Monte Carlo importance sampling

Let us begin with the one-dimensional case in Eq. (1.2). Suppose we want to compute the definite integral by Monte Carlo importance sampling with sampling points... 

Hence, it would clearly be advisable to carry out a non-uniform Monte Carlo importance sampling of configuration space with a w approximately proportional to the Boltzmann factor. 

MODELING STRATEGIES Monte Carlo Importance Sampling ... 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

Monte Carlo sampling is discussed extensively in Hammersley and Morton (1956), Hammersley and Handscomb (1964), Kloek and Van Dijk (1978), and Wilson (1984). For Monte Carlo results to be believable, the convergence properties of the Monte Carlo estimators must be met. Several statistical and practical limitations exist in this regard. The most important practical limitations of Monte Carlo are the following ... 

There are several sampling techniques in Monte Carlo analyses, the most common being random, median Latin hypercube and random Latin hypercube. Latin hypercube techniques are usually preferred because they need fewer iterations and thus are more efficient. They are, however, inferior to random sampling if high percentiles of the output are of interest and if the exact shape of the output distribution is important (Cullen and Frey 1999). 

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