# Computing reachable sets via barrier methods on SIMD architectures

L. Grüne, T. Jahn:
**Computing reachable sets via barrier methods on SIMD architectures**

in:
*Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012),*
*
J. Eberhardsteiner, H.J. Böhm, F.G. Rammerstorfer
(eds.)
*
Vienna University of Technology,
Vienna, Austria,
2012,
CD-ROM, Paper No. 1518, 20 pages

Smart-Link:
http://eccomas2012.conf.tuwien.ac.at/

ISBN/ISSN/ISMV Nummer: 978-3-9502481-9-7

Keywords: reachable set, feasibility problem, sparse linear equation system, Runge-Kutta method, CUDA, parallelization, lower arrow form

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## Abstract:

We consider the problem of computing reachable sets of ODE-based control systems parallely on CUDA hardware. To this end, we modify an existing algorithm based on solving optimal control problems.

The idea is to simplify the optimal control problems to pure feasibility problems instead of minimizing an objective function. We show that an interior point algorithm is well suited for solving the resulting feasibility problems and leads to a sequence of linear systems of equations with identical matrix layout. If the problem is defined properly, these matrices are sparse and can be transformed into a hierarchical lower arrow form which can be solved on CUDA hardware with sparse linear algebra and Cholesky’s method.

We demonstrate the performance of our new algorithm by computing the reachable sets of two test problems on a CPU implementation using several explicit and implicit Runge-Kutta methods of different order. The experiments reveal a significant speedup compared to the original optimal control algorithm.