An inverse coefficient problem with nonlinear parabolic equation

2009 ◽  
Vol 34 (1-2) ◽  
pp. 195-206 ◽  
Author(s):  
Li Jing ◽  
Xu Youjun
Keyword(s):  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Nonlinear Parabolic ◽  
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}.

Comparing numerical methods for inverse coefficient problem in parabolic equation

2006 ◽  
Vol 179 (2) ◽  
pp. 567-571 ◽  
Author(s):  
Afet Golayoglu Fatullayev ◽  
Emine Can ◽  
Nizami Gasilov
Keyword(s):  
Parabolic Equation ◽  
Numerical Methods ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Inverse Coefficient
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