82-XX STATISTICAL MECHANICS, STRUCTURE OF MATTER
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The topic of this thesis is the examination of an optimization model
which stems from the clustering process of non-reversible markov processes.
We introduce the cycle clustering problem und formulate it as a mixed
integer program (MIP).
We prove that this problem is N P-hard and discuss polytopal aspects
such as facets and dimension. The focus of this thesis is the development of
solving methods for this clustering problem. We develop problem specific
primal heuristics, as well as separation methods and an approximation
algorithm. These techniques are implemented in practice as an application
for the MIP solver SCIP.
Our computational experiments show that these solving methods result
in an average speedup of ×4 compared to generic solvers and that our
application is able to solve more instances to optimality within the given
time limit of one hour.
This thesis reports on simulating spin models on Nvidia graphics cards using the CUDA programming model; a particular approach for making GPGPU (General Purpose Computation on Graphics Processing Units) available for a wide range of software developers not necessarily acquainted with (massively) parallel programming. By comparing program execution times for simulations of the Ising model and the Ising spin glass by means of the Metropolis algorithm on Nvidia Tesla C1060 graphics cards and an Intel Core i7-920 quad-core x86 CPU (we used OpenMP to make our simulations run on all 4 execution units of the CPU), we noticed that the Tesla C1060 performed about a factor 5-10 faster than the Core i7-920, depending on the particular model and the accuracy of the calculations (32-bit or 64-bit). We also investigated the reliability of GPGPU computations, especially with respect to the occurrence of soft-errors as suggested in [23]. We noticed faulty program outputs during long-time simulations of the Ising model on ''large'' lattices. We were able to link these problems to overheating of the corresponding graphics cards. Doing Monte Carlo simulations on parallel computer architectures, as was the case in this thesis, suggests to also generate random numbers in a parallel manner. We present implementations of the random number generators Ranlux and Mersenne Twister. In addition, we give an alternative and very efficient approach for producing parallel random numbers on Nvidia graphics cards. We successfully tested all random number generators used in this thesis for their quality by comparing Monte Carlo estimates against exact calculations.