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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 paper aims to practice the use of Monte Carlo methods to simulate stock prices in order to price European call options using control variates. American put options are priced using the binomial model separately. Finally, we use the information to form a portfolio position using an Interactive Brokers paper trading account.

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.

<p>This thesis considers several different techniques for optimizing Monte Carlo simulations. The Monte Carlo system used is Penelope but most of the techniques are applicable to other systems. The two mayor techniques are the usage of the graphics card to do geometry calculations, and raytracing. Using graphics card provides a very efficient way to do fast ray and triangle intersections. Raytracing provides an approximation of Monte Carlo simulation but is much faster to perform. A program was also written in order to have a platform for Monte Carlo simulations where the different techniques were implemented and tested. The program also provides an overview of the simulation setup, were the user can easily verify that everything has been setup correctly. The thesis also covers an attempt to rewrite Penelope from FORTAN to C. The new version is significantly faster and can be used on more systems. A distribution package was also added to the new Penelope version. Since Monte Carlo simulations are easily distributed, running this type of simulations on ten computers yields ten times the speedup. Combining the different techniques in the platform provides an easy to use and at the same time efficient way of performing Monte Carlo simulations.</p>

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.

Although entanglement is a well studied property in the context of quantum systems, the ability to measure it in Monte Carlo methods is relatively new. Through measures of the Renyi entanglement entropy and mutual information one is able to examine and characterize criticality, pinpoint phase transitions, and probe universality. We describe the most basic algorithms for calculating these quantities in straightforward Monte Carlo methods and state of the art techniques used in high performance computing. This description emphasizes the core principal of these measurements and allows one to both build an intuition for these quantities and how they are useful in numerical studies. Using the Renyi entanglement entropy we demonstrate the ability to detect thermal phase transitions in the Ising model and XY model without use of an order parameter. The scaling near the critical point also shows signatures identifying the universality class of the model. Improved methods are explored using extended ensemble techniques that can increase calculation efficiency, and show good agreement with the standard approach. We explore the "ratio trick" at finite temperature and use it to explore the quantum critical fan of the one dimensional transverse field Ising model, showing agreement with finite temperature and finite size scaling from field theory. This same technique is used at zero temperature to explore the geometric dependence of the entanglement entropy and examine the universal scaling functions in the two dimensional transverse field Ising model. All of this shows the multitude of ways in which the study of the Renyi entanglement entropy can be efficiently and practically used in conventional and exotic condensed matter systems, and should serve as a reference for those wishing to use it as a tool.