A simple scheme for the parallelization of particle filters and its application to the tracking of complex stochastic systems

Considerable effort has been devoted in the past decade to the design of schemes for the parallel implementation of particle filters. The approaches vary from the totally heuristic to the mathematically well-principled. In this paper we investigate the use of possibly the simplest scheme for the parallelizaton of the standard particle filter, that consists in splitting the computational budget into $M$ fully independent particle filters with $N$ particles each, and then obtaining the desired estimators by averaging over the $M$ independent outcomes of the filters. This approach minimizes the parallelization overhead yet displays highly desirable theoretical properties. In particular, we prove that the mean square estimation error (of integrals with respect to the filter measure) vanishes asymptotically with the same rate, proportional to $1 / MN$, as the standard (centralized) particle filter with the same total number of particles. Parallelization, therefore, has the obvious advantage of dividing the running times while preserving the (asymptotic) performance. We also propose a time-error index to quantify this improvement and to compare schemes with different degrees of parallelization. As a side result, we show that the expected value of the random probability measure output by each independent filter converges in total variation distance to the true posterior with rate of order $1 / N$ (note that the average measure over the $M$ filters is just a sample-mean estimate of this expected measure). Finally, we provide an illustrative numerical example for the tracking of a Lorenz 65 chaotic system with dynamical noise and partial (noisy) observations. Our computer simulations show how, in a practical application, the proposed paralelization scheme can attain the same approximation accuracy as the corresponding centralized particle filter with only a small fraction of the running time.