Determination of the Space-Dependent Source Term in a Fourth-Order Parabolic Problem

The reconstruction of an unknown space-dependent source term in a fourth-order parabolic problem is investigated from the given final or time-integral measured data. The solvability of such inverse problem is proved with final or integral observation, and under some conditions on the input data, the uniqueness of the solution is guaranteed globally. In practice the measured data usually containing noise, then the inverse problem becomes ill-posed. Two schemes shall be applied to obtain the unknown quantity: non-local boundary value method and optimization method. The non-local boundary value method is employed to recover the unknown source term with the final data or integral measurement, and the convergence estimate is derived. For the inverse problem with final or integral measurement, the minimizer of the corresponding optimization method is utilized to approximate the solution. The convergence rates of the minimizer are obtained. Meanwhile, the objective functional is proved to be Fréchet differentiable and convex, then the conjugate gradient method (CGM) is established to numerically determine the source term. Two numerical experiments illustrate both algorithms can be utilized to recover the source term accurately and stably.