Monte Carlo methods for matrix computations on the grid

Many scientific and engineering applications involve inverting large matrices or solving systems of linear algebraic equations. Solving these problems with proven algorithms for direct methods can take very long to compute, as they depend on the size of the matrix. The computational complexity of the stochastic Monte Carlo methods depends only on the number of chains and the length of those chains. The computing power needed by inherently parallel Monte Carlo methods can be satisfied very efficiently by distributed computing technologies such as Grid computing. In this paper we show how a load balanced Monte Carlo method for computing the inverse of a dense matrix can be constructed, show how the method can be implemented on the Grid, and demonstrate how efficiently the method scales on multiple processors. (C) 2007 Elsevier B.V. All rights reserved.

Noisy Monte Carlo: convergence of Markov chains with approximate transition kernels

Monte Carlo algorithms often aim to draw from a distribution π by simulating a Markov chain with transition kernel P such that π is invariant under P. However, there are many situations for which it is impractical or impossible to draw from the transition kernel P. For instance, this is the case with massive datasets, where is it prohibitively expensive to calculate the likelihood and is also the case for intractable likelihood models arising from, for example, Gibbs random fields, such as those found in spatial statistics and network analysis. A natural approach in these cases is to replace P by an approximation Pˆ. Using theory from the stability of Markov chains we explore a variety of situations where it is possible to quantify how ’close’ the chain given by the transition kernel Pˆ is to the chain given by P . We apply these results to several examples from spatial statistics and network analysis.

Parallel Hybrid Monte Carlo Algorithms for Matrix Computations

In this paper we consider hybrid (fast stochastic approximation and deterministic refinement) algorithms for Matrix Inversion (MI) and Solving Systems of Linear Equations (SLAE). Monte Carlo methods are used for the stochastic approximation, since it is known that they are very efficient in finding a quick rough approximation of the element or a row of the inverse matrix or finding a component of the solution vector. We show how the stochastic approximation of the MI can be combined with a deterministic refinement procedure to obtain MI with the required precision and further solve the SLAE using MI. We employ a splitting A = D – C of a given non-singular matrix A, where D is a diagonal dominant matrix and matrix C is a diagonal matrix. In our algorithm for solving SLAE and MI different choices of D can be considered in order to control the norm of matrix T = D –1C, of the resulting SLAE and to minimize the number of the Markov Chains required to reach given precision. Further we run the algorithms on a mini-Grid and investigate their efficiency depending on the granularity. Corresponding experimental results are presented.

Incorporating technology buying behaviour into UK-based long term domestic stock energy models to provide improved policy analysis

The UK has a target for an 80% reduction in CO2 emissions by 2050 from a 1990 base. Domestic energy use accounts for around 30% of total emissions. This paper presents a comprehensive review of existing models and modelling techniques and indicates how they might be improved by considering individual buying behaviour. Macro (top-down) and micro (bottom-up) models have been reviewed and analysed. It is found that bottom-up models can project technology diffusion due to their higher resolution. The weakness of existing bottom-up models at capturing individual green technology buying behaviour has been identified. Consequently, Markov chains, neural networks and agent-based modelling are proposed as possible methods to incorporate buying behaviour within a domestic energy forecast model. Among the three methods, agent-based models are found to be the most promising, although a successful agent approach requires large amounts of input data. A prototype agent-based model has been developed and tested, which demonstrates the feasibility of an agent approach. This model shows that an agent-based approach is promising as a means to predict the effectiveness of various policy measures.

A combined Bayesian Markovian approach for behaviour recognition

Numerous techniques exist which can be used for the task of behavioural analysis and recognition. Common amongst these are Bayesian networks and Hidden Markov Models. Although these techniques are extremely powerful and well developed, both have important limitations. By fusing these techniques together to form Bayes-Markov chains, the advantages of both techniques can be preserved, while reducing their limitations. The Bayes-Markov technique forms the basis of a common, flexible framework for supplementing Markov chains with additional features. This results in improved user output, and aids in the rapid development of flexible and efficient behaviour recognition systems.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

Bayesian estimation and selection of nonlinear vector error correction models: The case of the sugar-ethanol-oil nexus in Brazil

Nonlinear adjustment toward long-run price equilibrium relationships in the sugar-ethanol-oil nexus in Brazil is examined. We develop generalized bivariate error correction models that allow for cointegration between sugar, ethanol, and oil prices, where dynamic adjustments are potentially nonlinear functions of the disequilibrium errors. A range of models are estimated using Bayesian Monte Carlo Markov Chain algorithms and compared using Bayesian model selection methods. The results suggest that the long-run drivers of Brazilian sugar prices are oil prices and that there are nonlinearities in the adjustment processes of sugar and ethanol prices to oil price but linear adjustment between ethanol and sugar prices.

