FPGA-accelerated Monte-Carlo integration using stratified sampling and Brownian bridges

Abstract

Monte-Carlo Integration (MCI) is a numerical technique for evaluating integrals which have no closed form solution. Naive MCI randomly samples the integrand at uniformly distributed points. This naive approach converges very slowly. Stratified sampling can be used to concentrate the samples on segments of the integration domain where the integrand has the highest variance. Even with stratified sampling, MCI converges very slowly for multidimensional integrals. In this work, we implement an FPGA-accelerated design for MISER, a widely used adaptive MCI algorithm applying stratified sampling. We show how to eliminate the recursion from MISER and partition the algorithm between CPUs and FPGAs. The CPUs manage the control-heavy stratification strategy, while the FPGA is responsible for sampling the integrand. The integrand is compiled into a deep pipeline on the FPGA, producing one function evaluation per clock cycle. We demonstrate the FPGA-accelerated design by pricing a path dependent financial derivative called an Asian option. To make optimal use of the stratification, we implement a Brownian bridge on the FPGA that produces one entire bridge per clock cycle. The FPGA-accelerated design is up to 880 times faster compared to a software reference using the GSL implementation of MISER. Compared to naive MCI in software, our design even requires up to 3572 times less execution time to achieve the same accuracy.