Affiliation:
1. Kocaeli Universitesi , Izmir , Turkey
Abstract
Abstract
This paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional
J
(
k
)
:=
(
1
/
2
)
∥
u
(
0
,
⋅
;
k
)
-
f
∥
L
2
(
0
,
T
)
2
{J(k):=(1/2)\lVert u(0,\cdot\,;k)-f\rVert^{2}_{L^{2}(0,T)}}
corresponding to an inverse coefficient problem for the
1
D
{1D}
parabolic equation
u
t
=
(
k
(
x
)
u
x
)
x
{u_{t}=(k(x)u_{x})_{x}}
with the Neumann boundary conditions
-
k
(
0
)
u
x
(
0
,
t
)
=
g
(
t
)
{-k(0)u_{x}(0,t)=g(t)}
and
u
x
(
l
,
t
)
=
0
{u_{x}(l,t)=0}
. In addition, compactness and Lipschitz continuity of the input-output operator
Φ
[
k
]
:=
u
(
x
,
t
;
k
)
|
x
=
0
+
,
Φ
[
⋅
]
:
𝒦
⊂
H
1
(
0
,
l
)
↦
H
1
(
0
,
T
)
,
\Phi[k]:=u(x,t;k)\lvert_{x=0^{+}},\quad\Phi[\,\cdot\,]:\mathcal{K}\subset H^{1%
}(0,l)\mapsto H^{1}(0,T),
as well as solvability of the regularized inverse problem and the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional are proved. Furthermore, relationships between the sufficient conditions for the Lipschitz continuity of the Fréchet gradient and the regularity of the weak solution of the direct problem as well as the measured output
f
(
t
)
:=
u
(
0
,
t
;
k
)
{f(t):=u(0,t;k)}
are established. One of the derived lemmas also introduces a useful application of the Lipschitz continuity of the Fréchet gradient. This lemma shows that an important advantage of gradient methods comes when dealing with the functionals of class
C
1
,
1
(
𝒦
)
{C^{1,1}(\mathcal{K})}
. Specifically, this lemma asserts that if
J
∈
C
1
,
1
(
𝒦
)
{J\in C^{1,1}(\mathcal{K})}
and
{
k
(
n
)
}
⊂
𝒦
{\{k^{(n)}\}\subset\mathcal{K}}
is the sequence of iterations obtained by the Landweber iteration algorithm
k
(
n
+
1
)
=
k
(
n
)
+
ω
n
J
′
(
k
(
n
)
)
{k^{(n+1)}=k^{(n)}+\omega_{n}J^{\prime}(k^{(n)})}
, then for
ω
n
∈
(
0
,
2
/
L
g
)
{\omega_{n}\in(0,2/L_{g})}
, where
L
g
>
0
{L_{g}>0}
is the Lipschitz constant, the sequence
{
J
(
k
(
n
)
)
}
{\{J(k^{(n)})\}}
is monotonically decreasing and
lim
n
→
∞
∥
J
′
(
k
(
n
)
)
∥
=
0
{\lim_{n\to\infty}\lVert J^{\prime}(k^{(n)})\rVert=0}
.
Reference11 articles.
1. R. Bellman,
Asymptotic series for the solutions of linear differential-difference equations,
Rend. Circ. Mat. Palermo (2) 7 (1958), 261–269.
2. J. R. Cannon and P. DuChateau,
An inverse problem for a nonlinear diffusion equation,
SIAM J. Appl. Math. 39 (1980), no. 2, 272–289.
3. P. DuChateau,
Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems,
SIAM J. Math. Anal. 26 (1995), no. 6, 1473–1487.
4. H. W. Engl, M. Hanke and A. Neubauer,
Regularization of Inverse Problems,
Math. Appl. 375,
Kluwer Academic Publishers, Dordrecht, 1996.
5. L. C. Evans,
Partial differential equations, 2nd ed.,
Grad. Stud. Math. 19,
American Mathematical Society, Providence, 2010.
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献