This text describes several methods to approximate multivariate integrals. Cubature formulae that are exact for a space of polyno-mials and Monte Carlo methods are the best known. More recently developed methods such as quasi-Monte Carlo methods (including lattice rules), Smolyak rules and stochastic integration rules are also described. This short note describes the contents of a session keyno...

In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers and a a particular Monte Carlo method numerically computes the Riemann integral. Whereas other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integral...

While the Quasi-Monte Carlo method of numerical integration achieves smaller integration error than standard Monte Carlo, its use in particle physics phenomenology has been hindered by the abscence of a reliable way to estimate that error. The standard Monte Carlo error estimator relies on the assumption that the points are generated independently of each other and, therefore, fails to account ...

* In this article we show the deep connections between discrepancy theory on the one hand and quasi-Monte Carlo integration on the other. Discrepancy theory was established as an area of research going back to the seminal paper by Weyl (1916), whereas Monte Carlo (and later quasi-Monte Carlo) was invented in the 1940s by John von Neumann and Stanislaw Ulam to solve practical problems. The conne...

As opposed to Monte Carlo integration the quasi-Monte Carlo method does not allow for an error estimate from the samples used for the integral approximation and the deterministic error bound is not accessible in the setting of computer graphics, since usually the integrands are of unbounded variation. We investigate the application of randomized quasi-Monte Carlo integration to bidirectional pa...

in this paper, the pricing of a european call option on the underlying asset is performed by using a monte carlo method, one of the powerful simulation methods, where the price development of the asset is simulated and value of the claim is computed in terms of an expected value. the proposed approach, applied in monte carlo simulation, is based on the black-scholes equation which generally def...

The splitting of Quasi-Monte Carlo (QMC) point sequences into blocks or interleaved substreams has been suggested to raise the speed of distributed numerical integration and to lower to traffic on the network. The usefulness of this approach in GRID environments is discussed. After specifying requirements for using QMC techniques in GRID environments in general we review and evaluate the propos...

Monte Carlo integration is a powerful method for computing the value of complex integrals using probabilistic techniques. This document explains the math involved in Monte Carlo integration. First I give an overview of discrete random variables. Then I show how concepts from discrete random variables can be combined with calculus to reason about continuous random variables. Finally, with a know...

The details of doing a Monte Carlo direct lighting calculation are presented. For direct lighting from multiple luminaires, a method of sending one shadow ray per viewing ray is presented, and it is argued that this is preferable for scenes with many luminaires. Some issues of the design of probability densities on unions of luminaire surfaces are discussed.

We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deterministic methods we prove that adaptive Monte Carlo methods are much better. Abstract. We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deter-ministic ...