Mixture density network training by computation in parameter space

Training Mixture Density Network (MDN) configurations within the NETLAB framework takes time due to the nature of the computation of the error function and the gradient of the error function. By optimising the computation of these functions, so that gradient information is computed in parameter space, training time is decreased by at least a factor of sixty for the example given. Decreased training time increases the spectrum of problems to which MDNs can be practically applied making the MDN framework an attractive method to the applied problem solver.

Time complexity and convergence analysis of domain theoretic Picard method

We present an implementation of the domain-theoretic Picard method for solving initial value problems (IVPs) introduced by Edalat and Pattinson [1]. Compared to Edalat and Pattinson's implementation, our algorithm uses a more efficient arithmetic based on an arbitrary precision floating-point library. Despite the additional overestimations due to floating-point rounding, we obtain a similar bound on the convergence rate of the produced approximations. Moreover, our convergence analysis is detailed enough to allow a static optimisation in the growth of the precision used in successive Picard iterations. Such optimisation greatly improves the efficiency of the solving process. Although a similar optimisation could be performed dynamically without our analysis, a static one gives us a significant advantage: we are able to predict the time it will take the solver to obtain an approximation of a certain (arbitrarily high) quality.

Numerically optimized implementations, mitigation and simulations of polarisation mode dispersion in real-time transmission systems

The aim of this thesis is to present numerical investigations of the polarisation mode dispersion (PMD) effect. Outstanding issues on the side of the numerical implementations of PMD are resolved and the proposed methods are further optimized for computational efficiency and physical accuracy. Methods for the mitigation of the PMD effect are taken into account and simulations of transmission system with added PMD are presented. The basic outline of the work focusing on PMD can be divided as follows. At first the widely-used coarse-step method for simulating the PMD phenomenon as well as a method derived from the Manakov-PMD equation are implemented and investigated separately through the distribution of a state of polarisation on the Poincaré sphere, and the evolution of the dispersion of a signal. Next these two methods are statistically examined and compared to well-known analytical models of the probability distribution function (PDF) and the autocorrelation function (ACF) of the PMD phenomenon. Important optimisations are achieved, for each of the aforementioned implementations in the computational level. In addition the ACF of the coarse-step method is considered separately, based on the result which indicates that the numerically produced ACF, exaggerates the value of the correlation between different frequencies. Moreover the mitigation of the PMD phenomenon is considered, in the form of numerically implementing Low-PMD spun fibres. Finally, all the above are combined in simulations that demonstrate the impact of the PMD on the quality factor (Q=factor) of different transmission systems. For this a numerical solver based on the coupled nonlinear Schrödinger equation is created which is otherwise tested against the most important transmission impairments in the early chapters of this thesis.

