An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions

2011 ◽  
Vol 34 (6) ◽  
pp. 692-702 ◽  
Author(s):  
Mansur I. Ismailov ◽  
Fatma Kanca
Keyword(s):  
Parabolic Equation ◽  
Nonlocal Boundary ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Inverse Coefficient

Lipschitz continuity of the Fréchet gradient in an inverse coefficient problem for a parabolic equation with Dirichlet measured output

2018 ◽  
Vol 26 (3) ◽  
pp. 349-368 ◽  
Author(s):  
Alemdar Hasanov
Keyword(s):  
Parabolic Equation ◽  
Lipschitz Continuity ◽  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Gradient Methods ◽  
Neumann Boundary ◽  
Iteration Algorithm ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Inverse Coefficient

AbstractThis paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional {J(k):=(1/2)\lVert u(0,\cdot\,;k)-f\rVert^{2}_{L^{2}(0,T)}} corresponding to an inverse coefficient problem for the {1D} parabolic equation {u_{t}=(k(x)u_{x})_{x}} with the Neumann boundary conditions {-k(0)u_{x}(0,t)=g(t)} and {u_{x}(l,t)=0}. In addition, compactness and Lipschitz continuity of the input-output operator\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)} 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}(\mathcal{K})}. Specifically, this lemma asserts that if {J\in C^{1,1}(\mathcal{K})} and {\{k^{(n)}\}\subset\mathcal{K}} is the sequence of iterations obtained by the Landweber iteration algorithm {k^{(n+1)}=k^{(n)}+\omega_{n}J^{\prime}(k^{(n)})}, then for {\omega_{n}\in(0,2/L_{g})}, where {L_{g}>0} is the Lipschitz constant, the sequence {\{J(k^{(n)})\}} is monotonically decreasing and {\lim_{n\to\infty}\lVert J^{\prime}(k^{(n)})\rVert=0}.

scholarly journals Inverse Coefficient Problem of the Parabolic Equation with Periodic Boundary and Integral Overdetermination Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Fatma Kanca
Keyword(s):  
Parabolic Equation ◽  
Continuous Dependence ◽  
Periodic Boundary ◽  
Fourier Method ◽  
Time Dependent ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Numerical Examples ◽  
Integral Overdetermination ◽  
Inverse Coefficient

This paper investigates the inverse problem of finding a time-dependent diffusion coefficient in a parabolic equation with the periodic boundary and integral overdetermination conditions. Under some assumption on the data, the existence, uniqueness, and continuous dependence on the data of the solution are shown by using the generalized Fourier method. The accuracy and computational efficiency of the proposed method are verified with the help of the numerical examples.

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