Weighted Monte Carlo Method Research Articles

Optimization of Weighted Monte Carlo Methods with Respect to Auxiliary Variables

We consider the problems whose mathematical model is determined by some Markov chain terminating with probability one; moreover, we have to estimate linear functionals of a solution to an integral equation of the second kind with the corresponding substochastic kernel and free term [1]. To construct weighted modifications of numerical statistical models, we supplement the coordinates of the phase space with auxiliary variables whose random values functionally define the transitions in the initial chain. Having implemented each auxiliary random variable, we multiply the weight by the ratio of the corresponding densities of the initial and numerically modeled distributions. We solve the minimization problem for the variances of estimators of linear functionals by choosing the modeled distribution of the first auxiliary random variable.

Corrections to the Article “Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation”

Corrections to the Article “Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation”

Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation

We construct weighted modifications of statistical modeling of an ensemble of interacting particles which is connected with approximate solution of a nonlinear Boltzmann equation.

SOLVING THE MULTIDIMENSIONAL DIFFERENCE BIHARMONIC EQUATION BY THE MONTE CARLO METHOD

We construct and justify new weighted Monte Carlo methods for estimation of a solution to the Dirichlet problem for the multidimensional difference biharmonic equation by modeling a “random walk by a grid.” Vector versions of our algorithms extend to the difference metaharmonic equations, with the shape of unbiasedness conditions of estimators preserved together with boundedness of their variances. In this connection, we construct a simple algorithm for estimation of the first eigenvalue of the multidimensional difference Laplace operator. Moreover, we construct special algorithms of a “random walk by a grid” which under certain conditions allow us to estimate solutions of the Dirichlet problem for the biharmonic equation with a weak nonlinearity as well as solutions to problems with mixed boundary conditions, the Neumann condition inclusively.

Using the weighted Monte Carlo method for solving nonlinear integral equations

Article Using the weighted Monte Carlo method for solving nonlinear integral equations was published on January 1, 1994 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 9, issue 2).

Weighted Monte Carlo for electron transport in semiconductors

A new powerful method, the weighted Monte Carlo method, for the solution of electron transport problems in semiconductors is presented. The method is formally derived from an iterative expansion of the integral form of the Boltzmann equation and results in a simulation of electron dynamics where arbitrary probabilities for the various events which generate electron trajectories are used. It is much more flexible than the standard algorithm that can be derived by it as a very special case. The method is applied here to study the tail of the distribution function in a simple silicon model.

Minimax theory of weighted Monte Carlo methods

Minimax theory of weighted Monte Carlo methods