Mathematical modelling and control of agitated extraction columns

Liquid-liquid extraction has long been known as a unit operation that plays an important role in industry. This process is well known for its complexity and sensitivity to operation conditions. This thesis presents an attempt to explore the dynamics and control of this process using a systematic approach and state of the art control system design techniques. The process was studied first experimentally under carefully selected. operation conditions, which resembles the ranges employed practically under stable and efficient conditions. Data were collected at steady state conditions using adequate sampling techniques for the dispersed and continuous phases as well as during the transients of the column with the aid of a computer-based online data logging system and online concentration analysis. A stagewise single stage backflow model was improved to mimic the dynamic operation of the column. The developed model accounts for the variation in hydrodynamics, mass transfer, and physical properties throughout the length of the column. End effects were treated by addition of stages at the column entrances. Two parameters were incorporated in the model namely; mass transfer weight factor to correct for the assumption of no mass transfer in the. settling zones at each stage and the backmixing coefficients to handle the axial dispersion phenomena encountered in the course of column operation. The parameters were estimated by minimizing the differences between the experimental and the model predicted concentration profiles at steady state conditions using non-linear optimisation technique. The estimated values were then correlated as functions of operating parameters and were incorporated in·the model equations. The model equations comprise a stiff differential~algebraic system. This system was solved using the GEAR ODE solver. The calculated concentration profiles were compared to those experimentally measured. A very good agreement of the two profiles was achieved within a percent relative error of ±2.S%. The developed rigorous dynamic model of the extraction column was used to derive linear time-invariant reduced-order models that relate the input variables (agitator speed, solvent feed flowrate and concentration, feed concentration and flowrate) to the output variables (raffinate concentration and extract concentration) using the asymptotic method of system identification. The reduced-order models were shown to be accurate in capturing the dynamic behaviour of the process with a maximum modelling prediction error of I %. The simplicity and accuracy of the derived reduced-order models allow for control system design and analysis of such complicated processes. The extraction column is a typical multivariable process with agitator speed and solvent feed flowrate considered as manipulative variables; raffinate concentration and extract concentration as controlled variables and the feeds concentration and feed flowrate as disturbance variables. The control system design of the extraction process was tackled as multi-loop decentralised SISO (Single Input Single Output) as well as centralised MIMO (Multi-Input Multi-Output) system using both conventional and model-based control techniques such as IMC (Internal Model Control) and MPC (Model Predictive Control). Control performance of each control scheme was. studied in terms of stability, speed of response, sensitivity to modelling errors (robustness), setpoint tracking capabilities and load rejection. For decentralised control, multiple loops were assigned to pair.each manipulated variable with each controlled variable according to the interaction analysis and other pairing criteria such as relative gain array (RGA), singular value analysis (SVD). Loops namely Rotor speed-Raffinate concentration and Solvent flowrate Extract concentration showed weak interaction. Multivariable MPC has shown more effective performance compared to other conventional techniques since it accounts for loops interaction, time delays, and input-output variables constraints.

Dynamic simulation of chemical processes

This thesis describes the design and implementation of an interactive dynamic simulator called DASPRII. The starting point of this research has been an existing dynamic simulation package, DASP. DASPII is written in standard FORTRAN 77 and is implemented on universally available IBM-PC or compatible machines. It provides a means for the analysis and design of chemical processes. Industrial interest in dynamic simulation has increased due to the recent increase in concern over plant operability, resiliency and safety. DASPII is an equation oriented simulation package which allows solution of dynamic and steady state equations. The steady state can be used to initialise the dynamic simulation. A robust non linear algebraic equation solver has been implemented for steady state solution. This has increased the general robustness of DASPII, compared to DASP. A graphical front end is used to generate the process flowsheet topology from a user constructed diagram of the process. A conversational interface is used to interrogate the user with the aid of a database, to complete the topological information. An original modelling strategy implemented in DASPII provides a simple mechanism for parameter switching which creates a more flexible simulation environment. The problem description generated is by a further conversational procedure using a data-base. The model format used allows the same model equations to be used for dynamic and steady state solution. All the useful features of DASPI are retained in DASPII. The program has been demonstrated and verified using a number of example problems, Significant improvements using the new NLAE solver have been shown. Topics requiring further research are described. The benefits of variable switching in models has been demonstrated with a literature problem.

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.

Dynamic simulation of chemical plant

This thesis describes the design and implementation of a new dynamic simulator called DASP. It is a computer program package written in standard Fortran 77 for the dynamic analysis and simulation of chemical plants. Its main uses include the investigation of a plant's response to disturbances, the determination of the optimal ranges and sensitivities of controller settings and the simulation of the startup and shutdown of chemical plants. The design and structure of the program and a number of features incorporated into it combine to make DASP an effective tool for dynamic simulation. It is an equation-oriented dynamic simulator but the model equations describing the user's problem are generated from in-built model equation library. A combination of the structuring of the model subroutines, the concept of a unit module, and the use of the connection matrix of the problem given by the user have been exploited to achieve this objective. The Executive program has a structure similar to that of a CSSL-type simulator. DASP solves a system of differential equations coupled to nonlinear algebraic equations using an advanced mixed equation solver. The strategy used in formulating the model equations makes it possible to obtain the steady state solution of the problem using the same model equations. DASP can handle state and time events in an efficient way and this includes the modification of the flowsheet. DASP is highly portable and this has been demonstrated by running it on a number of computers with only trivial modifications. The program runs on a microcomputer with 640 kByte of memory. It is a semi-interactive program, with the bulk of all input data given in pre-prepared data files with communication with the user is via an interactive terminal. Using the features in-built in the package, the user can view or modify the values of any input data, variables and parameters in the model, and modify the structure of the flowsheet of the problem during a simulation session. The program has been demonstrated and verified using a number of example problems.

