A Stochastic Algorithm for Parametric Sensitivity in Smoluchowski's Coagulation Equation

Abstract

In this article a stochastic particle system approximation to the parametric sensitivity in the Smoluchowski coagulation equation is introduced. The parametric sensitivity is the derivative of the solution to the equation with respect to some parameter, where the coagulation kernel depends on this parameter. It is proved that the particle system converges weakly to the sensitivity as the number of particles $\mathrm{N}$ increases. A Monte Carlo algorithm is developed and variance reduction techniques are applied. Numerical experiments are conducted for two kernels: the additive kernel and one which has been used for studying soot formation in a free molecular regime. It is shown empirically that the techniques for variance reduction are indeed very effective and that the order of convergence is $\mathrm{O}(1/\mathrm{N})$. The algorithm is then compared to an algorithm based on a finite difference approximation to the sensitivity, and it is found that the variance of the sensitivity estimators are considerably lower than that for the finite difference approach. Furthermore, two methods of establishing “efficiency” are considered and the new algorithm is found to be significantly more efficient.