A comparison of variational and Markov chain Monte Carlo methods for inference in partially observed stochastic dynamic systems

In recent work we have developed a novel variational inference method for partially observed systems governed by stochastic differential equations. In this paper we provide a comparison of the Variational Gaussian Process Smoother with an exact solution computed using a Hybrid Monte Carlo approach to path sampling, applied to a stochastic double well potential model. It is demonstrated that the variational smoother provides us a very accurate estimate of mean path while conditional variance is slightly underestimated. We conclude with some remarks as to the advantages and disadvantages of the variational smoother. © 2008 Springer Science + Business Media LLC.

Gaussian process approximations of stochastic differential equations

Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.

Numerical implementation of the coarse step method with a varying differential group delay

The effect of having a fixed differential group delay term in the coarse step method results in a periodic pattern in the inserting a varying DGD term at each integration step, according to a Gaussian distribution. Simulation results are given to illustrate the phenomenon and provide some evidence about its statistical nature.

Analysis and simulation of computer controlled cellular mobile radio systems

Cellular mobile radio systems will be of increasing importance in the future. This thesis describes research work concerned with the teletraffic capacity and the canputer control requirements of such systems. The work involves theoretical analysis and experimental investigations using digital computer simulation. New formulas are derived for the congestion in single-cell systems in which there are both land-to-mobile and mobile-to-mobile calls and in which mobile-to-mobile calls go via the base station. Two approaches are used, the first yields modified forms of the familiar Erlang and Engset formulas, while the second gives more complicated but more accurate formulas. The results of computer simulations to establish the accuracy of the formulas are described. New teletraffic formulas are also derived for the congestion in multi -cell systems. Fixed, dynamic and hybrid channel assignments are considered. The formulas agree with previously published simulation results. Simulation programs are described for the evaluation of the speech traffic of mobiles and for the investigation of a possible computer network for the control of the speech traffic. The programs were developed according to the structured progranming approach leading to programs of modular construction. Two simulation methods are used for the speech traffic: the roulette method and the time-true method. The first is economical but has some restriction, while the second is expensive but gives comprehensive answers. The proposed control network operates at three hierarchical levels performing various control functions which include: the setting-up and clearing-down of calls, the hand-over of calls between cells and the address-changing of mobiles travelling between cities. The results demonstrate the feasibility of the control netwvork and indicate that small mini -computers inter-connected via voice grade data channels would be capable of providing satisfactory control

Simulation and control of a stepping motor

Open-loop operatlon of the stepping motor exploits the inherent advantages of the machine. For near optimum operation: in this mode, however, an accurate system model is required to facilitate controller design. Such a model must be comprehensive and take account of the non-linearities inherent in the system. The result is a complex formulation which can be made manageable with a computational aid. A digital simulation of a hybrid type stepping motor and its associated drive circuit is proposed. The simulation is based upon a block diagram model which includes reasonable approximations to the major non-linearities. The simulation is shown to yield accurate performance predictions. The determination of the transfer functions is based upon the consideration of the physical processes involved rather than upon direct input-outout measurements. The effects of eddy currents, saturation, hysteresis, drive circuit characteristics and non-linear torque displacement characteristics are considered and methods of determining transfer functions, which take account of these effects, are offered. The static torque displacement characteristic is considered in detail and a model is proposed which predicts static torque for any combination of phase currents and shaft position. Methods of predicting the characteristic directly from machine geometry are investigated. Drive circuit design for high efficiency operation is considered and a model of a bipolar, bilevel circuit is proposed. The transfers between stator voltage and stator current and between stator current and air gap flux are complicated by the effects of eddy currents, saturation and hysteresis. Frequency response methods, combined with average inductance measurements, are shown to yield reasonable transfer functions. The modelling procedure and subsequent digital simulation is concluded to be a powerful method of non-linear analysis.

