On the numerical sensitivity of computer simulations on hybrid and parallel computing systems

Simulation results depend not only on the precision of the floating point arithmetic with respect to the numerical accuracy of the results. They are also sensitive to differences of floating point arithmetic implementations of different hybrid and parallel computing systems such as CPUs, CPUs, dedicated processors like the Cell processor or the GRAPE special-purpose computer with the same precision. As floating point operations may not maintain basic properties like associative or distributive properties of the underlying mathematical operations, the numerical values computed by simulations may become dependent on the hardware platform and the specific run of the program. Numerical accuracy control of the simulation would identify significant variations of the simulation results due to these numerical effects. For this purpose, the numerical accuracy is controlled in this paper by a method for rounding error estimation based on the discrete stochastic arithmetic (DSA). This method, which is investigated on both CPUs and CPUs here, is generally applicable independent of the algorithm and can provide a tight estimation of the rounding errors while increasing the computational time only by a factor of approximately 3 in the ideal case. It is shown that the method can be applied automatically without modifying source code. Furthermore, performance improvements compared to the numerical accuracy control based on higher precision arithmetic can be obtained.