MSU, East Lansing, Michigan
MSU, East Lansing, Michigan
The code COSY INFINITY supports rigorous computations with numerical verification based on Taylor models, a tool developed by us that can be viewed as an extension of the differential algebraic methods that also determines rigorous Taylor remainder bounds. Such verified computation techniques can be utilized for global optimization tasks, resulting in a guarantee that the true optimum over a given domain is found. The method of Taylor models has a high order scaling property, suppressing the problem of over-estimation that is a common problem of reliable computational methods. We have applied the method to some of typical optimization tasks in accelerator physics such as lattice design parameter optimizations and the Lyapunov function based long-term stability estimates for storage rings. The implementation of Taylor models in COSY INFINITY has inherited all the advantageous features of the implementation of differential algebras in the code, resulting in very efficient execution. COSY-GO, the Taylor model based rigorous global optimizer of COSY INFINITY, can run either on a single processor or in a multi processor mode based on MPI. We present various results of optimization problems run on more than 2,000 processors at NERSC operated by the US Department of Energy. Specifically, we discuss rigorous long-term stability estimates of the Tevatron, as well as high-dimensional rigorous design optimization of RIA fragment separators.
MSU, East Lansing, Michigan
We will demonstrate the code COSY INFINITY Version 9. Besides the known feature of computations of high order Taylor transfer maps based on differential algebras, the latest version has many new features, many of them using algorithms only possible with differential algebras and Taylor models. Aside from conventional beam dynamics design and optimization tools, we will focus on new features, including rigorous global optimization, computation of remainder bounds for high order maps, minimal symplectic tracking in the EXPO framework, and the ability to integrate high-order maps through user-specified fields. Specific applications will focus on maps of absorbers, wedges, and novel non-cylindrical multipole elements.
MSU, East Lansing, Michigan
We show how the differential algebraic methods for ODEs and the resulting high order map computation can be generalized for solving certain PDEs. The entire PDE solving problem is cast in the form of an implicit constraint satisfaction problem, which is solved via differential algebraic partial inversion methods. As a result, it is possible to describe the solutions of the PDE locally as a very high order expression in the independent variables. Because of the high orders, it is possible to choose the size of the finite elements to be large, which leads to a very favorable behavior in high dimensions. The approach can be parallelized, and as such allows the solution of complicated high-dimensional PDEs in a reasonably efficient way. Furthermore, utilizing remainder differential algebraic methods, it is possible to provide rigorous and reasonably sharp error estimates of the entire procedure. We apply the methods to the study of the Vlasov equation describing the evolution of a beam under internal and external electromagnetic fields. In the case of this particular PDE, it is possible to perform time stepping to arbitrary order with a similar ease as in the case of the corresponing map computation case. Various examples will be given to illustrate the practical behavior of the method.