Monte Carlo simulation speeding

Dynamic Monte Carlo simulations were first used by Verdier and Stockmayer (5) for lattice polymers. An alternative dynamical Monte Carlo method has been developed by Ceperley, Kalos and Lebowitz (6) and applied to the study of single, three dimensional polymers. In addition to the dynamic Monte Carlo studies, molecular dynamics methods have been used. Ryckaert and Bellemans (7) and Weber (8) have studied liquid n-butane. Solvent effects have been probed by Bishop, Kalos and Frisch (9), Rapaport (10), and Rebertus, Berne and Chandler (11). Multichain systems have been simulated by Curro (12), De Vos and Bellemans (13), Wall et al (14), Okamoto (15), Kranbu ehl and Schardt (16), and Mandel (17). Curro s study was the only one without a lattice but no dynamic properties were calculated because the standard Metropolis method was employed. De Vos and Belleman, Okamoto, and Kranbuehl and Schardt studies included dynamics by using the technique of Verdier and Stockmayer. Wall et al and Mandel introduced a novel mechanism for speeding relaxation to equilibrium but no dynamical properties were studied. These investigations indicated that the chain contracted and the chain dynamic processes slowed down in the presence of other polymers. 

D. Frenkel (2004) Speed-up of Monte Carlo simulations by sampling of rejected states. Proc. Nat. Acad. Sci., 101, p. 17571... 

In closing, both molecular dynamics and Monte Carlo simulations can be used effectively to study micellar systems. The recent advances in the speed of computers and parallel processing enable the use of MD simulations for these studies, although they remain computationally expensive. Hence, MD simulations are still not effective for the study of equilibrium properties. Therefore, many studies have been published using MC... 

One of the biggest drawbacks to Monte Carlo simulation is the long run times that are often needed in order to achieve stable results. It is therefore useful to use techniques that speed up the simulations without causing the quality of the randomness of the numbers used internally to deteriorate. 

The stochastic analysis framework, that has shown its value in financial mathematics (e.g. Glasserman, 2004), is exploited by the TOPAZ methodology to develop Monte Carlo simulation models and appropriate speed-up factors by risk decomposition. The power of these stochastic analysis tools lies in their capability to model and analyse in a proper way the arbitrary stochastic event sequences (including dependent events) and the conditional probabilities of such event sequences in stochastic dynamic processes (Blom et al., 2(X)3c Krystul Blom, 2004). By using these tools from stochastic analysis, a Monte Carlo simulation based risk assessment can mathematically be decomposed into a well-defined sequence of conditional Monte Carlo simulations together with a subsequent composition of the total risk out of these conditional simulation results. The latter composition typically consists of a tree with conditional probabilities to be assessed at the leaves, and nodes which either add or multiply the probabilities coming from the subbranches of that node. Within TOPAZ such a tree is referred to as a collision risk tree (Blom et al., 2001, 2003). 

Future needs for a clean environment will lead to increasingly higher standards for air and water pollutants. These challenges require better sorbents that are not conunerciaUy available. Traditionally, sorbents were developed based on empiricism. To meet the new challenges, tailored sorbents need to be developed based on fundamental principles. Theoretical tools, such as ab initio molecular orbital theory and Monte Carlo simulations can be used to speed up the sorbent design. It is one of the goals of this book to help put sorbent design on a more rational basis. 

The Monte Carlo based model was built of the clearance between ships and the span. The random variables and their parameters were estimated he the base results obtained in the project The creation of the method of dynamic and probabilistic underkeel clearance estimation [L. Gucma, 2008d]. To build example solution the example bridge on the fairway Szczecin-Swinoujscie with the height above water level of H = 36 m have been chosen. The maximum ship, which can enter to Szczecin now is ship with following parameters L = 160 m, T = 9,15 m, and A = 35 m. The horizontal clearance have been calculated for two ship speeds 8 and 4 kn. Input data to the model one are presented in the Table 1. Monte Carlo simulations... 

The conditions have been chosen such that each P( Conditionjc) can be determined analytically as a function of one or more DCPN-based model parameter values. Hence, Monte Carlo simulations are only needed to determine the P(Safety Event I Conditionjc) for all k, after which all results are combined using the formula above. This significantly speeds up the Monte Carlo simulations by several orders of magnitude. 

Perez, D., Gamier, R., Chevalier, M. Signoret, IP, 1996. A comprehensive method to speed up generalised stochastic Petri net Monte Carlo simulation for non-Markovian systems. In Proceedings of ESREU96-PSAM III, vol. 3, pg 1941-1946, June 1996, Crete, Greece. 

Monte Carlo simulations [102]. are generally less precise in regard to the real reaction time and dynamics of a system compared to MD simulations since MC simulations base on statistics and not real speed of the particles [103]. MC simulations are often lattice based, are easier to program and are also well suited for discontinuous or not differentiable energy functions as well as to define structural equilibration and static properties [103]. With such properties, MC simulations are... 

Let us comment on the developed schematic Monte Carlo (MC) code. Of course there are many clever ways to improve the execution speed. There is the important concept of neighbor lists, of clever implementation of numerical instructions, and so on. In developing a computer simulation one usually plugs in these enhancing concepts bit by bit. 

The rapid rise in computer speed over recent years has led to atom-based simulations of liquid crystals becoming an important new area of research. Molecular mechanics and Monte Carlo studies of isolated liquid crystal molecules are now routine. However, care must be taken to model properly the influence of a nematic mean field if information about molecular structure in a mesophase is required. The current state-of-the-art consists of studies of (in the order of) 100 molecules in the bulk, in contact with a surface, or in a bilayer in contact with a solvent. Current simulation times can extend to around 10 ns and are sufficient to observe the growth of mesophases from an isotropic liquid. The results from a number of studies look very promising, and a wealth of structural and dynamic data now exists for bulk phases, monolayers and bilayers. Continued development of force fields for liquid crystals will be particularly important in the next few years, and particular emphasis must be placed on the development of all-atom force fields that are able to reproduce liquid phase densities for small molecules. Without these it will be difficult to obtain accurate phase transition temperatures. It will also be necessary to extend atomistic models to several thousand molecules to remove major system size effects which are present in all current work. This will be greatly facilitated by modern parallel simulation methods that allow molecular dynamics simulations to be carried out in parallel on multi-processor systems [115]. 

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