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

Lipschitz continuity of the Fréchet gradient in an inverse coefficient problem for a parabolic... AbstractThis paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional J⁢(k):=(1/2)⁢∥u⁢(0,⋅;k)-f∥L2⁢(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 ut=(k⁢(x)⁢ux)x{u_{t}=(k(x)u_{x})_{x}}with the Neumann boundary conditions -k⁢(0)⁢ux⁢(0,t)=g⁢(t){-k(0)u_{x}(0,t)=g(t)}and ux⁢(l,t)=0{u_{x}(l,t)=0}. In addition, compactness and Lipschitz continuity of the input-output operatorΦ[k]:=u(x,t;k)|x=0+,Φ[⋅]:𝒦⊂H1(0,l)↦H1(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 C1,1⁢(𝒦){C^{1,1}(\mathcal{K})}. Specifically, this lemma asserts that if J∈C1,1⁢(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)+ωn⁢J′⁢(k(n)){k^{(n+1)}=k^{(n)}+\omega_{n}J^{\prime}(k^{(n)})}, then for ωn∈(0,2/Lg){\omega_{n}\in(0,2/L_{g})}, where Lg>0{L_{g}>0}is the Lipschitz constant, the sequence {J⁢(k(n))}{\{J(k^{(n)})\}}is monotonically decreasing and limn→∞⁡∥J′⁢(k(n))∥=0{\lim_{n\to\infty}\lVert J^{\prime}(k^{(n)})\rVert=0}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Inverse and III-posed Problems de Gruyter

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

, Volume 26 (3): 20 – Jun 1, 2018
20 pages

Publisher
de Gruyter
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1569-3945
eISSN
1569-3945
D.O.I.
10.1515/jiip-2017-0106
Publisher site
See Article on Publisher Site

### Abstract

AbstractThis paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional J⁢(k):=(1/2)⁢∥u⁢(0,⋅;k)-f∥L2⁢(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 ut=(k⁢(x)⁢ux)x{u_{t}=(k(x)u_{x})_{x}}with the Neumann boundary conditions -k⁢(0)⁢ux⁢(0,t)=g⁢(t){-k(0)u_{x}(0,t)=g(t)}and ux⁢(l,t)=0{u_{x}(l,t)=0}. In addition, compactness and Lipschitz continuity of the input-output operatorΦ[k]:=u(x,t;k)|x=0+,Φ[⋅]:𝒦⊂H1(0,l)↦H1(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 C1,1⁢(𝒦){C^{1,1}(\mathcal{K})}. Specifically, this lemma asserts that if J∈C1,1⁢(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)+ωn⁢J′⁢(k(n)){k^{(n+1)}=k^{(n)}+\omega_{n}J^{\prime}(k^{(n)})}, then for ωn∈(0,2/Lg){\omega_{n}\in(0,2/L_{g})}, where Lg>0{L_{g}>0}is the Lipschitz constant, the sequence {J⁢(k(n))}{\{J(k^{(n)})\}}is monotonically decreasing and limn→∞⁡∥J′⁢(k(n))∥=0{\lim_{n\to\infty}\lVert J^{\prime}(k^{(n)})\rVert=0}.

### Journal

Journal of Inverse and III-posed Problemsde Gruyter

Published: Jun 1, 2018

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