Evolutionary Optimization And Its Applications In Inverse Problems Of Differential Equations

The dissertation consists of two parts: in Part One the author designs as well as analyzes evolutionary algorithms and applies the designed algorithms to optimization problems. The application, to a great extent, reflects effectiveness of the algorithms. In Part Two the author talks about the application of evolutionary computation to solving mathematical and physical inverse problems -- parameters identification of differential equations. The main research work is listed as follows:(1) The dissertation reviews history and development in terms of national as well as international evolutionary computation, the prospect of its application, basic characteristics and fundamental procedure that one has to follow when designing evolutionary algorithms. As a general term for population guidance random search techniques, evolutionary computation is based, to a certain degree, on the evolution of biological life in the natural world. Darwin's"the fittest survives"thought is adopted in the process of population evolution. Thanks to its strong points such as stability and generalization, and its intelligence characteristics such as self-organization, self-adaptation and self-learning, evolutionary computation has been extensively used in a variety of fields, for example, machine learning, process control, economy prediction, and engineering optimization. In addition, population search of evolutionary computation enables it to be suitable for large-scale parallelism, which motivates the development of parallel computer and parallel algorithms.(2) A multi-parent elite evolutionary operator as well as its respective elite evolutionary algorithm is designed. The dissertation overviews the numerical optimization problem, and then classifies as well as analyzes the ways in solving optimization problems. It also points out the disadvantages that exist in traditional numerical approaches. In this chapter, the author also illustrates the advantages of evolutionary algorithms in solving optimization problems, and proposes some urgent problems which need working out when designing such types of algorithms (such as constraint processing technology and coding strategy). Furthermore, the author constructs elite evolutionary algorithm to solving numerical optimization problems. As we know, optimization theory and methods are termed as a research area with a long history and vigilance. Since 1930s and 1940s, it has been booming on account of urgent needs from military, astronomic and economic areas. With its extensive application, a vast majority of problems tend to resort to optimization problems such as the problems raised in the areas of system engineering, control engineering, statistics and inverse problems. For the above mentioned problems, over a long period of time, many kinds of numerical methods based on iterative principle such as Newton Method, Simplicial Method, Conjugate Gradient Method, Homotopy Algorithm, the Fastest Descent Method and Penalty Function Algorithm have been widely applied. However, there exist some weak points in these traditional approaches, for example, intensified restriction on target function, tremendous dependence on the choice of initial value from algorithm results, likelihood of local minimum of algorithm, lack of simplicity as well as generalization, difficult handle of constrained problems and so on. As the modern optimized approach, evolutionary algorithm turns out to have achieved satisfactory results when it is used in global optimization problems under such circumstances such as large-scale, multi-modal and multi-state functions and discrete variables. Since it is far superior to common approaches in terms of solving speed and quality, evolutionary algorithm can be regarded as super approximate algorithm. In recent years, great success achieved by evolutionary computation in the area of optimization has been generally acknowledged by many researchers. In this chapter, the author further develops its successful experience obtained from the previous research and designs a multi-parent elite evolutionary operator as well as its respective elite evolutionary algorithm. When executing selecting operation, the author improves convergence of the algorithm by choosing elite individuals to join crossover operation. It turns out that the algorithm enjoys strong robustness and adaptation.(3) The advantages of the algorithms proposed by the dissertation are illustrated by means of solving some typical problems. In the case of single-modal function optimization problems, great improvement has been seen in the respect of convergence brought about by elite strategies. The author also analyzes and experiments the influence on the nature of algorithm made by parameter setup in the elite evolutionary algorithm. So far as multi-modal function optimization problems are concerned, the author constitutes two-stage global-local mixed evolutionary (GLME) algorithm. The evolution is divided into two stages. On the first stage—global evolution stage, evolution is conducted in the global search domain. The non-elite evolutionary algorithm is adopted in order that the diversity of population can be maintained. The global evolution is followed by the selection of some different individuals in global domain to form some niches. On the second stage-local evolution stage, evolution