An Approach to Parameter Estimation and Model Selection in Differential Equations

Two distinct and essentially independent sources of error occur in the parameter estimation problem – the error due to noisy observations, and the error due to approximation or discretization effects in the computational procedure. These give contributions of different orders of magnitude so the problem is essentially a two grid problem and there is scope for balancing these to minimize computational effort. Here the underlying computing procedures that determine these errors are reviewed. There is considerable structure in the integration of the differential system, and the role of cyclic reduction in unlocking this is discribed. The role of the stochastic effects in the optimization component of the computation is critically important in understanding the success of the Gauss-Newton algorithm, and the importance both of adequate data and of a true model is stressed. If the true model must be sought among a range of competitors then a stochastic embedding technique is suggested that converts under-specified models into “non-physical” consistent models for which the Gauss-Newton algorithm can be used.