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There are few reliable computational techniques applicable to the problem of structural phase behaviour. This is starkly emphasised by the fact that there are still a number of unanswered questions concerning the solid state of some of the simplest models of matter. To determine the phase behaviour of a given system we invoke the machinery of statistical physics, which identifies the equilibrium phase as that which minimises the free-energy. This type of problem can only be dealt with fully via numerical simulation, as any less direct approach will involve making some uncontrolled approximation. In particular, a numerical simulation can be used to evaluate the free-energy difference between two phases if the simulation is free to visit them both. However, it has proven very difficult to find an algorithm which is capable of efficiently exploring two different phases, particularly when one or both of them is a crystalline solid. This thesis builds on previous work (Physical Review Letters 79 p.3002), exploring a new Monte Carlo approach to this class of problem. This new simulation technique uses a global coordinate transformation to switch between two different crystalline structures. Generally, this `lattice switch' is found to be extremely unlikely to succeed in a normal Monte Carlo simulation. To overcome this, extended-sampling techniques are used to encourage the simulation to visit `gateway' microstates where the switch will be successful. After compensating for this bias in the sampling, the free-energy difference between the two structures can be evaluated directly from their relative probabilities. As concrete examples on which to base the research, the lattice-switch Monte Carlo method is used to determine the free-energy difference between the face-centred cubic (fcc) and hexagonal close-packed (hcp) phases of two generic model systems --- the hard-sphere and Lennard-Jones potentials. The structural phase behaviour of the hard-sphere solid is determined at densities near melting and in the close-packed limit. The factors controlling the efficiency of the lattice-switch approach are explored, as is the character of the `gateway' microstates. The face-centred cubic structure is identified as the thermodynamically stable phase, and the free-energy difference between the two structures is determined with high precision. These results are shown to be in complete agreement with the results of other authors in the field (published during the course of this work), some of whom adopted the lattice-switch method for their calculations. Also, the results are favourably compared against the experimentally observed structural phase behaviour of sterically-stabilised colloidal dispersions, which are believed to behave like systems of hard spheres. The logical extension of the hard sphere work is to generalise the lattice-switch technique to deal with `softer' systems, such as the Lennard-Jones solid. The results in the literature for the structural phase behaviour of this relatively simple system are found to be completely inconsistent. A number of different approaches to this problem are explored, leading to the conclusion that these inconsistencies arise from the way in which the potential is truncated. Using results for the ground-state energies and from the harmonic approximation, we develop a new truncation scheme which allows this system to be simulated accurately and efficiently. Lattice-switch Monte Carlo is then used to determine the fcc-hcp phase boundary of the Lennard-Jones solid in its entirety. These results are compared against the experimental results for the Lennard-Jones potential's closest physical analogue, the rare-gas solids. While some of the published rare-gas observations are in approximate agreement with the lattice-switch results, these findings contradict the widely held belief that fcc is the equilibrium structure of the heavier rare-gas solids for all pressures and temperatures. The possible reasons for this disagreement are discussed. Finally, we examine the pros and cons of the lattice-switch technique, and explore ways in which it can be extended to cover an even wider range of structures and interactions.

This report describes our work in pricing options using computational methods. First, I collected the historical asset prices for assets in four economic sectors to estimate model parameters, such as asset returns and covariances. Then I used these parameters to model asset prices using multiple geometric Brownian motion and simulate new asset prices. Using the generated prices, I used Monte Carlo methods and control variates to price call options. Next I used the binomial tree model to price put options, which I was introduced to in the course Math 571: Financial Mathematics I. Using the estimated put and call option prices together with some stocks, I formed a portfolio in an Interactive Brokers paper account . This project was done a part of the masters capstone course Math 573: Computational Methods of Financial Mathematics.

by Tang Hing Sing. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1989. / Bibliography: leaves 92-96.

Markov chain Monte Carlo (MCMC) is one of the most popular statistical inference methods in machine learning. Recent work shows that a significant improvement of the statistical efficiency of MCMC on complex distributions can be achieved by exploiting geometric properties of the target distribution. This is known as geometric MCMC. However, many such methods, like Riemannian manifold Hamiltonian Monte Carlo (RMHMC), are computationally challenging to scale up to high dimensional distributions. The primary goal of this thesis is to develop novel geometric MCMC methods applicable to large-scale problems. To overcome the computational bottleneck of computing second order derivatives in geometric MCMC, I propose an adaptive MCMC algorithm using an efficient approximation based on Limited memory BFGS. I also propose a simplified variant of RMHMC that is able to work effectively on larger scale than the previous methods. Finally, I address an important limitation of geometric MCMC, namely that is only available for continuous distributions. I investigate a relaxation of discrete variables to continuous variables that allows us to apply the geometric methods. This is a new direction of MCMC research which is of potential interest to many applications. The effectiveness of the proposed methods is demonstrated on a wide range of popular models, including generalised linear models, conditional random fields (CRFs), hierarchical models and Boltzmann machines.