is conducted in these niches and the elite evolutionary algorithm is adopted. Finally, the multi-global optimal solutions can be obtained by means of collecting individuals in these niches and removing the same ones. The experiment results indicate that all the global optimal solutions can be discovered in one run or a few runs. Based on the GLME algorithm and enlightened by domain decomposition, a global-local mixed evolutionary algorithm based on domain decomposition (GLME_DD) is constructed. The major principle of this algorithm consists in the space division. The global search domain in divided into some smaller domains, and then the GLME algorithm is conducted in each smaller domain. This algorithm not only benefits parallelism, but also contributes a lot to constraining the blindness in searching. Even if it is executed on a serial computer, the algorithm is very excellent in terms of the convergence rate, the solution precision and the search for multi solutions. For solving system of non-linear equations, the problem is converted to a relevant optimization problem and then this optimization problem is solved by the elite evolutionary algorithm. Satisfactory results have been achieved and the fairly precise solutions can be obtained within a short period of time. For the non-linear equation set which holds multi solutions, we also can convert the original problem to a multi-modal function optimization problem and then solve it by GLME algorithm. The experiment result shows that the all solutions can be achieved in one run.(4) An evolutionary algorithm to solve inverse problems of differential equations by involving the elite evolutionary strategy is proposed, which makes the key part in our research area--- application. Inverse problems in nature science, to a certain degree, can be also termed as Data Mining or KDD. With its wide application in industry as well as engineering, for example, earthquake exploration, studies on underground water quality and determination of crack inside material, inverse problems have been a hot research issue both at home and abroad. On the other hand, it is also a highly crossover discipline because of practical need from a variety of research fields and subjects. Due to the ill-posed nature of inverse problems, in particular, the high instability of solution's dependence on experiments and observation data, namely, slight perturbation produced from observation and experiments data, which is likely to result in huge difference in terms of solutions to inverse problems may cause extreme instability of the numerical solving processes of inverse problems. That is also the reason why inverse problems remain a challenging research issue in the field of computer science, engineering and industrial application. For theory and numerical algorithms, much work has been done by researchers at home and abroad and a lots of traditional mathematic or physical methods have been worked out. But these traditional methods have certain drawbacks and limitations. To contain solution's reliability and stability the regularization is often adopted. The effectiveness of a regularization method depends strongly on the choice of a good regularization parameter. There exists a significant amount of research on the development of appropriate strategies for selecting regularization parameters but it appears that very few of the strategies are utilized for practical application. As a result, it is urgent to do research on this issue so as to make breakthrough against the traditional methods and to search for a new means of solving inverse problems. Evolutionary computation, with its potential application value, is now viewed as an original approach to solving inverse problems. In recent years, some algorithms have been worked out about the application of evolutionary computation for solving inverse problems, but in these algorithms some strong constrains are put on the problems or the noise has not been considered. The dissertation also proposes an evolutionary algorithm for solving the inverse problems of differential equations by combining the evolutionary computation and regularization method for solving inverse problems. As for the parameter identification problems in differential equations, an evolutionary algorithm is designed by the means of constructing unique fitness function and evolutional operators. In this algorithm, the noise in observation data has been, for the first time, taken into consideration in comparison with similar types of research work. And besides, some numerical experiments are conducted on parameter identification problems in elliptic and parabolic differential equations. The experimental results indicate that the new algorithm can work out these problems effectively. The new algorithm is capable of resisting the noise. When no noise exists or the noise is very small, the identified parameter is almost same as the original one. Amazingly, even when the intensified level of the noise increases to 10%, satisfactory results can also be obtained, which had never happened in terms of traditional methods. The experiment results reported in this dissertation put a solid basis for further researching other kinds of inverse problems.(5) At last, the research work in the dissertation is summarized. The author suggests that more research work in this area should be done in the future and predicts bright prospect of future research.