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.

Egységes kockázatkezelési módszertan kialakítása a villamosenergia-ipari átviteli rendszerirányító társaságnál (Development of an integrated risk management methodology for an electricity transmission system operator (TSO) company)

A felelős vállalatirányítás egyik stratégiai jelentőségű tényezője a vállalati szintű kockázatkezelés, mely napjaink egyik legnagyobb kihívást jelentő területe a vállalatvezetés számára. A hatékony vállalati kockázatkezelés nem valósulhat meg kizárólag az általános, nemzetközi és hazai szakirodalomban megfogalmazott kockázatkezelési alapelvek követése mentén, a kockázatkezelési rendszer kialakítása során figyelembe kell venni mind az iparági, mind az adott vállalatra jellemző sajátosságokat. Mindez különösen fontos egy olyan speciális tevékenységet folytató vállalatnál, mint a villamosenergia-ipari átviteli rendszerirányító társaság (transmission system operator, TSO). A cikkben a magyar villamosenergia-ipari átviteli rendszerirányító társasággal együttműködésben készített kutatás keretében előálló olyan komplex elméleti és gyakorlati keretrendszert mutatnak be a szerzők, mely alapján az átviteli rendszerirányító társaság számára kialakítottak egy új, területenként egységes kockázatkezelési módszertant (fókuszban a kockázatok azonosításának és számszerűsítésének módszertani lépéseivel), mely alkalmas a vállalati szintű kockázati kitettség meghatározására. _______ This study handles one of today’s most challenging areas of enterprise management: the development and introduction of an integrated and efficient risk management system. For companies operating in specific network industries with a dominant market share and a key role in the national economy, such as electricity TSO’s, risk management is of stressed importance. The study introduces an innovative, mathematically and statistically grounded as well as economically reasoned management approach for the identification, individual effect calculation and summation of risk factors. Every building block is customized for the organizational structure and operating environment of the TSO. While the identification phase guarantees all-inclusivity, the calculation phase incorporates expert techniques and Monte Carlo simulation and the summation phase presents an expected combined distribution and value effect of risks on the company’s profit lines based on the previously undiscovered correlations between individual risk factors.

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.

Study of the kinetics and mechanism of rapid self-assembly in block copolymer thin films during solvo-microwave annealing

Microwave annealing is an emerging technique for achieving ordered patterns of block copolymer films on substrates. Little is understood about the mechanisms of microphase separation during the microwave annealing process and how it promotes the microphase separation of the blocks. Here, we use controlled power microwave irradiation in the presence of tetrahydrofuran (THF) solvent, to achieve lateral microphase separation in high- lamellar-forming poly(styrene-b-lactic acid) PS-b-PLA. A highly ordered line pattern was formed within seconds on silicon, germanium and silicon on insulator (SOI) substrates. In-situ temperature measurement of the silicon substrate coupled to condition changes during "solvo-microwave" annealing allowed understanding of the processes to be attained. Our results suggest that the substrate has little effect on the ordering process and is essentially microwave transparent but rather, it is direct heating of the polar THF molecules that causes microphase separation. It is postulated that the rapid interaction of THF with microwaves and the resultant temperature increase to 55 degrees C within seconds causes an increase of the vapor pressure of the solvent from 19.8 to 70 kPa. This enriched vapor environment increases the plasticity of both PS and PLA chains and leads to the fast self-assembly kinetics. Comparing the patterns formed on silicon, germanium and silicon on insulator (SOI) and also an in situ temperature measurement of silicon in the oven confirms the significance of the solvent over the role of substrate heating during "solvo-microwave" annealing. Besides the short annealing time which has technological importance, the coherence length is on a micron scale and dewetting is not observed after annealing. The etched pattern (PLA was removed by an Ar/O-2 reactive ion etch) was transferred to the underlying silicon substrate fabricating sub-20 nm silicon nanowires over large areas demonstrating that the morphology is consistent both across and through the film.

Advanced Applications of 3D Dosimetry and 3D Printing in Radiation Therapy

As complex radiotherapy techniques become more readily-practiced, comprehensive 3D dosimetry is a growing necessity for advanced quality assurance. However, clinical implementation has been impeded by a wide variety of factors, including the expense of dedicated optical dosimeter readout tools, high operational costs, and the overall difficulty of use. To address these issues, a novel dry-tank optical CT scanner was designed for PRESAGE 3D dosimeter readout, relying on 3D printed components and omitting costly parts from preceding optical scanners. This work details the design, prototyping, and basic commissioning of the Duke Integrated-lens Optical Scanner (DIOS).

The convex scanning geometry was designed in ScanSim, an in-house Monte Carlo optical ray-tracing simulation. ScanSim parameters were used to build a 3D rendering of a convex ‘solid tank’ for optical-CT, which is capable of collimating a point light source into telecentric geometry without significant quantities of refractive-index matched fluid. The model was 3D printed, processed, and converted into a negative mold via rubber casting to produce a transparent polyurethane scanning tank. The DIOS was assembled with the solid tank, a 3W red LED light source, a computer-controlled rotation stage, and a 12-bit CCD camera. Initial optical phantom studies show negligible spatial inaccuracies in 2D projection images and 3D tomographic reconstructions. A PRESAGE 3D dose measurement for a 4-field box treatment plan from Eclipse shows 95% of voxels passing gamma analysis at 3%/3mm criteria. Gamma analysis between tomographic images of the same dosimeter in the DIOS and DLOS systems show 93.1% agreement at 5%/1mm criteria. From this initial study, the DIOS has demonstrated promise as an economically-viable optical-CT scanner. However, further improvements will be necessary to fully develop this system into an accurate and reliable tool for advanced QA.

Pre-clinical animal studies are used as a conventional means of translational research, as a midpoint between in-vitro cell studies and clinical implementation. However, modern small animal radiotherapy platforms are primitive in comparison with conventional linear accelerators. This work also investigates a series of 3D printed tools to expand the treatment capabilities of the X-RAD 225Cx orthovoltage irradiator, and applies them to a feasibility study of hippocampal avoidance in rodent whole-brain radiotherapy.

As an alternative material to lead, a novel 3D-printable tungsten-composite ABS plastic, GMASS, was tested to create precisely-shaped blocks. Film studies show virtually all primary radiation at 225 kVp can be attenuated by GMASS blocks of 0.5cm thickness. A state-of-the-art software, BlockGen, was used to create custom hippocampus-shaped blocks from medical image data, for any possible axial treatment field arrangement. A custom 3D printed bite block was developed to immobilize and position a supine rat for optimal hippocampal conformity. An immobilized rat CT with digitally-inserted blocks was imported into the SmART-Plan Monte-Carlo simulation software to determine the optimal beam arrangement. Protocols with 4 and 7 equally-spaced fields were considered as viable treatment options, featuring improved hippocampal conformity and whole-brain coverage when compared to prior lateral-opposed protocols. Custom rodent-morphic PRESAGE dosimeters were developed to accurately reflect these treatment scenarios, and a 3D dosimetry study was performed to confirm the SmART-Plan simulations. Measured doses indicate significant hippocampal sparing and moderate whole-brain coverage.