Estimation of sample mean and variance for Monte-Carlo simulations

Monte-Carlo simulations are able to provide estimates of solutions for problems that are otherwise intractable, by examining the aggregate behaviour of large numbers of random simulations. Because these simulations are independent and embarrassingly parallel, FPGAs are increasingly being used to implement Monte-Carlo applications. However, as the number of simulation runs increases, the problem of accurately accumulating the aggregate statistics, such as the mean and variance, becomes very difficult. This paper examines three accumulation methods, adapting them for use in FPGA applications. In particular, we develop a mean and variance calculator based on cascading accumulators, which is able to process streams of floating-point data in one pass, while operating in fixed-point internally. This method has the advantage that it calculates the exact sample mean and an accurate numerically stable sample variance, while using few logic resources and providing performance to match commercial floating-point operators: clock rates of 434 MHz are achieved in Virtex-5, 1.46 times faster than a double-precision accumulator, while using one eighth of the resources.