ANALYSIS OF COSTATE DISCRETIZATIONS IN PARAMETER ESTIMATION FOR LINEAR EVOLUTION EQUATIONS

摘要

A widely used approach to parameter identification is the output least-squares formulation. Numerical methods for solving the resulting minimization problem almost invariably require the computation of the gradient of the output least-squares functional. When the identification problem involves time-dependent distributed parameter systems (or approximations thereof), numerical evaluation of the gradient can be extremely time consuming. The costate method can greatly reduce the cost of computing these gradients. However, questions have been raised concerning the accuracy and convergence of costate approximations, even when the numerical methods being used are known to converge rapidly on the forward problem. In this paper it is shown that the use of time-marching schemes that yield high-order accuracy on the forward problem does not necessarily lead to high-order accurate costate approximations. In fact, in some cases these approximations do not converge at all. However, under certain circumstances, rapidly converging gradient approximations do result because of rapid weak-star-type convergence of the costate approximations. These issues are treated both theoretically and numerically. [References: 9]