The objective of this research was to develop a complex MCNPX model of the human head to predict absorbed dose distributions during proton therapy of ocular tumors. Absorbed dose distributions using the complex geometry were compared to a simple MCNPX model of the human eye developed by Oertli. The proton therapy beam used at Laboratori Nazionali del Sud-INFN was chosen for comparison. Dose calculations included dose due to proton and secondary interactions, multiple coulombic energy scattering, elastic and inelastic scattering, and non-elastic nuclear reactions. Benchmarking MCNPX was accomplished using the proton simulations outlined by Oertli. Once MCNPX was properly benchmarked, the proton beam and MCNPX models were combined to predict dose distributions for three treatment scenarios. First, an ideal treatment scenario was modeled where the dose was maximized to the tumor volume and minimized elsewhere. The second situation, a worst case scenario, mimicked a patient starring directly into the treatment beam during therapy. During the third simulation, the treatment beam was aimed into the bone surrounding the eye socket to estimate the dose to the vital regions of the eye due to scattering. Dose distributions observed for all three cases were as expected. Superior dose distributions were observed with the complex geometry for all tissues of the phantom and the tumor volume. This study concluded that complex MCNPX geometries, although initially difficult to implement, produced superior dose distributions when compared to simple models.

Monte Carlo method is an invaluable tool in the field of radiation protection, used to calculate shielding effectiveness, as well as dose for medical applications. With few exceptions, most of the objects currently simulated have been homogeneous materials that vary in density by a factor of 3 or less. In the irradiation of very heterogeneous objects, particularly layered or leafy food items, one will encounter air pockets within the bundle as a matter of course. These pockets will cause variations in density of up to three orders of magnitude. Air pockets in a tissue equivalent phantom were found to produce “hot spots” in the dose distribution, and introduced significant deviations between the calculated and measured distribution of dose to the phantom. To date, very little published work had been done in the area of Monte-Carlo simulation of objects of such disparate density. Before Monte Carlo methods can be used successfully in this regime, further code development and experimental validation will be necessary, of which this work is just a beginning. Phantoms were made of corrugated low-Z material similar in electron density to plant based material. These phantoms incorporated air gaps of comparable size to those found in the leafy objects of interest. Dimensions were chosen to bracket electron ranges in the material of the objects modeled. Monte Carlo analysis will provide a reasonable qualitative picture of the dose distribution, but such a picture is not yet sufficiently accurate in a quantitative sense. Air gaps within the plant material produced large discrepancies between calculation and measurement. Smaller air gaps were observed to produce greater discrepancy between calculation and measurement.

The objective of this research was to develop a complex MCNPX model of the human head to predict absorbed dose distributions during proton therapy of ocular tumors. Absorbed dose distributions using the complex geometry were compared to a simple MCNPX model of the human eye developed by Oertli. The proton therapy beam used at Laboratori Nazionali del Sud-INFN was chosen for comparison. Dose calculations included dose due to proton and secondary interactions, multiple coulombic energy scattering, elastic and inelastic scattering, and non-elastic nuclear reactions. Benchmarking MCNPX was accomplished using the proton simulations outlined by Oertli. Once MCNPX was properly benchmarked, the proton beam and MCNPX models were combined to predict dose distributions for three treatment scenarios. First, an ideal treatment scenario was modeled where the dose was maximized to the tumor volume and minimized elsewhere. The second situation, a worst case scenario, mimicked a patient starring directly into the treatment beam during therapy. During the third simulation, the treatment beam was aimed into the bone surrounding the eye socket to estimate the dose to the vital regions of the eye due to scattering. Dose distributions observed for all three cases were as expected. Superior dose distributions were observed with the complex geometry for all tissues of the phantom and the tumor volume. This study concluded that complex MCNPX geometries, although initially difficult to implement, produced superior dose distributions when compared to simple models.

A turbomolecular pump (TMP) with good performance must have higher compress ratio and higher pumping speed. At the same time, the performance of turbomolecular pump depends on blade geometries and the rotational speed. When design the blade of Turbomolecular Pump, the blade geometries including, the blade angle, the blade spacing, the blade chord, the spacing-chord ratio, the tip diameter, the root diameter, and the number of blades and as well as the rotational speed of the rotor must be considered. In this paper the simulation for gas molecular behavior is obtained by the Monte Carlo method. Therefore, a Maxwellian distribution of particles at the inlet and outlet of the flow region and diffuse reflection for the particles that collide with the walls are assumed. Models of this type have been applied to the two-dimensional case. The most important result is to compare the performance between turbomolecular pumps with curve style and plane style of blades. Furthermore, that direct multi-stage simulation (DMS) by Monte Carlo method is used in this paper. The compression ratio multiplication (CRM) method is the improved due to the considering the change of velocity distribution of molecular at the adjacent stages. From results of the simulation, the effect upon the geometric parameters of the blades and the arrangement in the multi-stage are concluded, that are very useful in designing the turbomolecular.