Time-dependent approximation schemes for some problems of parameter estimation in distributed systems

A parameter estimation method that can be used to estimate functional parameters in delay differential equations and moving boundary problems is discussed. In either problem, the original model equation (which is infinite-dimensional) is approximated by a system of ordinary differential equations that can be solved numerically in an efficient way. The approximation scheme is based on time-dependent spline elements. For the delay equation with time-varying delay, convergence results are presented that indicate the estimates obtained using the approximating system. Numerical test examples converge in some sense to a best-fit parameter for the original system are included by means of which time-varying and state-dependent delays and a time-varying diffusion coefficient in a one-phase, one-dimensional Stefan problem are estimated.