TITLE:
On the Regularization Method for Solving Ill-Posed Problems with Unbounded Operators
AUTHORS:
Nguyen Van Kinh
KEYWORDS:
Ill-Posed Problem, Regularization Method, Unbounded Linear Operator
JOURNAL NAME:
Open Journal of Optimization,
Vol.11 No.2,
June
15,
2022
ABSTRACT: Let be a linear, closed, and densely defined unbounded operator, where X and Y are Hilbert spaces. Assume that A is not boundedly invertible. Suppose the equation Au=f is solvable, and instead of knowing exactly f only know its approximation satisfies the condition: In this paper, we are interested a regularization method to solve the approximation solution of this equation. This approximation is a unique global minimizer of the functional , for any , defined as follows: . We also study the stability of this method when the regularization parameter is selected a priori and a posteriori. At the same time, we give an application of this method to the weak derivative operator equation in Hilbert space.