Bayesian methods for estimation, optimization and experimental design

Mathematical models often contain parameters that need to be calibrated from measured data. The emergence of efficient Markov Chain Monte Carlo (MCMC) methods has made the Bayesian approach a standard tool in quantifying the uncertainty in the parameters. With MCMC, the parameter estimation problem can be solved in a fully statistical manner, and the whole distribution of the parameters can be explored, instead of obtaining point estimates and using, e.g., Gaussian approximations. In this thesis, MCMC methods are applied to parameter estimation problems in chemical reaction engineering, population ecology, and climate modeling. Motivated by the climate model experiments, the methods are developed further to make them more suitable for problems where the model is computationally intensive. After the parameters are estimated, one can start to use the model for various tasks. Two such tasks are studied in this thesis: optimal design of experiments, where the task is to design the next measurements so that the parameter uncertainty is minimized, and model-based optimization, where a model-based quantity, such as the product yield in a chemical reaction model, is optimized. In this thesis, novel ways to perform these tasks are developed, based on the output of MCMC parameter estimation. A separate topic is dynamical state estimation, where the task is to estimate the dynamically changing model state, instead of static parameters. For example, in numerical weather prediction, an estimate of the state of the atmosphere must constantly be updated based on the recently obtained measurements. In this thesis, a novel hybrid state estimation method is developed, which combines elements from deterministic and random sampling methods.

Software for design of experiments and response modelling of cake ltration applications

Filtration is a widely used unit operation in chemical engineering. The huge variation in the properties of materials to be ltered makes the study of ltration a challenging task. One of the objectives of this thesis was to show that conventional ltration theories are di cult to use when the system to be modelled contains all of the stages and features that are present in a complete solid/liquid separation process. Furthermore, most of the ltration theories require experimental work to be performed in order to obtain critical parameters required by the theoretical models. Creating a good overall understanding of how the variables a ect the nal product in ltration is somewhat impossible on a purely theoretical basis. The complexity of solid/liquid separation processes require experimental work and when tests are needed, it is advisable to use experimental design techniques so that the goals can be achieved. The statistical design of experiments provides the necessary tools for recognising the e ects of variables. It also helps to perform experimental work more economically. Design of experiments is a prerequisite for creating empirical models that can describe how the measured response is related to the changes in the values of the variable. A software package was developed that provides a ltration practitioner with experimental designs and calculates the parameters for linear regression models, along with the graphical representation of the responses. The developed software consists of two software modules. These modules are LTDoE and LTRead. The LTDoE module is used to create experimental designs for di erent lter types. The lter types considered in the software are automatic vertical pressure lter, double-sided vertical pressure lter, horizontal membrane lter press, vacuum belt lter and ceramic capillary action disc lter. It is also possible to create experimental designs for those cases where the variables are totally user de ned, say for a customized ltration cycle or di erent piece of equipment. The LTRead-module is used to read the experimental data gathered from the experiments, to analyse the data and to create models for each of the measured responses. Introducing the structure of the software more in detail and showing some of the practical applications is the main part of this thesis. This approach to the study of cake ltration processes, as presented in this thesis, has been shown to have good practical value when making ltration tests.

Software for design of experiments and response modelling of cake filtration applications

