Note on a Paper by J. Spanier Concerning Monte Carlo Estimators

The Monte Carlo method

When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to angular flux / radiance because the macroscopic cross-section $\Sigma_t(s)$ depends on the distance $s$ to the previous collision or boundary. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green's functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green's functions are derived, which reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.

A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation.

The statistical error and computational efficiency of Monte Carlo reaction rate estimation depends crucially on which Monte Carlo procedures are used. We present a systematic comparison of the variance and computational cost of the most common estimation procedures and several others, and we show a non-trivial dependence of the choice of the optimal estimator on the problem parameters. The comparison is based on an invariant imbedding procedure, in which systems of ordinary differential equations (ODEs) are derived that quantify the statistical error and computational cost of each estimator. We analyze in detail a scenario with forward-backward scattering in a one-dimensional slab. For this test case, we perform a parametric study of the expected statistical error and computational cost by numerically solving the ODEs.

methods

An analytical study of Monte Carlo source term estimators in plasma edge simulations of fusion reactors

The distributed activation energy model (DAEM) is widely used in chemical kinetics to statistically describe the occurrence of numerous independent parallel reactions. In this article, we suggest a rethink in the context of a Monte Carlo integral formulation to compute the conversion rate at any time without approximation. After the basics of the DAEM are introduced, the considered equations (under isothermal and dynamic conditions) are respectively expressed into expected values, which in turn are transcribed into Monte Carlo algorithms. To describe the temperature dependence of reactions under dynamic conditions, a new concept of null reaction, inspired from null-event Monte Carlo algorithms, has been introduced. However, only the first-order case is addressed for the dynamic mode due to strong nonlinearities. This strategy is then applied to both analytical and experimental density distribution functions of the activation energy. We show that the Monte Carlo integral formulation is efficient in solving the DAEM without approximation and that it is well-adapted due to the possibility of using any experimental distribution function and any temperature profile. Furthermore, this work is motivated by the need for coupling chemical kinetics and heat transfer in a single Monte Carlo algorithm.

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Note on a Paper by J. Spanier Concerning Monte Carlo Estimators

Citations

The Monte Carlo method

A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation.

An analytical study of Monte Carlo source term estimators in plasma edge simulations of fusion reactors

A Monte Carlo-Based Strategy to Assess Complex Kinetics: Application of the Null-Reaction Method to DAEM.

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