A gradient-based approach for identification of the zeroth-order coefficient 𝑝(𝑥) in the parabolic equation 𝑢𝑡 = (𝑘(𝑥)𝑢𝑥)𝑥 − 𝑝(𝑥)𝑢 from Dirichlet-type measured output

EgeUn###
This paper is devoted to the inverse problem of identifying an unknown spacewise-dependent zeroth-order coefficient p(x) in the 1D diffusion equation u(t) = (k(x)u(x))(x) - p(x)u from boundary Dirichlet measured output f(t) := u(0, t), t is an element of [0, T]. Compactness and Lipschitz continuity of the input-output operator Phi[p] := u(x, t; p)vertical bar x=0(+), Phi[ . ] : p subset of H-1(0, l) bar right arrow L-2(0, T) are proved. Then an existence of a quasi-solution of the inverse problem is obtained. We prove Frechet differentiability of the Tikhonov functional and derive an explicit gradient formula for the Frechet gradient through the solutions of the direct and corresponding adjoint problems solutions. This allows to use gradient-type algorithms for the numerical solution of the considered inverse problem.