Coupling Algorithms for Calculating Sensitivities of Smoluchowski's Coagulation Equation

In this paper, two new stochastic algorithms for calculating parametric derivatives of the solution to the Smoluchowski coagulation equation are presented. It is assumed that the coagulation kernel is dependent upon these parameters. The new algorithms (called “Single” and “Double”) work by coupling two Marcus-Lushnikov processes in such a way as to reduce the difference between their trajectories, thereby significantly reducing the variance of central difference estimators of the parametric derivatives. In the numerical results, the algorithms are shown to have an $O(1/N)$ order of convergence as expected, where $N$ is the initial number of particles. It was also found that the Single and Double Algorithms provide much smaller variances. Furthermore, a method for establishing “efficiency” is considered, which takes into account the variances as well as CPU run times, and the “Double” is significantly more “efficient” compared to the “Independent” Algorithm in most cases.