14557

13128

كارشناسي ارشد

آناليز عددي

اصفهان : دانشگاه صنعتي اصفهان

1397

سيزده، 90 ص، مصور، جدول، نمودار

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رضا مختاري

فريد بهرامي

سيد محمد حسيني- نادر كريمي- سيد محمود منجگاني

1398/02/04

كتابنامه

رياضي كاربردي

رياضي

1398/02/04

2529172

Investigating numerical solution of some ill posed problems with application to image restoration Farideh Sadat Shari fs shari @math iut ac ir 12 01 2018 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari mokhtari@cc iut ac ir Advisor Dr Farid Bahrami fbahrami@cc iut ac ir 2010 MSC 65F 22 65F 08 94A08 Keywords preconditioner for iterative method ill posed problems regularization reorthogonalization image restoration AbstractStudying regularization methods as well as preconditioning techniques in solving ill posedproblems with application to image restoration is the main goal of this thesis Investigatingof this kind of problems has shown that a great number of problems from various branchesof classical mathematics such as integral and ordinary partial di erential equations may beclassi ed as ill posed problems Systems of linear equations with large coe cient matricesmay arise from the discretization of ill posed problems For example discretization of the rst kind Fredholm integral equation in two dimensional spaces with application to imagerestoration leads to such linear system The history of inverse problems and ill posedproblems returns to about the middle of the 20th century However concept of inverseand ill posed problems goes back to the Hadamard s works at the beginning of the lastcentury According to the Hadamard s de nition a problem is ill posed when its solution is notunique or does not depend continuously on data Moreover the history of image restorationreturns to 1950 s Image restoration often leads to the solving of large linear systems ofequations with very ill conditioned probably singular matrices and error contaminated right hand sides The attainment of a valuable restoration from an available image requires aregularization method In this thesis at rst a brief description of Kronecker product its properties innerproduct and required preliminaries are represented It is supposed that the blurringmatrix has a Kronecker product structure and there is a need to investigate some methods forimage restoration E ective restored images are obtained by using Tikhonov regularizationmethods based on the global Lanczos method and by using the relation between this methodand Gauss type quadrature rules Regularization methods such as Tikhonov regularizationlead to reduction the right hand side error di usion in the computed solution Then anothermethod based on the global Golub Kahan bidiagonalization is also investigated for imagerestoration The relation between the global Golub Kahan bidiagonalization and the Gauss type quadrature rules is utilized to cheaply compute bounds which are useful for determin ing the regularization parameter We also exploit the discrepancy principle for calculating

رضا مختاري

فريد بهرامي

سيد محمد حسيني- نادر كريمي- سيد محمود منجگاني