Monte Carlo theory and practice

Fits of Monte Carlo distributions to data

“…In the case addressed by this paper, the distribution of the real events is the sum of contributions of N different processes, and the predictions for each of these processes are given in terms of a distribution of Monte Carlo events [2]. Let rnij be the number of Monte Carlo entries generated for process i in bin j.…”

Reference functions to decrease errors in Monte Carlo integrals

“…where x is a multidimensional variable of dimension M:c (large), f( x) is a function of the variable x that can be expressed in closed form, 1 and pM 0 (x) is a function of x that can be simulated by a distribution of points generated by Monte Carlo. The distribution pM 0 (x) may not be expressed in closed form but a Monte Carlo algorithm exists that generates points k of coordinates ek with a probability pM 0 (x)dx for each point k to fall in the 1 In this context, "closed form" means the possibility for an expression to be calculated to the computer accuracy in a few machine cycles, i.e., in a time negligible with respect to the time needed to compute integrals numerically.…”

“…The distribution pM 0 (x) may not be expressed in closed form but a Monte Carlo algorithm exists that generates points k of coordinates ek with a probability pM 0 (x)dx for each point k to fall in the 1 In this context, "closed form" means the possibility for an expression to be calculated to the computer accuracy in a few machine cycles, i.e., in a time negligible with respect to the time needed to compute integrals numerically. Note that limits of integrations have not been spelled out in Eqs.…”

“…Monte Carlo techniques are used to estimate quantities that are difficult to calculate by analytic means [1,2]. In general, the problem can be formulated in terms of evaluating some integrals.…”

Formulation of a Stochastic Semi Analytical Finite Element Method for Uncertain Media

“…In probabilistic methods, uncertainty in the parameters is considered and is represented by a random variable or field. Among the different probabilistic methods in mechanics, direct Monte Carlo simulation, which involves sampling and estimation, is an example for a frequently used statistical approach [6,7,8]. There are also many no statistical methods in probabilistic mechanics.…”