Statistical Analysis Method For Electromagnetic Inverse Problems

Electromagnetic inverse problems are always the research focus in the field of computational electromagnetism. From the point of view of application, it can be classified into two major categories: one is parameter identification problem; the other is optimization design problem. For the former problem, the main task is to reconstruct the source parameters and their physical properties under the given experimental results and parameters. And the essential problem of electromagnetic optimization design is a parameters optimization process under the expecting performance index of electromagnetic system. The main purpose of this thesis is to develop a set of statistical inference methods (including classic statistical methods and modern Bayesian statistical methods) and design of experiment techniques to the electromagnetic inverse problems. And with the proposed methods, we can lay a solid foundation for the efficient and robust analysis of all kinds of engineering inverse problems.Firstly, for the parameter identification problem, we convert this problem into the framework of multivariate linear inverse problem. There are two main research contents; one is the classical statistical estimation method for model parameter, such as maximum likelihood estimator, least square estimator and its weighted form, linear unbiased estimator and minimum norm estimator. The other is the Bayesian statistical estimation method for model parameter, such as Bayesian estimator, maximum a posterior estimator, linear minimum mean square error estimator. All these estimators are derived from different starting points and model assumptions.In this section, we also give a detailed discussion about the prior information and noise information for the Bayesian method. Then we investigate the particle size distribution estimating problem of magnetic particles in feerfluids with Monte Carlo method. The experiment results demonstrate that all the proposed methods can be easily implemented and can induce satisfied results. Meanwhile, the results given by Bayesian method are better than that of classical methods, which are more sensitively to the noise. In summary, all these methods can be seen as effective direct extraction procedures of parameter information from the experimental data directly; and they can be widely employed in many electromagnetic inverse problems.Secondly, for the optimization design problem, we first give a discussion about the statistical approximate models and the techniques of design of experiment. The reconstruction theory and method will be fully discussed, which includes the following approximate models: parameter models (including response surface model and radial basis function model), semi-parameter model (Kriging model). Moreover, we will consider the model selection strategies for the practical problems. The background of these methods is the classical statistical theory and the design and analysis techniques of experiment.Then, as models and algorithms were almost discussed separately in traditional optimization methods and these methods may waste a lot of computation cost; we present a new efficient global optimization method, sequential optimization method (SOM), in this paper. SOM is a sequential sampling process; it only needs a small sample data, and the overall computational effort needed is much less than that by direct optimization method. To illustrate the performance of the proposed methods, two analytic test functions and IEEE TEAM Workshop Problems 22 and 25 are investigated. Experimental results of test function demonstrate that SOM can obtain satisfactory solutions; and the number of finite element sample points needed is less than 1/10 compared with that by direct optimization method.Finally, dimension reduction optimization method (DROM) based on SOM is presented for high dimensional optimization design problems of electromagnetic devices. Using DROM, a high dimensional problem can be converted into a low dimensional problem with expert experience or some design of experiment techniques. Then three engineering problems are investigated to illustrate the efficiency of the proposed methods. From the experimental results, we can see that the presented methods can obviously reduce the computational cost of finite element analysis, while the optimal results also satisfy design specification. The number of finite element sample points needed is less than 1/3 compared with that by direct optimization method. In summary, all these methods are very suit for the optimization design problems of electromagnetic devices, including discrete, continuous, mono-objective, multi-objective, low and high dimensional optimization design problems.