Optimization of chemical plant simulation using double collocation

A method has been constructed for the solution of a wide range of chemical plant simulation models including differential equations and optimization. Double orthogonal collocation on finite elements is applied to convert the model into an NLP problem that is solved either by the VF 13AD package based on successive quadratic programming, or by the GRG2 package, based on the generalized reduced gradient method. This approach is termed simultaneous optimization and solution strategy. The objective functional can contain integral terms. The state and control variables can have time delays. Equalities and inequalities containing state and control variables can be included into the model as well as algebraic equations and inequalities. The maximum number of independent variables is 2. Problems containing 3 independent variables can be transformed into problems having 2 independent variables using finite differencing. The maximum number of NLP variables and constraints is 1500. The method is also suitable for solving ordinary and partial differential equations. The state functions are approximated by a linear combination of Lagrange interpolation polynomials. The control function can either be approximated by a linear combination of Lagrange interpolation polynomials or by a piecewise constant function over finite elements. The number of internal collocation points can vary by finite elements. The residual error is evaluated at arbitrarily chosen equidistant grid-points, thus enabling the user to check the accuracy of the solution between collocation points, where the solution is exact. The solution functions can be tabulated. There is an option to use control vector parameterization to solve optimization problems containing initial value ordinary differential equations. When there are many differential equations or the upper integration limit should be selected optimally then this approach should be used. The portability of the package has been addressed converting the package from V AX FORTRAN 77 into IBM PC FORTRAN 77 and into SUN SPARC 2000 FORTRAN 77. Computer runs have shown that the method can reproduce optimization problems published in the literature. The GRG2 and the VF I 3AD packages, integrated into the optimization package, proved to be robust and reliable. The package contains an executive module, a module performing control vector parameterization and 2 nonlinear problem solver modules, GRG2 and VF I 3AD. There is a stand-alone module that converts the differential-algebraic optimization problem into a nonlinear programming problem.

Computer simulation of form-roll design

Changes in modern structural design have created a demand for products which are light but possess high strength. The objective is a reduction in fuel consumption and weight of materials to satisfy both economic and environmental criteria. Cold roll forming has the potential to fulfil this requirement. The bending process is controlled by the shape of the profile machined on the periphery of the rolls. A CNC lathe can machine complicated profiles to a high standard of precision, but the expertise of a numerical control programmer is required. A computer program was developed during this project, using the expert system concept, to calculate tool paths and consequently to expedite the procurement of the machine control tapes whilst removing the need for a skilled programmer. Codifying the expertise of a human and the encapsulation of knowledge within a computer memory, destroys the dependency on highly trained people whose services can be costly, inconsistent and unreliable. A successful cold roll forming operation, where the product is geometrically correct and free from visual defects, is not easy to attain. The geometry of the sheet after travelling through the rolling mill depends on the residual strains generated by the elastic-plastic deformation. Accurate evaluation of the residual strains can provide the basis for predicting the geometry of the section. A study of geometric and material non-linearity, yield criteria, material hardening and stress-strain relationships was undertaken in this research project. The finite element method was chosen to provide a mathematical model of the bending process and, to ensure an efficient manipulation of the large stiffness matrices, the frontal solution was applied. A series of experimental investigations provided data to compare with corresponding values obtained from the theoretical modelling. A computer simulation, capable of predicting that a design will be satisfactory prior to the manufacture of the rolls, would allow effort to be concentrated into devising an optimum design where costs are minimised.

Simulation of robot manipulator control strategies

The high capital cost of robots prohibit their economic application. One method of making their application more economic is to increase their operating speed. This can be done in a number of ways e.g. redesign of robot geometry, improving actuators and improving control system design. In this thesis the control system design is considered. It is identified in the literature review that two aspects in relation to robot control system design have not been addressed in any great detail by previous researchers. These are: how significant are the coupling terms in the dynamic equations of the robot and what is the effect of the coupling terms on the performance of a number of typical independent axis control schemes?. The work in this thesis addresses these two questions in detail. A program was designed to automatically calculate the path and trajectory and to calculate the significance of the coupling terms in an example application of a robot manipulator tracking a part on a moving conveyor. The inertial and velocity coupling terms have been shown to be of significance when the manipulator was considered to be directly driven. A simulation of the robot manipulator following the planned trajectory has been established in order to assess the performance of the independent axis control strategies. The inertial coupling was shown to reinforce the control torque at the corner points of the trajectory, where there was an abrupt demand in acceleration in each axis but of opposite sign. This reduced the tracking error however, this effect was not controllable. A second effect was due to the velocity coupling terms. At high trajectory speeds it was shown, by means of a root locus analysis, that the velocity coupling terms caused the system to become unstable.

