电磁逆问题的统计分析方法
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摘要
电磁场逆问题一直以来都是计算电磁学领域的研究热点。从应用的角度它可以分为两大类：参数辨识问题和优化设计问题。其中前者的本质是在给定的试验结果和试验参数的前提下，反演或重建出现这一结果的源参数及其电磁和物理特性；而后者的本质是给定某电磁系统期望的性能指标，然后通过参数的寻优来实现这一目标。本文的主要研究目的是为电磁场逆问题提供一套比较系统的统计推断方法和试验设计技术，包括经典的统计分析方法和现代的贝叶斯(Bayes)统计推断方法；从而为快速有效的分析电磁逆优化工程问题打下坚实的基础。
首先，针对参数辨识问题，我们将常见的电磁逆问题纳入到多元线性逆问题的模型框架中来。核心研究内容有两个：其一是模型参数的经典统计学估计方法，主要包括极大似然估计量、最小二乘估计量及其加权形式的估计量、线性无偏估计量和最小范数估计量。其二是模型参数的Bayes估计方法，主要包括一般Bayes估计量、极大后验估计量和最小线性均方误差估计量。所有的这些估计量都是从不同的出发点，针对不同的模型假设下得到的。在Bayes反演方法中，我们还将详细的说明参数的先验信息和噪声信息的估计方法。随后还将结合蒙特卡洛(Monte Carlo)方法对铁磁流体微粒尺寸分布的估计问题进行实例分析，分析结果显示上述的统计估计方法都是非常有效的。同时我们也发现Bayes方法对噪声项的敏感性显著的低于经典统计学方法，而且结果的精度也优于前者，是一种值得推广的解决参数辨识问题的统计方法。
其次，针对优化设计问题我们首先讨论了统计近似模型的拟合方法和试验技术的设计。其中近似模型部分主要从理论上研究了模型的重构理论与方法以及各个模型之间的关系：包括确定性参数的响应面模型、径向基函数模型和半参数的克热金(Kriging)模型。同时还将研究近似模型在我们提出的各种优化方法下的的最优选择策略问题。所有模型部分的理论基础均为经典统计学理论和试验设计与分析技术。
随后，针对传统的独立研究优化算法和模型的缺点，我们提出了一种新的优化方法：序贯优化方法。它本质上是一个序贯采样的优化过程，能同时执行模型的拟合优化与算法的参数优化过程。通过对两个IEEE标准工程测试问题(TEAMWorkshop Problems 22 and 25)和两个数学验证函数的考察及分析，我们发现序贯优化方法对这类问题非常有效，对模型的选取没有依赖性，能在同样满足求解精度的要求下，将有限元的计算量降到不足直接优化方法的1／10。
最后，针对高维电磁装置的优化设计问题，我们在序贯优化方法的基础上提出了降维优化方法。它通过两种降维技术(专家先验和试验设计技术)将一个高维的优化问题转化为一个只关于显著性参数的低维优化问题。从三个高维电磁装置的实际设计分析来看，这种新的方法同样非常有效，求解的精度也能达到设计的要求，同时能将有限元的计算量减少到约为直接优化方法的1／3左右。总之，序贯优化方法和降维优化方法对离散域、连续域，单目标、多目标，低维和高维的电磁装置优化设计问题都非常有效，可广泛的应用于工程电磁装置的优化设计问题之中。
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.
首先，针对参数辨识问题，我们将常见的电磁逆问题纳入到多元线性逆问题的模型框架中来。核心研究内容有两个：其一是模型参数的经典统计学估计方法，主要包括极大似然估计量、最小二乘估计量及其加权形式的估计量、线性无偏估计量和最小范数估计量。其二是模型参数的Bayes估计方法，主要包括一般Bayes估计量、极大后验估计量和最小线性均方误差估计量。所有的这些估计量都是从不同的出发点，针对不同的模型假设下得到的。在Bayes反演方法中，我们还将详细的说明参数的先验信息和噪声信息的估计方法。随后还将结合蒙特卡洛(Monte Carlo)方法对铁磁流体微粒尺寸分布的估计问题进行实例分析，分析结果显示上述的统计估计方法都是非常有效的。同时我们也发现Bayes方法对噪声项的敏感性显著的低于经典统计学方法，而且结果的精度也优于前者，是一种值得推广的解决参数辨识问题的统计方法。
其次，针对优化设计问题我们首先讨论了统计近似模型的拟合方法和试验技术的设计。其中近似模型部分主要从理论上研究了模型的重构理论与方法以及各个模型之间的关系：包括确定性参数的响应面模型、径向基函数模型和半参数的克热金(Kriging)模型。同时还将研究近似模型在我们提出的各种优化方法下的的最优选择策略问题。所有模型部分的理论基础均为经典统计学理论和试验设计与分析技术。
随后，针对传统的独立研究优化算法和模型的缺点，我们提出了一种新的优化方法：序贯优化方法。它本质上是一个序贯采样的优化过程，能同时执行模型的拟合优化与算法的参数优化过程。通过对两个IEEE标准工程测试问题(TEAMWorkshop Problems 22 and 25)和两个数学验证函数的考察及分析，我们发现序贯优化方法对这类问题非常有效，对模型的选取没有依赖性，能在同样满足求解精度的要求下，将有限元的计算量降到不足直接优化方法的1／10。
最后，针对高维电磁装置的优化设计问题，我们在序贯优化方法的基础上提出了降维优化方法。它通过两种降维技术(专家先验和试验设计技术)将一个高维的优化问题转化为一个只关于显著性参数的低维优化问题。从三个高维电磁装置的实际设计分析来看，这种新的方法同样非常有效，求解的精度也能达到设计的要求，同时能将有限元的计算量减少到约为直接优化方法的1／3左右。总之，序贯优化方法和降维优化方法对离散域、连续域，单目标、多目标，低维和高维的电磁装置优化设计问题都非常有效，可广泛的应用于工程电磁装置的优化设计问题之中。
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.
引文
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