dc.date.accessioned2011-05-26T14:17:50Z
dc.date.available2011-05-26T14:17:50Z
dc.date.graduationmonthAugust
dc.date.issued2011-05-26
dc.date.published2011
dc.description.abstractSeveral methods for a stable solution to the equation $F(u)=f$ have been developed. Here $F:H\to H$ is an operator in a Hilbert space $H$, and we assume that noisy data $f_\delta$, $\|f_\delta-f\|\le \delta$, are given in place of the exact data $f$. When $F$ is a linear bounded operator, two versions of the Dynamical Systems Method (DSM) with stopping rules of Discrepancy Principle type are proposed and justified mathematically. When $F$ is a non-linear monotone operator, various versions of the DSM are studied. A Discrepancy Principle for solving the equation is formulated and justified. Several versions of the DSM for solving the equation are formulated. These methods consist of a Newton-type method, a gradient-type method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation $F(u)=f$ is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to the equation $F(u)=f$ is proved. This dissertation consists of six chapters which are based on joint papers by the author and his advisor Prof. Alexander G. Ramm. These papers are published in different journals. The first two chapters deal with equations with linear and bounded operators and the last four chapters deal with non-linear equations with monotone operators.
dc.description.advisorAlexander G. Ramm
dc.description.degreeDoctor of Philosophy
dc.description.departmentDepartment of Mathematics
dc.description.levelDoctoral
dc.identifier.urihttp://hdl.handle.net/2097/9204
dc.language.isoen_US
dc.publisherKansas State University
dc.rights© the author. This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/
dc.subjectIll-posed problems
dc.subjectDynamical Systems Method
dc.subjectRegularization
dc.subjectDiscrepancy Principle
dc.subjectMonotone operators
dc.subject.umiMathematics (0405)
dc.titleNumerical solutions to some ill-posed problems
dc.typeDissertation

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