An iterative method for the solution of ill-posed parabolic systems

This note is concerned with the identification of the unknown initial condition for a parabolic system. It introduces an iterative algorithm that can be used to recover the initial condition. The algorithm assumes a function for the unknown initial condition and obtains a background field. It then obtains the error dynamics. The error dynamics is also ill-posed in the sense that the initial condition is unknown. Using the measurements obtained at the boundaries, the algorithm introduces two formulations for the error dynamics. These two formulations share the same initial condition. By equating the responses of these two formulations it is then possible to obtain an equation for the unknown initial condition. A number of numerical examples are also used to study the performance of the algorithm.