【作者】 陈宇；

【导师】 黄建国；

【作者基本信息】 上海交通大学 ， 计算数学， 2009， 博士

【摘要】 反问题在医学成像、无损探伤、气象预报、图像处理、计量经济、生命科学等领域都有着广泛的应用.因此对它的理论分析和数值求解有重大的研究意义和应用价值.本文的主要工作就是针对若干有实际应用背景的反问题模型,系统开展相关理论分析与数值求解研究工作.首先,讨论了一维情形利用带噪声均值数据进行函数重构的问题.提出了一种正则化方法.并证明了该方法解的存在唯一性,再由Banach空间中的优化定理给出一系列条件来刻画解的性质,从而发现解可由样条函数表示.通过插值算子的引进我们也给出了近似解的误差估计.最后用数值模拟通过图表说明该方法在实际计算时是可行的和有效的,同时提出了一些选取正则化参数的方法.进一步,将此正则化方法推广到了二维甚至高维情形.通过辅助空间的引入,同样证明了此方法解的存在唯一性,并且说明了此解可由格林函数表示.利用Poincar′e不等式给出了近似解的L2模误差估计.数值例子验证了该方法的有效性,其中正则化参数由多种不同的选取策略决定.接着,考虑了腐蚀探测问题.给出了薄管区域基于小参数展开方法的一个误差估计.假设管壁的厚度为a ,对于提出的两种方法其误差度量分别为O(a)和O(a2).同时,我们用优化控制的思想构造了一个非薄板区域腐蚀探测问题的数值求解方法.数值实验证明了该方法的可行及有效性.最后,本文讨论了二阶线性椭圆型方程的柯西问题.给出了一种数值方法来求解未知边界上的信息,该方法通过最优控制的思想将原来的柯西问题改写为等价的优化问题,并通过添加正则化项来克服离散后问题的病态性.证明了该数值方法的收敛性,给出了当正则化参数趋于零时数值解的收敛理论.同时提供了一系列的数值实验支持我们的理论结果.并且通过有限元形函数表示技巧将算法进一步重写,从而可以利用Hansen工具包来选取合适的正则化参数.该方法也被推广用于平面弹性力学柯西问题的数值求解.更多还原

【Abstract】 Inverse problems are often encountered in medical imaging, non-destructive detecting, weather forecast, image processing, parameteridentification in econometrics, life science and many other fields.Therefore, it is of great importance to investigate their theoreticalproperties and numerical solutions. This thesis will focus on thetheoretical study and algorithm design for some practical inverseproblems.First, we propose a regularization method for the function recon-struction from approximate average ?uxes in one dimension. Uniquesolvability of the method is proved and a number of conditions aregiven to characterize the solution by the optimization theory in Ba-nach spaces. The solution can be expressed in terms of some splinefunctions. Error estimates are established after the introduction ofsome interpolation operators. A series of numerical examples are pro-vided to illustrate the e?ectiveness and computational performance ofthe method. Some ideas for the choice of the regularization parameterare also suggested based on the computational experience.Then, we extend the previous method for the function recon- struction from noisy local averages to any dimension. After theintroduction of an auxiliary domain, the reconstruction function canbe described in an explicit form by some Green’s functions. Errorbounds for the approximate solution in L2-norm are derived by usinga novel Poicar′e inequality. Several numerical examples are providedto show computational performance of the method, with the regular-ization parameters selected by di?erent strategies.Next, we study an inverse problem in corrosion detection. Erroranalysis is developed for a parameter expansion method used todetermine the corrosion coe?cient in a pipe. It is shown that themagnitude of the errors is O(a) and O(a2) for the two proposedmethods respectively, where a stands for the thickness of the pipe.Also, a numerical method based on the optimal control approachis proposed for such problems in non-sheet case. Some numericalexamples are given to show the e?ciency of the method.Finally, we study a numerical method for solving the Cauchyproblem corresponding to an elliptic partial di?erential equation. Thenumerical method is based on a reformulation of the Cauchy problemthrough an optimal control approach coupled with a regularizationterm which is included to treat the severe ill-conditioning of thecorresponding discretized formulation. We prove convergence of thenumerical method and present theoretical results for the limiting behaviors of the numerical solution as the regularization parameterapproaches zero. Results from some numerical examples are reported.We also extend the method to solve a Cauchy problem for the planeelasticity.更多还原