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Transactions on Engineering and Computing Sciences - Vol. 11, No. 6
Publication Date: December 25, 2023
DOI:10.14738/tecs.116.16001.
Kinh, N. V. (2023). On the Regularization Method for Solving Ill-Posed Problems with Linear Closed Densely Operators. Transactions
on Engineering and Computing Sciences, 11(6). 67-79.
Services for Science and Education – United Kingdom
On the Regularization Method for Solving Ill-Posed Problems
with Linear Closed Densely Operators
Nguyen Van Kinh
Faculty of Applied Science, Ho Chi Minh University of Industry and Trade
140 Le Trong Tan Street, Ward Tay Thanh, District Tan Phu,
Ho Chi Minh City, Vietnam
ABSTRACT
Let
A D A X Y : ( )Ì ®
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 if
f f d
- £ d
then the following results are
provided:Problem
2 2
,
F u Au f u ( ): a d d = - + a
has a unique global minimizer
,
ua d
for
any
f
d
, and
( )
1
, Y u A AA I f a d d a
- * *
= +
. There is a function α(δ),
0
lim ( ) 0
d
a d
®
=
such that
( ), 0 0
lim 0 u x a d d d®
- =
, where
0
x
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 oprator. At the same time give an
application to the weak derivative operator equation.
Keywords: Ill-posed problem, regularization method, unbounded linear operator.
INTRODUCTION
Let
A D A X Y : ( )Ì ®
be a linear, closed, densely defined unbounded operator, where
X
and
Y
are Hilbert spaces. Consider the equation
Au f = (1)
Problem finding solution of (1) is called ill-posed in the sense of Hadamard [17] if
A
is not
boundedly invertible. This may happen if the null space
N A u Au ( ) : 0 = = { }
is not trivial, i.e.,
A
is not injective, or if
A
is injective but
1 A
-
is unbounded, i.e., the range of
A, R A( )
is not
closed [3].
If
A < ¥
, and
f
d
the noisy data, are given
f f ,
d
- £ d (2)