A fast stratified sampling simulation of coagulation processes

We develop a new version of the direct simulation Monte Carlo method [3] for coagulation processes governed by homogeneous Smoluchowsky equations. The method is based on a subdivision of the set of particle pairs into classes, and on an efficient algorithm for sampling from a discrete distribution, the so-called Walker's alias method [4]. The efficiency of the new method is drastically increased compared to the conventional methods, especially when the coagulation kernel is strongly varying. The method is applied to solving a problem of islands formation on a surface due to a diffusion controlled coagulation

Discrete random walk on large spherical grids generated by spherical means for PDEs

A new general stochastic-deterministic approach for a numerical solution of boundary value problems of potential and elasticity theories is suggested. It is based on the use of the Poisson-like integral formulae for overlapping spheres. An equivalent system of integral equations is derived and then approximated by a system of linear algebraic equations. We develop two classes of special Monte Carlo iterative methods for solving these systems of equations which are a kind of stochastic versions of the Chebyshev iteration method and successive overrelaxation method (SOR). In the case of classical potential theory this approach accelerates the convergence of the well known Random Walk on Spheres method (RWS). What is however much more important, this approach suggests a first construction of a fast convergent finite-variance Monte Carlo method for the system of Lam'e equations

Stochastic simulation method for a 2D elasticity problem with random loads

We develop a stochastic simulation method for a numerical solution of the Lam\'e equation with random loads. To treat the general case of large intensity of random loads, we use the Random Walk on Fixed Spheres (RWFS) method described in our paper \cite{sab-lev-shal-2006}. The vector random field of loads which stands in the right-hand-side of the system of elasticity equations is simulated by the Randomization Spectral method presented in \cite{sab-1991} and recently revised and generalized in \cite{kurb-sab-2006}. Comparative analysis of RWFS method and an alternative direct evaluation of the correlation tensor of the solution is made. We derive also a closed boundary value problem for the correlation tensor of the solution which is applicable in the case of inhomogeneous random loads. Calculations of the longitudinal and transverse correlations are presented for a domain which is a union of two arbitrarily overlapped discs. We also discuss a possibility to solve an inverse problem of determination of the elastic constants from the known longitudinal and transverse correlations of the loads

Stochastic simulation of flows and particle transport in porous tubes

A Monte Carlo method is developed for stochastic simulation of flows and particle transport in tubes filled with a porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modelled in a tube with prescribed boundary conditions. Numerical experiments are carried out by solving the random Darcy equation for each sample of the hydraulic conductivity by a SOR iteration method, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, diffusion coefficients, the mean and variance of Lagrangian trajectories, and discuss a ''stagnation" effect which was found in our simulations

Random walk on fixed spheres for Laplace and Lamé equations

The Random Walk on Fixed Spheres (RWFS) introduced in our previous paper is presented in details for Laplace and Lam'e equations governing static elasticity problems. The approach is based on the Poisson type integral formulae written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is equivalently reformulated in the form of a system of integral equations defined on the intersection surfaces (arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equations, a Random Walk procedure is constructed where the random walks are living on the intersecting surfaces. Since the spheres are fixed, it is convenient to construct also discrete random walk methods for solving the system of linear equations approximating the system of integral equations. We develop here two classes of special Monte Carlo iterative methods for solving these systems of linear algebraic equations which are constructed as a kind of randomized versions of the Chebyshev iteration method and Successive Over Relaxation (SOR) method. It is found that in this class of randomized SOR methods, the Gauss-Seidel method has a minimal variance. In our prevoius paper we have concluded that in the case of classical potential theory, the Random Walk on Fixed Spheres considerably improves the convergence rate of the standard Random Walk on Spheres method. More interesting, we succeeded there to extend the algorithm to the system of Lam'e equations which cannot be solved by the conventional Random Walk on Spheres method. We present here a series of numerical experiments for 2D domains consisting of 5, 10, and 17 discs, and analyze the dependence of the variance on the number of discs and elastic constants. Further generalizations to Neumann and Dirichlet-Neumann boundary conditions are also possible

