INFORMATION ON MASTER’THESIS
1. Full name: Vu Xuan Quynh
2. Sex: Male
3. Date of birth: 06/12/1990
4. Place of birth: Bac Giang
5. Changes in academic process: No
6. Official thesis title: "Browder-Tikhonov regularization method for nonlinear ill-posed
problems of accretive type."
7. Major: Applied Mathematics
8. Code: 60 46 01 12
9. Supervisors: Professor Dr. Nguyen Buong, Institute of Information Technology,
Vietnam Academy of Science and Technology.
10. Summary of the results of the thesis:
Many problems in science, technology, economy, etc. lead to problems in which their
solutions are unstable with respect to initial data. When initial data changes a little,
problems may be not have solution or having solution which is different far from correct
solution. They are called ill-posed problems and we need to find stable methods to solve
these problems.
We consider this equation
,
X,
(1)
where A is operator from metric space X to metric space Y. The equation (1) is called
well-posed problem if :
a) For each
Y, there exist solution x( )
X of (1);
b) This solution is unique;
c) This solution depends continuously on data (A, ).
If one of three above conditions is not satisfied then the equation (1) is called ill-posed
problem. When Y is Banach space with norm ||.||, then the above problem can be
regularized by minimizing Tikhonov smooth functional
( )=
where
D(
+
),
) = D(A), with choosing a parameter α = α (h,
approximation of (A, ),
> 0 is a regularize parameter,
), (
,
) is a
) is stable function. In the
case, the problem is nonlinear then finding minima element of Tikhonov functional
become difficult. Therefore, to solve this problem, when A : X
is monotone
operator, Browder proposed a new Tikhonov regularization form by using a operator
having hemicontinous property and strongly monotone. Follow this idea, Alber used
general duality mapping to regularize this problem.
To finding solution of (1), we consider the following Browder-Tikhonov regularization
method :
A( ) + ( -
)=
,
(2)
where A : X -› X is operator of monontone type in Banach space has approximation
property. When normed duality mapping J is weak to weak continous and continous then
(2) has a unique solution
converges to
solution of (1). We also show this
convergence when J has no weak to weak continous property but satisfies two conditions:
||A( ) - A(
where
exist z
) – J* A'(
)*J ( -
)||
X, > 0, J* is normed duality mapping of
||A( ) - A(
,
)||,
(3)
is a solution of (1) and there
X such that
A'(
)z =
-
.
(4)
Finally, if J is not weak to weak continous and no satisfy the above two conditions (3)
and (4) then this method is also convergence.
Thesis is presented in two chapters :
Chapter 1 includes :
+ Some knowledge about functional analysis.
+ Introduction about ill-posed problem and regularization algorithm.
+ Equation with operator of accretive type.
Chapter 2 presents:
+ Some results of Browder-Tikhonov regularization method when normed duality
mapping J is weak to weak continous.
+ Consider this regularization method in the case J is not weak to weak continous.
+ Present some new results of Newton-Kantorovich iterative method combine with
Browder-Tikhonov regularization method.
September 25, 2015
Student
Vu Xuan Quynh