Sampling from the Multivariate Gaussian Distribution using Reconfigurable Hardware

Abstract

The multivariate Gaussian distribution models random processes as vectors of Gaussian samples with a fixed correlation matrix. Such distributions are useful for modelling real-world multivariate time-series such as equity returns, where the returns for businesses in the same sector are likely to be correlated. Generating random samples from such a distribution presents a computational challenge due to the dense matrix-vector multiplication needed to introduce the required correlations. This paper proposes a hardware architecture for generating random vectors, utilising the embedded block RAMs and multipliers found in contemporary FPGAs. The approach generates a new n dimensional random vector every n clock cycles, and has a raw generation rate over 200 times that of a single Opteron 2.2GHz using an optimised BLAS package for linear algebra computation. The generation architecture is an ideal source for both software simulations connected via high bandwidth connection, and for completely FPGA based simulations. Practical performance is explored in a case study in Delta-Gamma Value-at-Risk, where a standalone Virtex-4 xc4vsx55 solution at 400 MHz is 33 times faster than a quad Opteron 2.2GHz SMP. The FPGA solution also scales well for larger problem sizes, allowing larger portfolios to be simulation.