Quasi-Monte Carlo Methods for some Linear Algebra Problems. Convergence and Complexity

We present quasi-Monte Carlo analogs of Monte Carlo methods for some linear algebra problems: solving systems of linear equations, computing extreme eigenvalues, and matrix inversion. Reformulating the problems as solving integral equations with a special kernels and domains permits us to analyze the quasi-Monte Carlo methods with bounds from numerical integration. Standard Monte Carlo methods for integration provide a convergence rate of O(N^(−1/2)) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)^k)N^(−1)). We have shown theoretically and through numerical tests that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the considered Monte Carlo methods. We also analyze the complexity of considered quasi-Monte Carlo algorithms and compare them to the complexity of the analogous Monte Carlo and deterministic algorithms.

Monte Carlo Algorithms for Linear Problems

MSC Subject Classification: 65C05, 65U05.

Monte Carlo Study of Particle Transport Problem in Air Pollution

2002 Mathematics Subject Classification: 65C05.

Monte Carlo Method for Reconstruction of Densities

2000 Mathematics Subject Classification: 65C05

Variational Markov chain Monte Carlo for Bayesian smoothing of non-linear diffusions

In this paper we develop set of novel Markov Chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. The novel diffusion bridge proposal derived from the variational approximation allows the use of a flexible blocking strategy that further improves mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample applications the algorithm is accurate except in the presence of large observation errors and low to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient. © 2011 Springer-Verlag.

How much is corporate cash-pooling worth? Modelling and simulation

The paper analyzes a special corporate banking product, the so called cash-pool, which gained remarkable popularity in the recent years as firms try to centralize and manage their liquidity more efficiently. The novelty of this paper is the formalization of a valuation model which can serve as a basis for a Monte Carlo simulation to assess the most important benefits of the firms arising from the pooling of their cash holdings. The literature emphasizes several benefits of cash-pooling such as interest rate savings, economy of scale and reduced cash-flow volatility. The presented model focuses on the interest rate savings complemented with a new aspect: the reduced counterparty risk toward the bank. The main conclusion of the analysis is that the value of a cash-pool is higher in case of firms with large, diverse and volatile cash-flows having less access to the capital markets especially if the partner bank is risky and offers a high interest spread. It is also shown that cash-pooling is not the privilege of large multinational firms any more as the initial direct costs can be easily regained within a year even in the case of SMEs.

Análise da formulação em Monte Carlo dos pacotes de energia multiespectrais aplicada a meios constituídos por vapor d'água

The Monte Carlo method is accurate and is relatively simple to implement for the solution of problems involving complex geometries and anisotropic scattering of radiation as compared with other numerical techniques. In addition, differently of what happens for most of numerical techniques, for which the associated simulations computational time tends to increase exponentially with the complexity of the problems, in the Monte Carlo the increase of the computational time tends to be linear. Nevertheless, the Monte Carlo solution is highly computer time consuming for most of the interest problems. The Multispectral Energy Bundle model allows the reduction of the computational time associated to the Monte Carlo solution. The referred model is here analyzed for applications in media constituted for nonparticipating species and water vapor, which is an important emitting species formed during the combustion of hydrocarbon fuels. Aspects related to computer time optimization are investigated the model solutions are compared with benchmark line-by-line solutions

Monte Carlo determination of the phase diagram of the double-exchange model

We study the phase diagram of the double exchange model, with antiferromagnetic interactions, in a cubic lattice both at zero and finite temperature. There is a rich variety of magnetic phases, combined with regions where phase separation takes place. We identify phases, intrinsic to the cubic lattice, which are stable for realistic values of the interactions and dopings. Some of these phases break chiral symmetry, leading to unusual features.

Quelques contributions sur les méthodes de Monte Carlo

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

Collimation of a D-D Neutron Generator for Clinical Implementation of Neutron Stimulated Emission Computed Tomography: a Monte Carlo Study

This work is an investigation into collimator designs for a deuterium-deuterium (DD) neutron generator for an inexpensive and compact neutron imaging system that can be implemented in a hospital. The envisioned application is for a spectroscopic imaging technique called neutron stimulated emission computed tomography (NSECT).

Previous NSECT studies have been performed using a Van-de-Graaff accelerator at the Triangle Universities Nuclear Laboratory (TUNL) in Duke University. This facility has provided invaluable research into the development of NSECT. To transition the current imaging method into a clinically feasible system, there is a need for a high-intensity fast neutron source that can produce collimated beams. The DD neutron generator from Adelphi Technologies Inc. is being explored as a possible candidate to provide the uncollimated neutrons. This DD generator is a compact source that produces 2.5 MeV fast neutrons with intensities of 1012 n/s (4π). The neutron energy is sufficient to excite most isotopes of interest in the body with the exception of carbon and oxygen. However, a special collimator is needed to collimate the 4π neutron emission into a narrow beam. This work describes the development and evaluation of a series of collimator designs to collimate the DD generator for narrow beams suitable for NSECT imaging.

A neutron collimator made of high-density polyethylene (HDPE) and lead was modeled and simulated using the GEANT4 toolkit. The collimator was designed as a 52 x 52 x 52 cm3 HDPE block coupled with 1 cm lead shielding. Non-tapering (cylindrical) and tapering (conical) opening designs were modeled into the collimator to permit passage of neutrons. The shape, size, and geometry of the aperture were varied to assess the effects on the collimated neutron beam. Parameters varied were: inlet diameter (1-5 cm), outlet diameter (1-5 cm), aperture diameter (0.5-1.5 cm), and aperture placement (13-39 cm). For each combination of collimator parameters, the spatial and energy distributions of neutrons and gammas were tracked and analyzed to determine three performance parameters: neutron beam-width, primary neutron flux, and the output quality. To evaluate these parameters, the simulated neutron beams are then regenerated for a NSECT breast scan. Scan involved a realistic breast lesion implanted into an anthropomorphic female phantom.

This work indicates potential for collimating and shielding a DD neutron generator for use in a clinical NSECT system. The proposed collimator designs produced a well-collimated neutron beam that can be used for NSECT breast imaging. The aperture diameter showed a strong correlation to the beam-width, where the collimated neutron beam-width was about 10% larger than the physical aperture diameter. In addition, a collimator opening consisting of a tapering inlet and cylindrical outlet allowed greater neutron throughput when compared to a simple cylindrical opening. The tapering inlet design can allow additional neutron throughput when the neck is placed farther from the source. On the other hand, the tapering designs also decrease output quality (i.e. increase in stray neutrons outside the primary collimated beam). All collimators are cataloged in measures of beam-width, neutron flux, and output quality. For a particular NSECT application, an optimal choice should be based on the collimator specifications listed in this work.

Quelques contributions sur les méthodes de Monte Carlo

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.