Affiliation:
1. Muskhelishvili Institute of Computational Mathematics , Georgian Technical University , 4 G. Peradze Str. , Tbilisi 0159 , Georgia
Abstract
Abstract
We fix a continuous linear operator
A
:
H
→
M
{A:H\rightarrow M}
acting between the Hilbert spaces H and M that admits a singular value decomposition (SVD). We consider the following ill-posed problem: for an element
f
∈
M
{f\in M}
, find
u
∈
H
{u\in H}
such that
A
u
=
f
{Au=f}
and a generalized solution
in the sense of Moore–Penrose u is sought that satisfies the equation
A
*
A
u
=
A
*
f
{A^{*}Au=A^{*}f}
. Moreover, we fix an integer
n
∈
ℕ
0
=
{
0
,
1
,
2
,
…
}
{n\in\mathbb{N}_{0}=\{0,1,2,\dots\}}
and transfer this equation to a special Hilbert space
D
(
(
A
*
A
)
-
n
)
{D((A^{*}A)^{-n})}
of n-orbits. For an approximate solution of this equation in the case of a nonadaptive information on the right-hand side f, a linear spline algorithm is constructed.
The specificity of the considered norm is that the approximate solution is the truncated singular value decomposition (TSVD) and does not depend on n. In the case
n
=
0
{n=0}
, the space
D
(
(
A
*
A
)
-
n
)
{D((A^{*}A)^{-n})}
coincides with H and we obtain the results for the latter space. In the limiting case of the Fréchet–Hilbert space of all orbits
D
(
(
A
*
A
)
-
∞
)
{D((A^{*}A)^{-\infty})}
, the equation
A
*
A
u
=
A
*
f
{A^{*}Au=A^{*}f}
becomes well-posed and was considered in
[D. Ugulava and D. Zarnadze,
On a linear generalized central spline algorithm of computerized tomography,
Proc. A. Razmadze Math. Inst. 168 2015, 129–148].
It is also noted that the space
D
(
(
A
*
A
)
-
∞
)
{D((A^{*}A)^{-\infty})}
is the projective limit of the sequence of Hilbert spaces
{
D
(
(
A
*
A
)
-
n
)
}
{\{D((A^{*}A)^{-n})\}}
. The application of the obtained results for the computerized tomography problem, i.e., for the inversion of the Radon transform
ℜ
{\mathfrak{R}}
with the SVD of Louis
[A. K. Louis,
Orthogonal function series expansions and the null space of the Radon transform,
SIAM J. Math. Anal. 15 1984, 3, 621–633]
in the space
D
(
(
ℜ
*
ℜ
)
-
n
)
)
{D((\mathfrak{R}^{*}\mathfrak{R})^{-n}))}
is given.
Reference24 articles.
1. R. L. Dietz,
Die approximative Inverse als Rekonstruktionsmethode in der Röntgen-Computertomographie,
Ph.D. thesis, Universität des Saarlandes, Saarbrücken, 1999.
2. A. Hertle,
Continuity of the Radon transform and its inverse on Euclidean space,
Math. Z. 184 (1983), no. 2, 165–192.
3. M. A. Kon, K. Ritter and A. G. Werschulz,
On the average case solvability of ill-posed problems,
J. Complexity 7 (1991), no. 3, 220–224.
4. A. K. Louis,
Orthogonal function series expansions and the null space of the Radon transform,
SIAM J. Math. Anal. 15 (1984), no. 3, 621–633.
5. A. K. Louis,
Approximate inverse for linear and some nonlinear problems,
Inverse Problems 12 (1996), no. 2, 175–190.
Cited by
1 articles.
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