Filtration is a widely used unit operation in chemical engineering. The huge variation in the properties of materials to be ltered makes the study of ltration a challenging task. One of the objectives of this thesis was to show that conventional ltration theories are di cult to use when the system to be modelled contains all of the stages and features that are present in a complete solid/liquid separation process. Furthermore, most of the ltration theories require experimental work to be performed in order to obtain critical parameters required by the theoretical models. Creating a good overall understanding of how the variables a ect the nal product in ltration is somewhat impossible on a purely theoretical basis. The complexity of solid/liquid separation processes require experimental work and when tests are needed, it is advisable to use experimental design techniques so that the goals can be achieved. The statistical design of experiments provides the necessary tools for recognising the e ects of variables. It also helps to perform experimental work more economically. Design of experiments is a prerequisite for creating empirical models that can describe how the measured response is related to the changes in the values of the variable. A software package was developed that provides a ltration practitioner with experimental designs and calculates the parameters for linear regression models, along with the graphical representation of the responses. The developed software consists of two software modules. These modules are LTDoE and LTRead. The LTDoE module is used to create experimental designs for di erent lter types. The lter types considered in the software are automatic vertical pressure lter, double-sided vertical pressure lter, horizontal membrane lter press, vacuum belt lter and ceramic capillary action disc lter. It is also possible to create experimental designs for those cases where the variables are totally user de ned, say for a customized ltration cycle or di erent piece of equipment. The LTRead-module is used to read the experimental data gathered from the experiments, to analyse the data and to create models for each of the measured responses. Introducing the structure of the software more in detail and showing some of the practical applications is the main part of this thesis. This approach to the study of cake ltration processes, as presented in this thesis, has been shown to have good practical value when making ltration tests.

Simulation and optimization tools in paper machine concept design

The last decade has shown that the global paper industry needs new processes and products in order to reassert its position in the industry. As the paper markets in Western Europe and North America have stabilized, the competition has tightened. Along with the development of more cost-effective processes and products, new process design methods are also required to break the old molds and create new ideas. This thesis discusses the development of a process design methodology based on simulation and optimization methods. A bi-level optimization problem and a solution procedure for it are formulated and illustrated. Computational models and simulation are used to illustrate the phenomena inside a real process and mathematical optimization is exploited to find out the best process structures and control principles for the process. Dynamic process models are used inside the bi-level optimization problem, which is assumed to be dynamic and multiobjective due to the nature of papermaking processes. The numerical experiments show that the bi-level optimization approach is useful for different kinds of problems related to process design and optimization. Here, the design methodology is applied to a constrained process area of a papermaking line. However, the same methodology is applicable to all types of industrial processes, e.g., the design of biorefiners, because the methodology is totally generalized and can be easily modified.

Improving the competitiveness of electrolytic Zinc process by chemical reaction engineering approach

This doctoral thesis describes the development work performed on the leachand purification sections in the electrolytic zinc plant in Kokkola to increase the efficiency in these two stages, and thus the competitiveness of the plant. Since metallic zinc is a typical bulk product, the improvement of the competitiveness of a plant was mostly an issue of decreasing unit costs. The problems in the leaching were low recovery of valuable metals from raw materials, and that the available technology offered complicated and expensive processes to overcome this problem. In the purification, the main problem was consumption of zinc powder - up to four to six times the stoichiometric demand. This reduced the capacity of the plant as this zinc is re-circulated through the electrolysis, which is the absolute bottleneck in a zinc plant. Low selectivity gave low-grade and low-value precipitates for further processing to metallic copper, cadmium, cobalt and nickel. Knowledge of the underlying chemistry was poor and process interruptions causing losses of zinc production were frequent. Studies on leaching comprised the kinetics of ferrite leaching and jarosite precipitation, as well as the stability of jarosite in acidic plant solutions. A breakthrough came with the finding that jarosite could precipitate under conditions where ferrite would leach satisfactorily. Based on this discovery, a one-step process for the treatment of ferrite was developed. In the plant, the new process almost doubled the recovery of zinc from ferrite in the same equipment as the two-step jarosite process was operated in at that time. In a later expansion of the plant, investment savings were substantial compared to other technologies available. In the solution purification, the key finding was that Co, Ni, and Cu formed specific arsenides in the “hot arsenic zinc dust” step. This was utilized for the development of a three-step purification stage based on fluidized bed technology in all three steps, i.e. removal of Cu, Co and Cd. Both precipitation rates and selectivity increased, which strongly decreased the zinc powder consumption through a substantially suppressed hydrogen gas evolution. Better selectivity improved the value of the precipitates: cadmium, which caused environmental problems in the copper smelter, was reduced from 1-3% reported normally down to 0.05 %, and a cobalt cake with 15 % Co was easily produced in laboratory experiments in the cobalt removal. The zinc powder consumption in the plant for a solution containing Cu, Co, Ni and Cd (1000, 25, 30 and 350 mg/l, respectively), was around 1.8 g/l; i.e. only 1.4 times the stoichiometric demand – or, about 60% saving in powder consumption. Two processes for direct leaching of the concentrate under atmospheric conditions were developed, one of which was implemented in the Kokkola zinc plant. Compared to the existing pressure leach technology, savings were obtained mostly in investment. The scientific basis for the most important processes and process improvements is given in the doctoral thesis. This includes mathematical modeling and thermodynamic evaluation of experimental results and hypotheses developed. Five of the processes developed in this research and development program were implemented in the plant and are still operated. Even though these processes were developed with the focus on the plant in Kokkola, they can also be implemented at low cost in most of the zinc plants globally, and have thus a great significance in the development of the electrolytic zinc process in general.

