An inverse source problem with single Dirichlet type measured output data for a linear parabolic equation

The problem of determining an unknown source term in a linear parabolic equation u(t) = (k(x)u(x))(x) + F(x, t), (x, t) epsilon Omega(T), from the Dirichlet type measured output data h(t) := u(0, t) is studied. A formula for the Frechet gradient of the cost functional J(F) = parallel to u(0, t; f)-h(t)parallel to(2) is derived via the solution of the corresponding adjoint problem, within the weak solution theory for PDEs and the quasi-solution approach. The Lipschitz continuity of the gradient is proved. Based on the obtained results the convergence theorem for the gradient method is proposed. (c) 2011 Elsevier Ltd. All rights reserved.