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In this paper is described a general 2 nd order accurate (weak sense) procedure for stablizing Monte-Carlo simulations of Itô stochastic di erential equations. The splitting procedure includes explicit Runge-Kutta methods 1], semi-implicit methods 2] 3], and trapezoidal rule 1] 4]. We prove the semi-implicit method of Ottinger 3] and note that it may be generalized for arbitrary splittings.

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In this paper, we report on the theoretical foundations, empirical context and technical implementation of an agent-based modeling (ABM) framework, that uses a high-performance computing (HPC) approach to investigate human population dynamics on a global scale, and on evolutionary time scales. The ABM-HPC framework provides an in silico testbed to explore how short-term/small-scale patterns of individual human behavior and long-term/large-scale patterns of environmental change act together to influence human dispersal, survival and extinction scenarios. These topics are currently at the center of the Neanderthal debate, i.e., the question why Neanderthals died out during the Late Pleistocene, while modern humans dispersed over the entire globe. To tackle this and similar questions, simulations typically adopt one of two opposing approaches, top-down (equation-based) and bottom-up (agent-based) models of population dynamics. We propose HPC technology as an essential computational tool to bridge the gap between these approaches. Using the numerical simulation of worldwide human dispersals as an example, we show that integrating different levels of model hierarchy into an ABM-HPC simulation framework provides new insights into emergent properties of the model, and into the potential and limitations of agent-based versus continuum models.

It is amusing to note how the initial pessimism about vector/parallel methods for Monte Carlo simulations has now been replaced with requests for better and faster vcctorizcd random number generators for those purposes. In fact, much of the impetus for establishing the NSF supercomputer centers came from Nobel Laureate Kenneth G. Wilson's significant interests in Monte Carlo simulations of quantum field theories. The generality and usefulness of stochastic simulations has only begun to be explored and will likely be driven by the power of vector/parallel computer architectures [Monte Carlo 1983]. As the scope of these simulations increases, the need for fast vectorized random number generators with excellent statistical properties becomes more crucial. I hope that the three functions described below will move toward satisfying that need. The three functions, with their distributions and syntax for usage, are:1. Uniform distribution. The function is called RANQ and an example of usage is real u(n) cdir$ vfunction RANQ doli= 1,n u(i) = RANQ() 1 continue giving a uniformly distributed sequence { uilO. < ui< 1. }.2. Normal distribution. The function is called GAUSSQ and an example of usage is rcalv(n) cdir$ vfunction GAUSSQ doli= 1, n