Linguistic models for decision support

Linguistic modelling is a rather new branch of mathematics that is still undergoing rapid development. It is closely related to fuzzy set theory and fuzzy logic, but knowledge and experience from other fields of mathematics, as well as other fields of science including linguistics and behavioral sciences, is also necessary to build appropriate mathematical models. This topic has received considerable attention as it provides tools for mathematical representation of the most common means of human communication - natural language. Adding a natural language level to mathematical models can provide an interface between the mathematical representation of the modelled system and the user of the model - one that is sufficiently easy to use and understand, but yet conveys all the information necessary to avoid misinterpretations. It is, however, not a trivial task and the link between the linguistic and computational level of such models has to be established and maintained properly during the whole modelling process. In this thesis, we focus on the relationship between the linguistic and the mathematical level of decision support models. We discuss several important issues concerning the mathematical representation of meaning of linguistic expressions, their transformation into the language of mathematics and the retranslation of mathematical outputs back into natural language. In the first part of the thesis, our view of the linguistic modelling for decision support is presented and the main guidelines for building linguistic models for real-life decision support that are the basis of our modeling methodology are outlined. From the theoretical point of view, the issues of representation of meaning of linguistic terms, computations with these representations and the retranslation process back into the linguistic level (linguistic approximation) are studied in this part of the thesis. We focus on the reasonability of operations with the meanings of linguistic terms, the correspondence of the linguistic and mathematical level of the models and on proper presentation of appropriate outputs. We also discuss several issues concerning the ethical aspects of decision support - particularly the loss of meaning due to the transformation of mathematical outputs into natural language and the issue or responsibility for the final decisions. In the second part several case studies of real-life problems are presented. These provide background and necessary context and motivation for the mathematical results and models presented in this part. A linguistic decision support model for disaster management is presented here – formulated as a fuzzy linear programming problem and a heuristic solution to it is proposed. Uncertainty of outputs, expert knowledge concerning disaster response practice and the necessity of obtaining outputs that are easy to interpret (and available in very short time) are reflected in the design of the model. Saaty’s analytic hierarchy process (AHP) is considered in two case studies - first in the context of the evaluation of works of art, where a weak consistency condition is introduced and an adaptation of AHP for large matrices of preference intensities is presented. The second AHP case-study deals with the fuzzified version of AHP and its use for evaluation purposes – particularly the integration of peer-review into the evaluation of R&D outputs is considered. In the context of HR management, we present a fuzzy rule based evaluation model (academic faculty evaluation is considered) constructed to provide outputs that do not require linguistic approximation and are easily transformed into graphical information. This is achieved by designing a specific form of fuzzy inference. Finally the last case study is from the area of humanities - psychological diagnostics is considered and a linguistic fuzzy model for the interpretation of outputs of multidimensional questionnaires is suggested. The issue of the quality of data in mathematical classification models is also studied here. A modification of the receiver operating characteristics (ROC) method is presented to reflect variable quality of data instances in the validation set during classifier performance assessment. Twelve publications on which the author participated are appended as a third part of this thesis. These summarize the mathematical results and provide a closer insight into the issues of the practicalapplications that are considered in the second part of the thesis.