Volume 29, Issue 2

The purpose of this paper is to propose a time-step-robust cell-to-cell integration of particle trajectories in 3-D unstructured meshes in particle/mesh Lagrangian stochastic methods. The main idea is to dynamically update the mean fields used in the time integration by splitting, for each particle, the time step into sub-steps such that each of these sub-steps corresponds to particle cell residence times. This reduces the spatial discretization error. Given the stochastic nature of the models, a key aspect is to derive estimations of the residence times that do not anticipate the future of the Wiener process. To that effect, the new algorithm relies on a virtual particle, attached to each stochastic one, whose mean conditional behavior provides free-of-statistical-bias predictions of residence times. After consistency checks, this new algorithm is validated on two representative test cases: particle dispersion in a statistically uniform flow and particle dynamics in a non-uniform flow.

Finite-dimensional (FD) models X d ⁢ ( t ) X_{d}(t) , i.e., deterministic functions of time and finite sets of 𝑑 random variables, are constructed for stationary and nonstationary Gaussian processes X ⁢ ( t ) X(t) with continuous samples defined on a bounded time interval [ 0 , τ ] [0,\tau] . The basis functions of these FD models are finite sets of eigenfunctions of the correlation functions of X ⁢ ( t ) X(t) and of trigonometric functions. Numerical illustrations are presented for a stationary Gaussian process X ⁢ ( t ) X(t) with exponential correlation function and a nonstationary version of this process obtained by time distortion. It was found that the FD models are consistent with the theoretical results in the sense that their samples approach the target samples as the stochastic dimension is increased.

In this paper, we construct stochastic simulation algorithms for solving an elastostatics problem governed by the Lamé equation. Two different stochastic simulation methods are suggested: (1) a method based on a random walk on spheres, which is iteratively applied to anisotropic diffusion equations that are related through the mixed second-order derivatives (this method is meshless and can be applied to boundary value problems for complicated domains); (2) a randomized algorithm for solving large systems of linear algebraic equations that is the core of this method. It needs a mesh formation, but even for very fine grids, the algorithm shows a high efficiency. Both methods are scalable and can be easily parallelized.

In this paper, we introduce a new algorithm for estimating the lower bounds for the star discrepancy of any arbitrary point sets in [ 0 , 1 ] s [0,1]^{s} . Computing the exact star discrepancy is known to be an NP-hard problem, so we have been looking for effective approximation algorithms. The star discrepancy can be thought of as the maximum of a function called the local discrepancy, and we will develop approximation algorithms to maximize this function. Our algorithm is analogous to the random walk algorithm described in one of our previous papers [M. Alsolami and M. Mascagni, A random walk algorithm to estimate a lower bound of the star discrepancy, Monte Carlo Methods Appl. 28 (2022), 4, 341–348.]. We add a statistical technique to the random walk algorithm by implementing the Metropolis algorithm in random walks on each chosen dimension to accept or reject this movement. We call this Metropolis random walk algorithm. In comparison to all previously known techniques, our new algorithm is superior, especially in high dimensions. Also, it can quickly determine the precise value of the star discrepancy in most of our data sets of various sizes and dimensions, or at least the lower bounds of the star discrepancy.

This manuscript presents a short route to justify the widely used Monte Carlo Radiative Transfer (MCRT) algorithm straight from the Radiative Transfer Equation (RTE). In this regard, this paper starts deriving a probability measure obtained from the integral formulation of the RTE under a unidirectional point source in an infinite domain. This derivation only requires the analytical integration of the first two terms of a perturbation expansion. Although derivations have been devised to clarify the relationship between the MCRT and the RTE, they tend to be rather long and elaborate. Considering how simple it is to justify the MCRT from a loose probabilistic interpretation of the photon’s physical propagation process, the decay in popularity of former approaches relating MCRT to the RTE is entirely understandable. Unfortunately, all of this has given the false impression that MCRT and the RTE are not that closely related, to the point that recent works have explicitly stated that no direct link exists between them. This work presents a simpler route demonstrating how the MCRT algorithm emerges to statistically sample the RTE explicitly through Markov chains, further clarifying the method’s foundations. Although compact, the derivation proposed in this work does not skip any fundamental step, preserving mathematical rigor while giving specific expressions and functions. Thus, this derivation can help devise efficient ways to statistically sample the RTE for different scenarios or when coupling the MCRT method with other methods traditionally grounded in the RTE, such as the Spherical Harmonics and Discrete Ordinates methods.