TITLE:
On the Regularization Method to Stable Approximate Solution of Equations of the First Kind with Unbounded Operators
AUTHORS:
Nguyen Van Kinh
KEYWORDS:
Ill-Posed Problem, Regularization Method, Unbounded Linear Operator
JOURNAL NAME:
Open Journal of Optimization,
Vol.14 No.1,
January
21,
2025
ABSTRACT: Let
A:D(
A
)⊂X→Y
be a linear, closed, densely defined unbounded operator, where
X
and
Y
are Hilbert spaces. Assume that
A
is not boundedly invertible. If equation (1)
Au=f
is solvable, and
‖
f
δ
−f ‖≤δ
then the following results are provided: Problem
F
α,δ
(
u
):=
‖
Au−
f
δ
‖
2
+α
‖ u ‖
2
has a unique global minimizer
u
α,δ
for any
f
δ
∈Y
, and
u
α,δ
=
A
*
(
A
A
*
+α
I
Y
)
−1
f
δ
. Then there is a function
α(
δ
)
,
lim
δ→0
α(
δ
)=0
such that
lim
δ→0
‖
u
α(
δ
),δ
−
x
0
‖=0
, where
x
0
is the unique minimal-norm solution to (1). In this paper we introduce the regularization method solving Equation (1) with
A
being a linear, closed, densely defined unbounded operator. At the same time, an application is given to the weak derivative operator equation.