SEMINAR: Mathematics Colloquium

Abstract:

The Bock iteration (Bock, 1983) is a method for the minimization of an objective function in the form of a sum of squares subject to a set of equality constraints. It was formulated originally as a method for solving the problem of estimating parameters in a system of ordinary differential equations (ODE) posed in its simultaneous form, and this remains its principal application. The simultaneous method requires that the discretized ODE be imposed as equality constraints on the objective function of the data fitting problem. However, not only is the data fitting problem explicitly constrained, but the constraint system possesses the key additional feature that it becomes unbounded in size as the scale of the discretization $ elta t o 0, ; n o nfty$, where $n$ is the number of independent observations in the estimation problem. Constraints introduce Lagrange multipliers into the necessary conditions for the simultaneous method, and it is required to estimate these in order to carry out an asymptotic convergence rate analysis of the Bock iteration for large $n$. It is shown here that the Lagrange multipliers are $O eft( igma n^{-1/2} ight),; n o nfty$, where $ igma$ is the standard deviation of the measurement errors in the observations, and that a similar estimate is valid for the multiplier characterizing the first order convergence rate of the iteration. This shows that the Bock iteration has excellent convergence rate properties in large sample problems. An interesting feature of these estimates is that their derivation requires that the observational errors are normally distributed. The Lagrange multiplier estimates are obtained by interpreting the necessary conditions for an optimum of the estimation problem as a discretization of a stochastic ODE system. The convergence rate analysis must take into account the unbounded size of the constraint system. This is done by reducing the calculation of the multiplier characterizing the first order convergence rate of the Bock iteration to that of estimating the spectral radius of a matrix of fixed dimension independent of $n$.