Productivity analysis and measurement

The state of the art in productivity measurement and analysis shows a gap between simple methods having little relevance in practice and sophisticated mathematical theory which is unwieldy for strategic and tactical planning purposes, -particularly at company level. An extension is made in this thesis to the method of productivity measurement and analysis based on the concept of added value, appropriate to those companies in which the materials, bought-in parts and services change substantially and a number of plants and inter-related units are involved in providing components for final assembly. Reviews and comparisons of productivity measurement dealing with alternative indices and their problems have been made and appropriate solutions put forward to productivity analysis in general and the added value method in particular. Based on this concept and method, three kinds of computerised models two of them deterministic, called sensitivity analysis and deterministic appraisal, and the third one, stochastic, called risk simulation, have been developed to cope with the planning of productivity and productivity growth with reference to the changes in their component variables, ranging from a single value 'to• a class interval of values of a productivity distribution. The models are designed to be flexible and can be adjusted according to the available computer capacity expected accuracy and 'presentation of the output. The stochastic model is based on the assumption of statistical independence between individual variables and the existence of normality in their probability distributions. The component variables have been forecasted using polynomials of degree four. This model is tested by comparisons of its behaviour with that of mathematical model using real historical data from British Leyland, and the results were satisfactory within acceptable levels of accuracy. Modifications to the model and its statistical treatment have been made as required. The results of applying these measurements and planning models to the British motor vehicle manufacturing companies are presented and discussed.

The use of spot simulation imagery for forestry mapping and management

SPOT simulation imagery was acquired for a test site in the Forest of Dean in Gloucestershire, U.K. This data was qualitatively and quantitatively evaluated for its potential application in forest resource mapping and management. A variety of techniques are described for enhancing the image with the aim of providing species level discrimination within the forest. Visual interpretation of the imagery was more successful than automated classification. The heterogeneity within the forest classes, and in particular between the forest and urban class, resulted in poor discrimination using traditional `per-pixel' automated methods of classification. Different means of assessing classification accuracy are proposed. Two techniques for measuring textural variation were investigated in an attempt to improve classification accuracy. The first of these, a sequential segmentation method, was found to be beneficial. The second, a parallel segmentation method, resulted in little improvement though this may be related to a combination of resolution in size of the texture extraction area. The effect on classification accuracy of combining the SPOT simulation imagery with other data types is investigated. A grid cell encoding technique was selected as most appropriate for storing digitised topographic (elevation, slope) and ground truth data. Topographic data were shown to improve species-level classification, though with sixteen classes overall accuracies were consistently below 50%. Neither sub-division into age groups or the incorporation of principal components and a band ratio significantly improved classification accuracy. It is concluded that SPOT imagery will not permit species level classification within forested areas as diverse as the Forest of Dean. The imagery will be most useful as part of a multi-stage sampling scheme. The use of texture analysis is highly recommended for extracting maximum information content from the data. Incorporation of the imagery into a GIS will both aid discrimination and provide a useful management tool.

