Volume 23, Issue 4

A new Monte Carlo algorithm for solving the Robin boundary-value problem is described and applied to the calculation of the electron beam induced current in a simplified model of the imaging measurements.

In this paper we present a numerical scheme for the Random Cloud Model (RCM) on a bounded domain which approximates the solution of the time-dependent Schrödinger equation. The RCM is formulated as a Markov jump process on a particle number state space. Based on this process a stochastic algorithm is developed. It is shown that the algorithm reproduces the dynamics of the time-dependent Schrödinger equation for exact initial conditions on a bounded domain. The algorithm is then tested for two different cases. First, it is shown that the RCM reproduces the analytic solution for a particle in a potential well with infinite potential. Second, the RCM is used to study three cases with finite potential walls. It is found that the potential triggers processes, which produces RCM particles at a high rate that annihilate each other.

We propose a new numerical scheme for Backward Stochastic Differential Equations (BSDEs) based on branching processes. We approximate an arbitrary (Lipschitz) driver by local polynomials and then use a Picard iteration scheme. Each step of the Picard iteration can be solved by using a representation in terms of branching diffusion systems, thus avoiding the need for a fine time discretization. In contrast to the previous literature on the numerical resolution of BSDEs based on branching processes, we prove the convergence of our numerical scheme without limitation on the time horizon. Numerical simulations are provided to illustrate the performance of the algorithm.

We are interested in the numerical solution of the ordinary differential equation y′⁢(t)=f⁢(t,y⁢(t)){y^{\prime}(t)=f(t,y(t))} when f is smooth in y but lacks regularity in t . We describe a family of methods akin to the Runge–Kutta family. It involves Monte Carlo simulation of integrals. We focus on third-order schemes which use random samples in dimension three. We give error bounds in terms of the step size and the discrepancy of the set used for the Monte Carlo approximations. We solve a model problem in which f undergoes rapid time variations. It is shown for this example that, by using a quasi-random point set in place of pseudo-random samples, we are able to obtain reduced errors.

In this paper, we construct models that approximate the Gaussian stationary Ornstein–Uhlenbeck process with given reliability 1-δ{1-\delta}, 0<δ<1{0<\delta<1}, and accuracy β>0{\beta>0} in the space C⁢([0,T]){C([0,T])}.

One of the most used variants of hidden Markov models (HMMs) is the standard case where the time is discrete and the state spaces (hidden and observed spaces) are finite. In this framework, we are interested in HMMs whose emission process results from a combination of independent Markov chains. Principally, we assume that the emission process evolves as follows: given a hidden state realization k at time t , an emission is a realization of a Markov chain Ytk{Y_{t}^{k}} at time t , and for two different hidden states k and k′{k^{\prime}}, Ytk{Y_{t}^{k}} and Ytk′{Y_{t}^{k^{\prime}}} are assumed independent. Given the hidden process, the considered emission process selects its realizations from independent and homogeneous Markov chains evolving simultaneously. In this paper, we propose to study the three known basic problems of such an HMM variant, by deriving corresponding formulas and algorithms. This allows us to enrich the set of application scenarios of HMMs. Numerical examples are presented to show the applicability of our proposed approach by deriving statistical estimations.