Random Variate Generation and Markov Chain Monte Carlo

Austrian Science Fund

Stochastic simulation is a tool of well known and appreciated importance in many fields of research and application. To use stochastic simulation the generation of random variates from different distributions is a necessary prerequisite.
Therefore the first considerations how to generate random numbers started already in the fifties of the last century. Many methods for sampling from standard distributions have been proposed. The design goals for these algorithms are speed and little memory requirements. If random variates from non-standard distributions or for special simulation problems are required new algorithms are necessary. In the last decade so-called automatic methods have been developed that allow to sample from a large class of distributions with a single piece of code. These algorithms are efficient and have some advantages that makes them attractive even for sampling from standard distributions. The price for using such algorithms is that a setup is required that might be quite expensive for some methods/distributions. Although some research was done there are still a lot of open generation problems. In the framework of Markov Chain Monte Carlo (MCMC) sampling routines with a fast setup are needed. Methods for
sampling independent points from multivariate distributions are important but existing algorithms are slow or work only for low dimensions. For quasi-Monte Carlo methods (QMC) the development of fast automatic algorithms to generate non-uniformly distributed low-discrepancy sequences (quasi-random variates) is required. Another problem is that the mathematical theory behind such automatic methods is based on real numbers, whereas the environment in wich the algorithms are implemented uses so-called floating point numbers which have a limited precision, usually 16 decimal digits. The aim of this project is to find some solutions to these and similar problems.