A splitting method for a backward parabolic equation with time-dependent coefficients

In this paper we propose a stable numerical method for an ill-posed backward parabolic equation with time-dependent coefficients in a parallelepiped. The problem is reformulated as an ill-posed least squares problem which is solved by the conjugate gradient method with an a posteriori stopping rule. The least squares problem is discretized by a splitting method which reduces the large dimensions of the discretized problem. We calculate the gradient of the objective functional of the discretized least squares problem by the aid of an adjoint discretized problem which enhances its accuracy. The algorithm is tested on several examples, that proves its efficiency.