Problem solving in architectural design

This thesis is presented in two parts. The first part is an attempt to set out a framework of factors influencing the problem solving stage of the architectural design process. The discussion covers the nature of architectural problems and some of the main ways in which they differ from other types of design problems. The structure of constraints that both the problem and the architect impose upon solutions are seen as of great importance in defining the type of design problem solving situation. The problem solver, or architect, is then studied. The literature of the psychology of thinking is surveyed for relevant work . All of the traditional schools of psychology are found wanting in terms of providing a comprehensive theory of thinking. Various types of thinking are examined, particularly structural and productive thought, for their relevance to design problem solving. Finally some reported common traits of architects are briefly reviewed. The second section is a report of u~o main experiments which model some aspects of architectural design problem solving. The first experiment examines the way in which architects come to understand the structure of their problems. The performances of first and final year architectural students are compared with those of postgraduate science students and sixth form pupils. On the whole these groups show significantly different results and also different cognitive strategies. The second experiment poses design problems which involve both subjective and objective criteria, and examines the way in which final year architectural students are able to relate the different types of constraint produced. In the final section the significance of all the results is suggested. Some educational and methodological implications are discussed and some further experiments and investigations are proposed.

Generalized methods and solvers for noise removal from piecewise constant signals. I. Background theory

Removing noise from piecewise constant (PWC) signals is a challenging signal processing problem arising in many practical contexts. For example, in exploration geosciences, noisy drill hole records need to be separated into stratigraphic zones, and in biophysics, jumps between molecular dwell states have to be extracted from noisy fluorescence microscopy signals. Many PWC denoising methods exist, including total variation regularization, mean shift clustering, stepwise jump placement, running medians, convex clustering shrinkage and bilateral filtering; conventional linear signal processing methods are fundamentally unsuited. This paper (part I, the first of two) shows that most of these methods are associated with a special case of a generalized functional, minimized to achieve PWC denoising. The minimizer can be obtained by diverse solver algorithms, including stepwise jump placement, convex programming, finite differences, iterated running medians, least angle regression, regularization path following and coordinate descent. In the second paper, part II, we introduce novel PWC denoising methods, and comparisons between these methods performed on synthetic and real signals, showing that the new understanding of the problem gained in part I leads to new methods that have a useful role to play.

Generalized methods and solvers for noise removal from piecewise constant signals. II. New methods

Removing noise from signals which are piecewise constant (PWC) is a challenging signal processing problem that arises in many practical scientific and engineering contexts. In the first paper (part I) of this series of two, we presented background theory building on results from the image processing community to show that the majority of these algorithms, and more proposed in the wider literature, are each associated with a special case of a generalized functional, that, when minimized, solves the PWC denoising problem. It shows how the minimizer can be obtained by a range of computational solver algorithms. In this second paper (part II), using this understanding developed in part I, we introduce several novel PWC denoising methods, which, for example, combine the global behaviour of mean shift clustering with the local smoothing of total variation diffusion, and show example solver algorithms for these new methods. Comparisons between these methods are performed on synthetic and real signals, revealing that our new methods have a useful role to play. Finally, overlaps between the generalized methods of these two papers and others such as wavelet shrinkage, hidden Markov models, and piecewise smooth filtering are touched on.

Highly nonlinear chalcogenide suspended-core fibers for applications in the mid-infrared

Due to their unique dispersion and nonlinear properties, chalcogenide suspended-core fibers, characterized by a few micrometer-sized core suspended between large air-holes by few small glaß struts, are excellent candidates for mid-infrared applications. In the present study the influence of the main croß-section characteristics of the chalcogenide suspended-core fibers on the dispersion curve and on the position of the zero-dispersion wavelength has been thoroughly analyzed with a full-vector modal solver based on the finite element. In particular, the design of suspended-core fibers made of both As2S3 and As2Se3 has been optimized to obtain dispersion properties suitable for the supercontinuum generation in the mid-infrared.