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 ) ∈ Ω T , from the Dirichlet type measured output data h ( t ) : = u ( 0 , t ) is studied. A formula for the Fréchet gradient of the cost functional J ( F ) = ‖ u ( 0 , t ; F ) − h ( t ) ‖ 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.