Relative frequencies of constrained events in stochastic processes: An analytical approach

TL;DR

This work addresses the high computational cost of Monte Carlo methods for constrained stochastic processes by introducing an analytical route to compute asymptotic relative frequencies of events directly from constraints, avoiding stochastic sampling. The approach partitions event outcomes into constraint-defined subsets and uses SSA-based probabilities to derive closed-form expressions for asymptotic counts, enabling exact branching fractions without MC. It is validated on a simple two-outcome model and applied to Controlled Radical Polymerization (CRP) of acrylic monomers, where branching fractions align with MC results but with orders-of-magnitude speedups. The method is broadly applicable to constrained stochastic systems and can serve as a fast, accurate tool for parameter fitting of inter-event time PDFs in MC simulations.

Abstract

The stochastic simulation algorithm (SSA) and the corresponding Monte Carlo (MC) method are among the most common approaches for studying stochastic processes. They rely on knowledge of interevent probability density functions (PDFs) and on information about dependencies between all possible events. Analytical representations of a PDF are difficult to specify in advance, in many real life applications. Knowing the shapes of PDFs, and using experimental data, different optimization schemes can be applied in order to evaluate probability density functions and, therefore, the properties of the studied system. Such methods, however, are computationally demanding, and often not feasible. We show that, in the case where experimentally accessed properties are directly related to the frequencies of events involved, it may be possible to replace the heavy Monte Carlo core of optimization schemes with an analytical solution. Such a replacement not only provides a more accurate estimation of the properties of the process, but also reduces the simulation time by a factor of order of the sample size (at least $\approx 10^4$). The proposed analytical approach is valid for any choice of PDF. The accuracy, computational efficiency, and advantages of the method over MC procedures are demonstrated in the exactly solvable case and in the evaluation of branching fractions in controlled radical polymerization (CRP) of acrylic monomers. This polymerization can be modeled by a constrained stochastic process. Constrained systems are quite common, and this makes the method useful for various applications.

Figures (6)

• Figure 1: Comparison between the analytical solution \ref{['eqn:solution_single_constraint']} (lines) and corresponding statistics (crosses) obtained by the Monte Carlo (MC) method proposed in TimeDelayedPDF. Five independent runs are performed for two different parameters set: $n_0=3$, $c_2/c_1=1$ (solid line) and $n_0=3$, $c_2/c_1=0.2$ (dashed line). The MC sample size is equal to $G=10^4$.
• Figure 2: Experimental branching fractions and corresponding uncertainty intervals LinExpPdf.
• Figure 3: Fitted data obtained by the analytical approach (crosses) and by the MC method (open circles) (MC sample size $G=10^4$) are presented for two polymerization reactions, bulk and solution. Both approaches use the linear exponential inter-events times pdf \ref{['eqn:expressionlinexppdf']}.
• Figure 4: Computational times required for the optimization routine (Nelder-Mead method NelderMeadmethod) performed with an increasing number of iterations for bulk and solution polymerization. The analytical approach (crosses and squares) speeds up the procedure by the factor of $10^4$ compared with the MC based optimization method (open circles and triangles) of the same level of accuracy.
• Figure 5: Fitted data obtained by the analytical approach (crosses) and by the MC method (open circles) (MC sample size $G=10^4$) are presented for two polymerization reactions, bulk and solution. Both approaches use exponential pdf's for propagation and deactivation, and a linear exponential pdf \ref{['eqn:expressionlinexppdf']} for backbiting.