Validation of transfer maps using Taylor models

Abstract

Summary form only given. High-order transfer maps of particle systems play an important. role in the design and optimization of particle optical systems, both for satisfying the basic design needs as well as for the correction of aberrations and non-linear effects. The differential algebraic method has proved useful for this problem, allowing the computation of maps of arbitrary order of systems described by arbitrary electromagnetic fields. Since the high-order maps represent an approximation of the motion, in particular in strongly nonlinear cases in which convergence of the maps may he slow, it is important to know the quality of the approximation. Recent work has shown that it is in principle possible to not only propagate the conventional differential algebraic high-order objects, but also adjoint. remainder terms that rigorously account for any errors made both by the integrator scheme and any possibly inaccuracy of the description of the system. In this paper we describe various recent enhancements of the original method of maps with remainder bounds. First we address a variable doubling scheme that allows the treatment of the remainder effect themselves in terms of a transfer map-like relationship instead of a mere box. Using suitable extensions of the DA and Taylor model operators used in the Schauder fixed point theorem formulation of the problem, this approach is in practice nearly as transparent as the original method and leads to a significant enhancement of sharpness. Under the presence of a scheme for effective treatment of sparsity in the DA vectors, such as the method available in the code COSY INFINITY, the additional resources necessary for this algorithm are very modest. Next we describe a method for the treatment of the one-step approximation errors of the integration scheme called shrink wrapping. It allows the re-absorption of these approximation errors that are initially merely given by an n-dimensional box into the remainder terms described by the variable doubling. Using the combination of these effects, the already rather tight bounds for the approximation errors of transfer maps can be further significantly reduced, and it is even possible to estimate bounds on the overestimation of those error inclusions themselves.