In this paper we extend the reuse of paths to the shot from a moving light source. In the classical algorithm new paths have to be cast from each new position of a light source. We show that we can reuse all paths for all positions, obtaining in this way a theoretical maximum speed-up equal to the average length of the shooting path. 1.

In this paper, we study discretizations of the integral transport equation obtained by restricting the kernel to a nite set of points in phase space. The resulting matrix equation may then be solved adaptively to obtain an approximation to the solution of the original problem. The technique may be viewed as a variant of Nystom's quadrature method [1], which is normally applied by using a regular grid as the set of nodal points. We prove a theorem that establishes the error resulting from such approximations. This theorem suggests that low discrepancy nodal sets will produce smaller errors than cartesian product and other grid choices, especially when the dimension of the phase space is moderate or high. Application of Monte Carlo simulation to the matrix problem (making use of discrete random walks) has been carried out in a companion paper [2] and an improved version of this technique is applied here. Numerical results indicate the potential of this new technique for solving integral...

. The pricing of options is a very important problem encountered in financial markets today. The famous Black-Scholes model provides explicit closed form solutions for the values of certain (European style) call and put options. But for many other options, either there are no closed form solutions, or if such closed form solutions exist, the formulas exhibiting them are complicated and difficult to evaluate accurately by conventional methods. In this case, Monte Carlo methods may prove to be valuable. In this paper, we illustrate two separate applications of Monte Carlo and/or quasi-Monte Carlo methods to the pricing of options: first, the method is used to estimate multiple integrals related to the evaluation of European style options; second, an adaptive Monte Carlo method is applied to a finite difference approximation of a partial differential equation formulation of a class of finance problems. Some of the advantages in using the Monte Carlo method for such problems are discussed....

this paper is based on new approach

We study sequential sampling methods based principally on ideas of Halton. Such methods are designed to build information drawn from early batches of random walk histories into the random walk process used to generate later histories in order to accelerate convergence. In previously published work, such methods have been applied within the pseudorandom, rather than the quasirandom, context and have been applied only to matrix problems. In this paper, more general sequential techniques are formulated in an abstract space, such as Banach space. The more general formulation enables applications to linear algebraic equations and to integral equations to be obtained as special cases through specification of the Banach space and the operator defined on it. In this initial paper we outline the ideas needed for consideration of the more general problem and exhibit greatly accelerated convergence for a simple matrix problem. In a companion paper in which similar ideas are applied t...

. In 1996 Los Alamos National Laboratory initiated an ambitious five year research program aimed at achieving geometric convergence for Monte Carlo solutions of difficult neutron and photon transport problems. Claremont students, working with the author in Mathematics Clinic projects that same year and subsequently, have been partners in this undertaking. This paper summarizes progress made at Claremont over the two year period, with emphasis on recent advances. The Claremont approach has been to maintain as much generality as possible, aiming ultimately at the Monte Carlo solution of quite general transport equations while using various model transport problems - both discrete and continuous - to establish feasibility. As far as we are aware, prior to this effort, only the discrete case had been seriously attacked by sequential sampling methods: by Halton beginning in 1962 [1] and subsequently by Kollman in his 1993 Stanford dissertation [2]. In work performed in Claremont, an adaptiv...

this article these random series possess the martingale property. Examples of processes which satisfy this definition include the sample mean process of a sequence of martingale di#erences, multivariate linear least squares estimators and the Robbins-Monro stochastic approximation algorithm

The problem of solving System of Linear Algebraic Equations (SLAE) by parallel Monte Carlo numerical methods is considered.

In this paper we consider Monte Carlo (MC) algorithms based on the use of the resolvent matrix for solving linear algebraic problems. Estimates for the speedup and efficiency of the algorithms are presented. Some numerical examples performed on cluster of workstations using MPI are given.