【作者】 邹学文；

【导师】 闵涛；

【作者基本信息】 西安理工大学 ， 应用数学， 2006， 硕士

【摘要】 从具体的实例入手，描述性地给出了数学物理反问题的一般概念。指出数学物理反问题一般是非线性及不适定问题。为了有效地求解反问题特别是克服不适定这个难点，讨论了吉洪诺夫正则化方法，其基本思想是：用一族与原问题相邻近的适定问题去逼近原问题的解。这一方法在Hilbert空间中更便于理论分析，得到了在Hilbert空间中正则化的一些结果及误差估计，本文还给出了求解反问题的另一种方法：离散正则化方法，先用投影方法将无限维的反问题，近似在有限维空间上，得到一个病态系统，再利用正则化求解该系统。通过分析，可以得到正则化理论和方法的关键是如何构造“邻近问题”而得到正则算子和正则参数、如何决定与原始资料误差水平相匹配的正则参数以及上述工作的数值实现。 关于正则参数的选取始终是一个重要而有魅力的问题。针对这一问题，首次提出了利用遗传算法计算正则参数的基本思想。遗传算法仿照生物进化和遗传的规律，利用复制、交换、突变等操作，使优胜者繁殖，败劣者淘汰，一代一代的重复同样的操作，最终找到最优解或接近最优解。特别是在处理复杂数据和非线性计算上遗传算法具有很强的适应性。本文利用遗传算法结合确定正则参数的准则，提出了具体算法，编制了计算程序，数值模拟结果表明，所得到的正则参数具有很高的精度。特别对于大规模的不适定问题，该方法显示出一定的优越性。克服了传统的迭代法由于初值选取不当及规模很大时难于计算的不足。更多还原

【Abstract】 This paper gives the generic concept of the mathematics and physics inverse problem from material examples. It shows that the mathematics and physics inverse problems are always nonlinear and ill-posed. For solving inverse problems efficiently and conquering ill-posed problem, we have discussed Tikhonov regularization method, it basic idea is that: using a series well posed problem which is appropriate to the original problem to approach the solution of the original problem. Using this method in Hilbert space is easy to get a theoretical analysis, and we have obtained some results and error estimation of Tikhonov regularization method in Hilbert space. This paper also gives another method to solve inverse problem: dispersed regularization method, which using projection method to approximate infinite dimensional inverse problem in finite dimension and gain a ill-posed system firstly, and then at last using the regularization method to solve the system. From analysis we know that the key to the regularization method and theory is how to construct "appropriate problem to gain regular operator and regularization parameter, and how to decide parameter which is suited to the error level of source material, and how to get the numerical implementation of the above work.The problem of how to choice the regularization parameter is not only important but also attractive. To this problem we provide the idea of using genetic algorithms to calculate regularization parameter for the first time. Genetic algorithms follows the organic evolution rules, through operations selection、crossover mutation to finish the survival of the fittest, and repeats the same operation from generation to generation, obtains optimum solution or closes to optimum solution at last. Genetic algorithms has strong adaptability when deals with complex date and nonlinear calculation. This paper has provided specific algorithm and worked out calculation program, which makes use of genetic algorithms together with the rules how to fix on the regularization parameter. The numerical result turns out that the regularization parameter has high precision which gains by using genetic algorithms. This method displays some superiority especially to large-scale ill-posed problem, which conquers the shortage of traditional iteration when deal with large-scale problem or the problems have not had a proper initial- value choice.更多还原