The ADEnODES Project
Automatic Differentiation Enabled ODE Solvers
Investigators:
Introduction:
Several numerical Ordinary Differential Equation (ODE) solution algorithms require derivative information for effective solution, particularly when a Newton or quasi-Newton solver is involved. For an ODE in standard form, dy/dt=f(t,y), it is frequently the Jacobian, J(t,y)=df(t,y)/dy, that is required. Since f(t,y) defines the ODE, and hence is problem dependent, the Jacobian must either be supplied by the user or determined by the ODE solver itself. Since users frequently cannot be troubled to supply the Jacobian (or perhaps find it difficult to do so), the default behaviour of most current ODE solvers is to approximate the Jacobian by finite-differences. In this project we investigate whether Automatic Differentiation (AD) might provide a more robust, efficient and user friendly approach to determining such information. In particular, we shall consider the ODE solvers provided by the MATLAB computing environment in conjunction with the MAD AD package.
Project Aim: To investigate whether automatic differentiation can robustly provide derivative information required for the numerical solution of ODEs in the MATLAB computing environment.
Progress: To date we have integrated the forward mode of AD provided by MAD's fmad class into the boundary value solver bvp4c. This allows bvp4cAD's quasi-Newton solve to use AD calculated Jacobians which improves robustness and, under MATLAB 6.5, efficiency. The resulting AD enabled solver has been named bvp4cAD (see the bvp4cAD homepage for more information and downloads).
Journal Papers: