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

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 ? [0, T]. Compactness and Lipschitz continuity of the input-output operator ?[p] := u(x, t; p)|x=0+ , ?[ · ] : P ? H1(0, l) › L2(0, T) are proved. Then an existence of a quasi-solution of the inverse problem is obtained. We prove Fréchet differentiability of the Tikhonov functional and derive an explicit gradient formula for the Fréchet 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. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.