Variational data assimilation problems are concerned with computing unknown initial values for the simulation and prediction of natural phenomena, most notably in weather prediction, and are usually solved via an ill-posed optimal control problem for the initial state at the time of the first available measurements. An alternative "forward" approach focuses on computation of the final state after this interval which is just as suitable for prediction purposes and is well-posed without additional regularization. Specifically, it is possible to compute projections of the unknown final state on all elements of an orthonormal basis, which theoretically allows for the complete reconstruction of the final state. In this paper, an efficient numerical method for linear evolution equations of diffusive type is presented, and convergence of the numerical approximation based on a discontinuous Galerkin discretization is proved. The key of this method is the computation of an adaptively ordered orthonormal basis using proper orthogonal decomposition. Numerical examples for a scalar convection-diffusion equation in two and three dimensions show the effectiveness of the method.