SUMMARY A population balance system that models the synthesis of urea is studied in this paper. The equations for the flow field, the mass and the energy balances are given in a three‐dimensional domain, while the equation for the particle size distribution is given in a four‐dimensional domain. This problem is convection‐dominated and aggregation‐driven. Both features require the application of appropriate numerical methods. This paper presents a numerical approach for simulating the population balance system, which is based on finite element schemes, a finite difference method and a modern method to evaluate convolutIon integrals that appear in the aggregation term. Two experiments are considered and the numerical results are compared with experimental data. Unknown parameters in the aggregation kernel have to be calibrated. For appropriately chosen parameters, good agreements are achieved of the experimental data and the numerical results computed with the proposed method. A detailed study of the computational results reveals the influence of different parts of the aggregation kernel. Copyright © 2011 John Wiley & Sons, Ltd.

This article reviews the application of advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance, by using the discretization of a associated probability law. The MLMC approach works with a hierarchy of biased approximations which become progressively more accurate and more expensive. Using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This article considers the case where such exact sampling is not currently possible. We consider Markov chain Monte Carlo and sequential Monte Carlo methods which have been introduced in the literature and we describe different strategies which facilitate the application of MLMC within these methods.

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