Random walk on fixed spheres method for electro- and elastostatics problems

Stochastic algorithms for solving Dirichlet boundary value problems for the Laplace and Lame equations governing 2D elasticity problems are developed. The approach presented is based on the Poisson integral formula written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is reformulated in the form of equivalent system of integral equations defined on the intersection surfaces, i.e., arcs in 2D. A Random Walk algorithm can be applied then directly to the obtained system of integral equations where the random walks are living on the intersecting surfaces. We develop also a discrete random walk technique for solving the system of linear equations approximating the system of integral equations. We construct a randomized version of the successive over relaxation (SOR) method. In [6] we have demonstrated that in the case of classical potential theory our method considerably improves the convergence rate of the standard Random Walk on Spheres method. In this paper we extend the algorithm to the system of Lame equations which cannot be solved by the conventional Random Walk on Spheres method. Illustrating computations for 2D Laplace and Lame equations, and comparative analysis of different stochastic algorithms are presented

Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height

Forward and backward stochastic Lagrangian trajectory simulation methods are developed to calculate the footprint and cumulative footprint functions of concentration and fluxes in the case when the ground surface has an abrupt change of the roughness height. The statistical characteristics to the stochastic model are extracted numerically from a closure model we developed for the atmospheric boundary layer. The flux footprint function is perturbed in comparison with the footprint function for surface without change in properties. The perturbation depends on the observation level as well as roughness change and distance from the observation point. It is concluded that the footprint function for horizontally homogeneous surface, widely used in estimation of sufficient fetch for measurements, can be seriously biased in many cases of practical importance

A Fast Stratified Sampling Simulation of Coagulation Processes

We develop a new version of the direct simulation Monte Carlomethod for coagu-lation processes governed by homogeneous Smoluchowsky equations. The method is based ona subdivision of the set of particle pairs into classes, and on an efficient algorithm for samplingfrom a discrete distribution, the so-called Walker’s aliasmethod. The efficiency of the newmethod is drastically increased compared to the conventional methods, especially when thecoagulation kernel is strongly varying. The method is applied to solving a problem of islandsformation on a surface due to a diffusion controlled coagulation

Stochastic simulation method for a 2D elasticity problem with random loads

We develop a stochastic simulation method for a numerical solution of the Lamé equation with random loads. To treat the general case of large intensity of random loads, we use the Random Walk on Fixed Spheres (RWFS) method described in our paper [Sabelfeld KK, Shalimova IA, Levykin AI. Discrete random walk over large spherical grids generated by spherical means for PDEs. Monte Carlo Methods and Applications 2006; 12(1): 55–93]. The vector random field of loads which stands on the right-hand side of the system of elasticity equations is simulated by the Randomization Spectral method presented in [Sabelfeld KK. Monte Carlo methods in boundary value problems. Berlin (Heidelberg, New York): Springer-Verlag; 1991] and recently revised and generalized in [Kurbanmuradov O, Sabelfeld KK. Stochastic spectral and Fourier-wavelet methods for vector Gaussian random field. Monte Carlo Methods and Applications 2006; 12(5–6): 395–445]. Comparative analysis of the RWFS method and an alternative direct evaluation of the correlation tensor of the solution is made. We derive also a closed boundary value problem for the correlation tensor of the solution which is applicable in the case of inhomogeneous random loads. Calculations of the longitudinal and transverse correlations are presented for a domain which is a union of two arbitrarily overlapped discs. We also discuss a possibility to solve an inverse problem of the determination of the elastic constants from the known longitudinal and transverse correlations of the loads, and give some relevant numerical illustrations

Discrete random walk on large spherical grids generated by spherical means for PDEs

A new general stochastic-deterministic approach for a numerical solution of boundary value problems of potential and elasticity theories is suggested. It is based on the use of the Poisson-like integral formulae for overlapping spheres. An equivalent system of integral equations is derived and then approximated by a system of linear algebraic equations. We develop two classes of special Monte Carlo iterative methods for solving these systems of equations which are a kind of stochastic versions of the Chebyshev iteration method and successive overrelaxation method (SOR). In the case of classical potential theory this approach accelerates the convergence of the well known Random Walk on Spheres method (RWS). What is however much more important, this approach suggests a first construction of a fast convergent finite-variance Monte Carlo method for the system of Lamé equations