Affiliation:
1. Department of Mathematics , 29583 Visvesvaraya National Institute of Technology Nagpur , Maharashtra 440010 , India
Abstract
Abstract
In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form
F
(
u
)
=
y
{F(u)=y}
, where
F
:
𝒟
(
F
)
⊂
𝖴
→
𝖸
{F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}}
is a nonlinear operator between Hilbert spaces
𝖴
{\mathsf{U}}
and
𝖸
{\mathsf{Y}}
. The method is defined as follows:
u
n
+
1
δ
=
u
n
δ
-
(
T
0
∗
T
0
+
α
n
I
)
-
1
T
0
∗
(
F
(
u
n
δ
)
-
y
δ
)
,
u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{%
\ast}(F(u_{n}^{\delta})-y^{\delta}),
where
T
0
=
F
′
(
u
0
)
{T_{0}=F^{\prime}(u_{0})}
and
T
0
∗
=
F
′
(
u
0
)
∗
{T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}}
. Here
F
′
(
u
0
)
{F^{\prime}(u_{0})}
denotes the Frèchet derivative of F at an initial guess
u
0
∈
𝒟
(
F
)
{u_{0}\in\mathcal{D}(F)}
for the exact solution
u
†
{u^{\dagger}}
,
F
′
(
u
0
)
∗
{F^{\prime}(u_{0})^{\ast}}
is the adjoint of
F
′
(
u
0
)
{F^{\prime}(u_{0})}
and
{
α
n
}
{\{\alpha_{n}\}}
is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element
u
0
-
u
†
{u_{0}-u^{\dagger}}
. Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.