Numerical Accuracy Analysis in Simulations on Hybrid High-Performance Computing Systems

It is often a challenge to estimate the numerical accuracy of computed results or their validity due to numerical effects in complex simulation software packages, especially if these packages have been developed over many years. Detailed numerical properties for large sections of the source code may not be known to the current developers, or have simply not been investigated during its initial implementation. In addition, many high-performance computing systems are based on graphics processing units (GPUs), field programmable gate arrays (FPGAs) and special purpose processors such as the Cell processor. In such systems the low precision (e.g. single precision) floating point (FP) performance is significantly higher than the traditional high precision (e.g. double precision) FP performance. Although the double precision performance compared to single precision performance is enhanced on GPUs in the newer architectures for scientific computing, its price is increased by a similar factor compared to standard GPUs such that the performance difference related to the costs between single and double precision remains unchanged. Due to this performance-price ratio, single precision arithmetic is preferable to double precision in many cases. Also mixed-precision approaches are becoming increasingly popular e.g. for solvers of linear equations. However, the impact of using low precision on the numerical accuracy of the result has to be verified. In addition, even for double precision there exist numerous cases in which the algorithm is unstable. In general, the numerical accuracy analysis becomes increasingly relevant in simulation technology, especially on hybrid high-performance computing systems because of the difference in the accuracy of FP arithmetic. Furthermore, rounding errors will scale up significantly with the increasing problem size in the upcoming exascale computing era. In this thesis, methods to estimate the numerical accuracy of computed results are studied with respect to their reliability, efficient implementations and applicability to complex simulation software on hybrid high-performance systems.