Computer simulation of moist agglomerate collisions

This thesis considers the computer simulation of moist agglomerate collisions using the discrete element method (DEM). The study is confined to pendular state moist agglomerates, at which liquid is presented as either absorbed immobile films or pendular liquid bridges and the interparticle force is modelled as the adhesive contact force and interstitial liquid bridge force. Algorithms used to model the contact force due to surface adhesion, tangential friction and particle deformation have been derived by other researchers and are briefly described in the thesis. A theoretical study of the pendular liquid bridge force between spherical particles has been made and the algorithms for the modelling of the pendular liquid bridge force between spherical particles have been developed and incorporated into the Aston version of the DEM program TRUBAL. It has been found that, for static liquid bridges, the more explicit criterion for specifying the stable solution and critical separation is provided by the total free energy. The critical separation is given by the cube root of liquid bridge volume to a good approximation and the 'gorge method' of evaluation based on the toroidal approximation leads to errors in the calculated force of less than 10%. Three dimensional computer simulations of an agglomerate impacting orthogonally with a wall are reported. The results demonstrate the effectiveness of adding viscous binder to prevent attrition, a common practice in process engineering. Results of simulated agglomerate-agglomerate collisions show that, for colinear agglomerate impacts, there is an optimum velocity which results in a near spherical shape of the coalesced agglomerate and, hence, minimises attrition due to subsequent collisions. The relationship between the optimum impact velocity and the liquid viscosity and surface tension is illustrated. The effect of varying the angle of impact on the coalescence/attrition behaviour is also reported. (DX 187, 340).

Investigating the molecular structures of solid dispersions by the simulated annealing method

Knowledge of the molecular structures of solid dispersions is vital, yet, despite thousands of reports in this area, it remains unclear. The aim of this research is to investigate the molecular structure of solid dispersions with hot melt preparation method by the simulated annealing method. Simulation results showed linear polymer chains form the random coils under heat and the drug molecules stick on the surface of polymer coils, while drug molecules are dispersed molecularly but irregularly within the amorphous low molecular weight carriers. This research presents more reasonable molecular images of solid dispersions than the existed theory.

A procedure for the reconstruction of a stochastic stationary temperature field

An iterative procedure is proposed for the reconstruction of a temperature field from a linear stationary heat equation with stochastic coefficients, and stochastic Cauchy data given on a part of the boundary of a bounded domain. In each step, a series of mixed well-posed boundary-value problems are solved for the stochastic heat operator and its adjoint. Well-posedness of these problems is shown to hold and convergence in the mean of the procedure is proved. A discretized version of this procedure, based on a Monte Carlo Galerkin finite-element method, suitable for numerical implementation is discussed. It is demonstrated that the solution to the discretized problem converges to the continuous as the mesh size tends to zero.

Selection of simulation tools for improving supply chain performance

Simulation is an effective method for improving supply chain performance. However, there is limited advice available to assist practitioners in selecting the most appropriate method for a given problem. Much of the advice that does exist relies on custom and practice rather than a rigorous conceptual or empirical analysis. An analysis of the different modelling techniques applied in the supply chain domain was conducted, and the three main approaches to simulation used were identified; these are System Dynamics (SD), Discrete Event Simulation (DES) and Agent Based Modelling (ABM). This research has examined these approaches in two stages. Firstly, a first principles analysis was carried out in order to challenge the received wisdom about their strengths and weaknesses and a series of propositions were developed from this initial analysis. The second stage was to use the case study approach to test these propositions and to provide further empirical evidence to support their comparison. The contributions of this research are both in terms of knowledge and practice. In terms of knowledge, this research is the first holistic cross paradigm comparison of the three main approaches in the supply chain domain. Case studies have involved building ‘back to back’ models of the same supply chain problem using SD and a discrete approach (either DES or ABM). This has led to contributions concerning the limitations of applying SD to operational problem types. SD has also been found to have risks when applied to strategic and policy problems. Discrete methods have been found to have potential for exploring strategic problem types. It has been found that discrete simulation methods can model material and information feedback successfully. Further insights have been gained into the relationship between modelling purpose and modelling approach. In terms of practice, the findings have been summarised in the form of a framework linking modelling purpose, problem characteristics and simulation approach.

Three-dimensional gas-liquid simulation of an airlift bubble column reactor

Basic hydrodynamic parameters of an airlift reactor with internal loop were estimated experimentally and simulated using commercially available CFD software from Fluent. Circulation velocity in a 32-dm(3)-airlift reactor was measured using the magnetic tracer method, meanwhile the gas hold-up was obtained by analysis of the pressure drop using the method of inverted U-tube manometers. Comparison of simulated (in two and three dimensions) and experimental data was performed at different superficial gas velocities in the riser.