Martin Berz

Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models

A method is developed that allows the verified integration of ODEs based on local modeling with high-order Taylor polynomials with remainder bound. The use of such Taylor models of order n allows convenient automated verified inclusion of functional dependencies with an accuracy that scales with the (n + 1)-st order of the domain and substantially reduces blow-up.Utilizing Schauders fixed point theorem on certain suitable compact and convex sets of functions, we show how explicit nth order integrators can be developed that provide verified nth order inclusions of a solution of the ODE. The method can be used not only for the computation of solutions through a single initial condition, but also to establish the functional dependency between initial and final conditions, the so-called flow of the ODE. The latter can be used efficiently for a substantial reduction of the wrapping effect.Examples of the application of the method to conventional initial value problems as well as flows are given. The orders of the integration range up to twelve, and the verified inclusions of up to thirteen digits of accuracy have been demanded and obtained.

Modern map methods in particle beam physics

Dynamics. Differential Algebraic Techniques. Fields. Maps: Calculation. Maps: Properties. Spectrometers. Repetitive Systems.

Computation and Application of Taylor Polynomials with Interval Remainder Bounds

The expansion of complicated functions of many variables in Taylor polynomials is an important problem for many applications, and in practice can be performed rather conveniently (even to high orders) using polynomial algebras. An important application of these methods is the field of beam physics, where often expansions in about six variables to orders between five and ten are used.However, often it is necessary to also know bounds for the remainder term of the Taylor formula if the arguments lie within certain intervals. In principle such bounds can be obtained by interval bounding of the (n + 1)-st derivative, which in turn can be obtained with polynomial algebra; but in practice the method is rather inefficient and susceptible to blow-up because of the need of repeated interval evaluations of the derivative. Here we present a new method that allows the computation of sharp remainder intervals in parallel with the accumulation derivatives up to order n.The method is useful for a variety of numerical problems, including the interval inclusion of very complicated functions prone to blow-up. To this end, the function is represented by a Taylor polynomial with remainder using the above method. Since at least for high orders, the remainder terms have a tendency to be very small, the problem is reduced to an interval evaluation of the Taylor polynomial. The method is used for guaranteed global optimization of blow-up prone functions and compared with some interval-based global optimization schemes.

Efficient Control of the Dependency Problem Based on Taylor Model Methods

It is shown how the Taylor Model approach allows the rigorous description of functional dependencies with far-reaching control of the dependency problem. The amount of overestimation decreases with a high power of the interval over which the information is required, at a computational expense that increases rather moderately with the dimensionality of the problem. This leads to the possibility of treating even cases with a very significant dependency problem that are intractable using conventional methods.

GIOS-BEAMTRACE -- A program package to determine optical properties of intense ion beams

Abstract A program package is presented which allows determination of the properties of optical systems for intense ion beams in which space-charge forces must be taken into account. GIOS is based on the algebraic determination of the elements of transfer matrices and allows an automatic optimization of the optical system under consideration. BEAMTRACE is based on the numerical solution of the equations of motion in assumed fields. In both cases the internal space-charge fields are added to the external electromagnetic fields.

Taylor models and floating-point arithmetic: Proof that arithmetic operations are validated in COSY

The goal of this paper is to prove that the implementation of Taylor models in COSY, based on floating-point arithmetic, computes results satisfyin- g the «containment property», i.e. guaranteed results. First, Taylor models are defined and their implementation in the COSY software by Makino and Berz is detailed. Afterwards IEEE-754 floating-point arithmetic is introduced. Then the core of this paper is given: the algorithms implemented in COSY for multiplying a Taylor model by a scalar, for adding or multiplying two Taylor models are given and are proven to return Taylor models satisfying the containment property.

The method of power series tracking for the mathematical description of beam dynamics

Abstract A new method to compute the properties of charged particle optics systems is presented. It is based on the application of operators and functions to a power series algebra instead of real numbers. The method is as versatile as numerical integration methods and comparable to matrix methods in speed and accuracy thus combining the advantages of both strategies. The method has been implemented through fifth order in the code POWERTRACK. Due to the generality of the method, the order can be increased with very little programming effort.

Computational aspects of optics design and simulation: COSY INFINITY

Abstract The new differential algebraic (DA) techniques allow very efficient treatment and understanding of nonlinear motion in optical systems as well as circular accelerators. To utilize these techniques in their most general way, a powerful software environment is essential. A language with structure elements similar to Pascal was developed. It has object oriented features to allow for a direct utilization of the elementary operations of the DA package. The compiler of the language is written in Fortran 77 to guarantee wide portability. The language was used to write a very general beam optics code, COSY INFINITY. At its lowest level, it allows the computation of the maps of standard beam line elements including fringe fields and system parameters to arbitrary order. The power of the DA approach coupled with an adequate language environment reveals itself in the very limited length of COSY INFINITY of only a few hundred lines. Grouping of elements as well as structures for optimization and study are readily available through the features of the language. Because of the openness of the approach, it offers a lot of power for more advanced purposes. For example, it is very easy to construct new particle optical elements. There are also many ways to efficiently manipulate and analyze the maps.

Arbitrary order description of arbitrary particle optical systems

Abstract The differential algebraic approach for the design and analysis of particle optical systems and accelerators is presented. It allows the computation of transfer maps to arbitrary orders for arbitrary arrangements of electromagnetic fields, including the dependence on system parameters. The resulting maps can be cast into different forms. In the case of a Hamiltonian system, they can be used to determine the generating function or Eikonal representation. Also various factored Lie operator representations can be determined directly. These representations for Hamiltonian systems cannot be determined with any other method beyond relatively low orders. In the case of repetitive systems, a combination of the power series representation and the Lie operator representation allows a nonlinear change of variables such that the motion is very simple and its long term behaviour can be studied very efficiently. Furthermore, it is now possible to compute quantities relevant to the study of circular machines like tune shifts and chromaticities much more efficiently. Besides these aspects, the ability to compute maps depending on parameters provides analytical insight into the system. In addition, this approach allows very efficient optimization, to the extent that in many cases it is almost completely analytic.

COSY 5.0 — The fifth order code for corpuscular optical systems

Abstract COSY 5.0 is a new computer code for the design of corpuscular optical systems based on the principle of transfer matrices. The particle optical calculations include all image aberrations through fifth order. COSY 5.0 uses canonical coordinates and exploits the symplectic condition to increase the speed of computation. COSY 5.0 contains a library for the computation of matrix elements of all commonly used corpuscular optical elements such as electric and magnetic multipoles and sector fields. The corresponding formulas were generated algebraically by the computer code HAMILTON. Care was taken that the optimization of optical elements is achieved with minimal numerical effort. Finally COSY 5.0 has a very general mnemonic input code resembling a higher programming language.