On Consistency Properties of Discrete Adjoint Linear Multistep Methods

In this paper we analyze the consistency properties of discrete adjoints of linear multistep methods. Discrete adjoints are very popular in optimization and control since they can be constructed automatically by reverse mode automatic differentiation. The consistency analysis reveals that the discrete linear multistep adjoints are, in general, inconsistent approximations of the adjoint ODE solution along the trajectory. However, the discrete adjoints at the initial time (and therefore the discrete adjoint gradients) converge to the adjoint ODE solution with the same order as the original linear multistep method. Discrete adjoints inherit the zero-stability properties of the forward method. Numerical results confirm the theoretical findings.