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New Approaches for the Validation of Transfer Maps using Remainder-Enhanced Differential Algebra


Abstract

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, and the differential algebraic (DA) method has proved useful for this problem. Since the high-order maps represent an approximation of the motion, in particular in strongly nonlinear cases in which convergence of the maps may be 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 by the approximation by the Taylor expansion over the domain of interest.

In this paper we describe various recent enhancements of the original method of computations with remainder bounds that allow the control of the errors made both by the integrator scheme and any possible inaccuracy of the description of the system.

Using suitable extensions of the DA and Taylor model operators used in the Schauder fixed point theorem formulation of the ODE problem leads to a very transparent approach for the calculation of enclosures for the intergration errors. 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.


M. Berz, K. Makino, Nuclear Instruments and Methods A519 (2004) 53-62


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