Some Examples of Parameter Estimation by Multiple Shooting

Abstract

This paper gives an outline of the multiple shooting code DD03, in the AERE Harwell subroutine library. Also, four parameter estimation problems are described, two from the field of fluid dynamics, and two concerned with nuclear or plasma physics, ranging from a simple eigenvalue problem on a semi-infinite interval to a complex plasma stability problem. The formulation of each is discussed, so that it may be solved by the standard program and the results obtained are outlined. For a linear Schrödinger equation on a semi-infinite interval, the efficiency of the program is ouite sensitive to the boundary condition used. The Orr-Sommerfeld equation is of eighth order, with a complex eigenvalue; it consumes a great deal of computer time for large Reynold’s numbers. A fifth order fluid dynamics problem suffers from singularities and multiple solutions as the Reynold’s number increases, but particular solutions can be traced by a continuation method on one of the physical parameters. A class of plasma stability problems can be reduced to a singular second order differential equation, with a potential function, a complex eigenvalue, and various parameters, between which relations must be determined; with an appropriate formulation, and careful estimation of the parameters, the problem becomes auite tractable.