Measuring departures from Hardy-Weinberg: a Markov chain Monte Carlo method for estimating the inbreeding coefficient

Many well-established statistical methods in genetics were developed in a climate of severe constraints on computational power. Recent advances in simulation methodology now bring modern, flexible statistical methods within the reach of scientists having access to a desktop workstation. We illustrate the potential advantages now available by considering the problem of assessing departures from Hardy-Weinberg (HW) equilibrium. Several hypothesis tests of HW have been established, as well as a variety of point estimation methods for the parameter which measures departures from HW under the inbreeding model. We propose a computational, Bayesian method for assessing departures from HW, which has a number of important advantages over existing approaches. The method incorporates the effects-of uncertainty about the nuisance parameters--the allele frequencies--as well as the boundary constraints on f (which are functions of the nuisance parameters). Results are naturally presented visually, exploiting the graphics capabilities of modern computer environments to allow straightforward interpretation. Perhaps most importantly, the method is founded on a flexible, likelihood-based modelling framework, which can incorporate the inbreeding model if appropriate, but also allows the assumptions of the model to he investigated and, if necessary, relaxed. Under appropriate conditions, information can be shared across loci and, possibly, across populations, leading to more precise estimation. The advantages of the method are illustrated by application both to simulated data and to data analysed by alternative methods in the recent literature.

A singular vector perspective of 4DVAR: The spatial structure and evolution of baroclinic weather systems

The extent to which the four-dimensional variational data assimilation (4DVAR) is able to use information about the time evolution of the atmosphere to infer the vertical spatial structure of baroclinic weather systems is investigated. The singular value decomposition (SVD) of the 4DVAR observability matrix is introduced as a novel technique to examine the spatial structure of analysis increments. Specific results are illustrated using 4DVAR analyses and SVD within an idealized 2D Eady model setting. Three different aspects are investigated. The first aspect considers correcting errors that result in normal-mode growth or decay. The results show that 4DVAR performs well at correcting growing errors but not decaying errors. Although it is possible for 4DVAR to correct decaying errors, the assimilation of observations can be detrimental to a forecast because 4DVAR is likely to add growing errors instead of correcting decaying errors. The second aspect shows that the singular values of the observability matrix are a useful tool to identify the optimal spatial and temporal locations for the observations. The results show that the ability to extract the time-evolution information can be maximized by placing the observations far apart in time. The third aspect considers correcting errors that result in nonmodal rapid growth. 4DVAR is able to use the model dynamics to infer some of the vertical structure. However, the specification of the case-dependent background error variances plays a crucial role.

A singular vector perspective of 4D-Var: Filtering and interpolation

Four-dimensional variational data assimilation (4D-Var) combines the information from a time sequence of observations with the model dynamics and a background state to produce an analysis. In this paper, a new mathematical insight into the behaviour of 4D-Var is gained from an extension of concepts that are used to assess the qualitative information content of observations in satellite retrievals. It is shown that the 4D-Var analysis increments can be written as a linear combination of the singular vectors of a matrix which is a function of both the observational and the forecast model systems. This formulation is used to consider the filtering and interpolating aspects of 4D-Var using idealized case-studies based on a simple model of baroclinic instability. The results of the 4D-Var case-studies exhibit the reconstruction of the state in unobserved regions as a consequence of the interpolation of observations through time. The results also exhibit the filtering of components with small spatial scales that correspond to noise, and the filtering of structures in unobserved regions. The singular vector perspective gives a very clear view of this filtering and interpolating by the 4D-Var algorithm and shows that the appropriate specification of the a priori statistics is vital to extract the largest possible amount of useful information from the observations. Copyright © 2005 Royal Meteorological Society

The development and application of capillary electrophoresis methods for food analysis

Capillary electrophoresis (CE) offers the analyst a number of key advantages for the analysis of the components of foods. CE offers better resolution than, say, high-performance liquid chromatography (HPLC), and is more adept at the simultaneous separation of a number of components of different chemistries within a single matrix. In addition, CE requires less rigorous sample cleanup procedures than HPLC, while offering the same degree of automation. However, despite these advantages, CE remains under-utilized by food analysts. Therefore, this review consolidates and discusses the currently reported applications of CE that are relevant to the analysis of foods. Some discussion is also devoted to the development of these reported methods and to the advantages/disadvantages compared with the more usual methods for each particular analysis. It is the aim of this review to give practicing food analysts an overview of the current scope of CE.

Strategies, effectiveness and rationale of vector-borne disease control in the pastoralist system of south-western Uganda

In Uganda, control of vector-borne diseases is mainly in form of vector control, and chemotherapy. There have been reports that acaricides are being misused in the pastoralist systems in Uganda. This is because of the belief by scientists that intensive application of acaricide is uneconomical and unsustainable particularly in the indigenous cattle. The objective of this study was to investigate the strategies, rationale and effectiveness of vector-borne disease control by pastoralists. To systematically carry out these investigations, a combination of qualitative and quantitative research methods was used, in both the collection and the analysis of data. Cattle keepers were found to control tick-borne diseases (TBDs) mainly through spraying, in contrast with the control of trypanosomosis for which the main method of control was by chemotherapy. The majority of herders applied acaricides weekly and used an acaricide of lower strength than recommended by the manufacturers. They used very little acaricide wash, and spraying was preferred to dipping. Furthermore, pastoralists either treated sick animals themselves or did nothing at all, rather than using veterinary personnel. Oxytetracycline (OTC) was the drug commonly used in the treatment of TBDs. Nevertheless, although pastoralists may not have been following recommended practices in their control of ticks and tick-borne diseases, they were neither wasteful nor uneconomical and their methods appeared to be effective. Trypanosomosis was not a problem either in Sembabule or Mbarara district. Those who used trypanocides were found to use more drugs than were necessary.