数学与统计学院知识库

本文从最优性分析的角度考虑了三类经典的逆边值问题，即逆热传导问题、反向热传导问题、Laplace方程Cauchy问题。他们都是严重不适定问题，且未知触越接近边界点，不适定性越强。因此恢复解的稳定性，尤其是解在边界上的稳定性不仅有明显的物理背景，且有重要的理论研究价值。到目前为止，很多方法仅能得到数值结果而未给出误差分析，还有些方法可以给出阶数最优的稳定性估计，而最优性分析方面的工作很少，在边界上的结果就更少了。本文第二章对无界带形域上的Laplace方程Cauchy问题在区间(0,1]上进行最优误差分析，得到了该问题的真实解与正则近似解在一般源条件下的最优误差界。同时还给出两种正则化方法（即广义Tikhonov正则化方法和一类广义奇异值分解法）以及正则化参数的最优选取规则，当正则化参数依这种规则选取时可实现最优误差界。此外第二章还就这个问题给出了滤波方法和Fourier方法。当正则化参数适当选取时，分别得到了问题的真实解与正则近似解之间的阶数最优的误差估计和对数型稳定估计。 第三章分别对Laplace方程Cauchy问题和逆热传导问题在边界上的稳定性进行最优性分析。均得到了 问题的真实解与正则近似解在对数源条件下的最优误差界。 本文第四章应用离散正则化方法对有界域上的一个Laplace方程Cauchy 问题和一个反向热传导问题进行研究， 给出了精确解与用谱截断方法、最小二乘法以及对偶最小二乘法所得到的正则近似解之间的对数型稳定估计。

In this paper,from the viewpoint of optimality analysis,we consider three classical inverse boundary value problems: the inverse heat conduction problem,backward heat conduction problem and Cauchy problem for Laplace equation.They are all severely ill-posed problems, and The ill-posedness becomes sharp as the unknown solution is closer to the boundary point. Therefore, restoring stability of the solution, especially restoring stability of the solution on the boundary is very important for practice background and theoretical research.So far,many regularized methods only have the numerical results without error estimates, there
are also some methods which can give the order optimal error estimate,but few work is denoted to the optimality§and the results on the boundary are less.
In Chapter 2, we discuss optimality for the Cauchy problem for Laplace equation in unbounded strip region in the nterval (0,1]. Meanwhile, we provide two regularization methods (i.e., the generalized Tikhonov regularization and the generalized singular value decomposition) and the choice rule of the regularization parameter. When the regularization parameter is chosen according to the rule, it can realize the optimal error bounds. Moreover, we give filtering method and Fourier regularization method about this problem. When the regularization parameters are chosen suitably, we obtain the order optimal error bound and logarithmic stable estimate between the exact solution and its approximation, respectively. In Chapter 3, respectively, we analyze optimality of error estimate for the Cauchy problem for Laplace equation and inverse heat conduction problem on the boundary. And we both obtain the optimal error bounds between the exact solutions and their regularized approximations under the logarithmic source conditions. In Chapter 4, by using the discretization regularization, we study Cauchy problem for Laplace equation and backward heat conduction problem in the bounded region. Here, we give the logarithmic stability error estimate between the exact solutions and their regularized approximations which are obtained by the spectral cutoff method, the least squares method and the dual least squares method.