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}.
Inverse Coefficient Problem for Grushin-Type Parabolic Operators
