Least Squares Smoothing of Nonlinear Systems

We consider the fixed interval smoothing problem for data from linear or nonlinear models where there is a priori information about the boundary values of the state process. The nonlinearities and boundary values preclude a stochastic approach so instead we use a least squares methodology. The resulting variational equations are a coupled system of ordinary differential equations for the state and costate involving boundary conditions. If the model is linear and the a priori information is only about the initial state then several authors have given methods for solving the resulting equations in two sweeps. If the model is linear but the a priori information is about both the initial and final states then direct methods have been proposed. If the state dimension is large these methods can be very expensive and moreover they don’t readily generalize to nonlinear models. Therefore we present an iterative method for solving both linear and nonlinear problems.