الفهرس 
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Abstract
This thesis is a contribution to the field of solution methods for linear
ill-posed problems. This field has been in the focus of mathematical literature
for many years. It OCcursin a wide variety of applications such
as geophysics [26], astrometry [5], mathematical biology [6], and image
restoration ([2], [10]). The notion of a well-posed problem and ill-Posed
problem goes back to a famous paper by Jacques Hadamard published in
1902 [15]. In many science and engineering applications it is necessary to
compute an approximate solution of the linear system;
• Ax == b, A E Rmxn, m > n.
(1)
Throughout this thesis usually, the right-hand side b is corrupted by noise,
r.e.:
b;= b + 15,
where 6accounts for noiseand measurement errors. According to Hadamard
[15J .. well- osed if the followingconditions are the problem (1) IS said to be p ., ique (c) the solution
. (b) the solution IS un ,
satisfied (a) the solution exists, . the system (1) is
h d t Othewse, we say
t’ ously depends on tea a. . ation solution to the system con mu . apprOJum . t
ill-posed system. To determme atn ly due to the noi.se on b, the direc tsho-
(1) after adding noi.se, unfortuna e. f the true soluti.on x, because e
I tion of (1) is a poor approximation 0d hence A is nearl•ysingu Iar , so the
suy.stem has.a huge condition number baen m. exact. This is quite un.fortunatIe
omputed solution in this case may uring errors and computers o~ y
c. b’ often affected by noise or meas 1 ing the system, the noise since IS . ion When so VI have finite floating pom. t precisw .
1
and rounding errors can disturb more and more, and the solution can easily
end up being completely useless. A lot of mathematical problems are
ill-posed. Among them there are the followingvery well known example:
the Fredholm integral equation in the first kind l K(8, t)f(t)dt = g(8), c:S; 8 < d, (2)
we can write this equation in the form
Kf=g,
where K is compact operator, this imply that K-1 is unbounded (if it
exists) and hence the solution f might not have a continuous dependence
on 9 f28]. The thesis intention is to present and analyze a recent methodcalled
dynamical systems method (DSM) for a stable solutions of linear
ill-posed problems. Also, we present one of the traditional stable method
for solving linear ill-posed problems, this method called Tikhonov or variational
regularization method. Comparison between the two methods is one
of the main goals of this thesis. On the other hand, we discuss truncated
singular value decomposition (TSYD) method to solve linear ill-posed
problems and we show that the idea of L-curve to improve Tikhonov regularization
is unstable.
The organization of this thesis is as follows:
In Chapter 1. Some basic notations and fundamentals results used.in
this thesis are introduced. Especial attentions are given to the singular
value decomposition (SYD) and generalized singular value decomposition
(GSYD). We elaborate the properties of ill-posed problems. At the
2
end of this chapter, we will briefly introduce the quadrature method, the
Galerkin method and the Gauss-Leguerre quadrature method for solving
equation (2). Also, we will briefly present Runge-Kutta method for solving
ordinary differential equations which will consider in this thesis.
In Chapter 2. The Tikhonov (or variational) regularization technique
which is one of a popular method for solving ill-posed problems is presented.
The basic idea of Tikhonov regularization is to find an optimal
approximation from a family of approximate solutions depending on”a
positive parameter called regularization parameter. Several methods have
been developed to find this regularization parameter such as the discrepancy
principle and the L-curve method but in this chapter we will focus
on •the second technique. Also, in this chapter, we study the truncated
singular value decomposition (TSVD) method as one of the simplest type
of regularization methods that can be applied to linear ill-posed problems.
This method is based on singular value decomposition (SVD)-type expansions
for the linear system solution.
In Chapter 3. The dynamical systems method (DSM) is presented as
a new technique for solving ill-posed problems with noise. This method is
introduced theoretically by A. G. Ramm. The DSM is based on an analysis
of the solution of Cauchy problem for linear and nonlinear differential
equations in Hilbert space. Such an analysis was done for well-posed
and ill-posed problems (see [37], and the references sited therein). N.
H. Sweilam and A. M. Nagy in [41] (see also [42]) successfully applied
DSM and variational regularization method to obtain numerical solutions
to some well known ill-posed problems with noise from literature. The