摘要
本文考虑了线性椭圆和抛物方程的几类不适定问题：包括抛物方程时间反向问题和椭圆方程Cauchy问题,这些问题都是严重不适定的。
对一些比较困难的变系数问题,我们给出了若干条件稳定性结果。
对大部分其他问题,我们均建立了相应的正则化方法,这些方法包括Tikhonov方法,截断方法和拟边界值方法。对每种方法,我们都给出了正则化参数的选取规则,给出了误差估计,提供了相应的算法,给出了若干数值例子。特别是对于拟边界值方法,我们还做了进一步的深入分析,从中总结出一些性质,这些性质可以用于其他不适定问题的研究。
在数值实现中我们主要使用了有限差分方法和快速Fourier变换技巧。对于具有变系数的椭圆方程Cauchy问题,由于其不适定程度更强以及离散后所得到的线性方程组的维数庞大使得数值实现变得尤为困难,所以我们对其三维情形的数值实现专门进行了研究。我们用左预处理的广义最小残量方法结合拟边界值方法对其进行处理,其中一个有效的预条件矩阵被构造,一个快速直接算法也被提出,从而使得广义最小残量方法能够很好的被使用。
本文的数值结果与理论结果是完全相符的,这些数值结果充分地体现了所给的正则化方法能够很好地求解这些不适定问题。
In this thesis,we consider some ill-posed problems of linear elliptic and parabolic equations,i.e.,Backward parabolic equation in time and Cauchy problems of elliptic equation.
For some difficult variable coefficient cases,several conditional stability results are given.
Some regularization methods including Tikhonov regularization method,Cut-Off regularization method and Quasi-Boundary-Value method are used for these ill-posed problems respectively. A-priori choice rule for all regularization methods and a-posteriori choice rule for some of them are given.All corresponding error estimates are obtained.About the Quasi-Boundary-Value method,we give some properties after observing its applications. These properties are helpful for dealing with other ill-posed problems.
In the numerical aspect,we use the finite difference method and the Fast Fourier Transform to implement all regularization methods for both the constant coefficient and variable coefficient cases.For the variable coefficient case of elliptic Cauchy problem,we consider its three dimensional numerical implementation since it is more ill-posed and the coefficient matrix of the linear system is huge.A Left-Preconditioned GMRES method is used.A good preconditioner is constructed and a fast direct solver is also given to make the GMRES method work well.
The numerical results are consistent with the theoretical results.These results show that our regularization methods for these ill-posed problems work effectively.
引文
1法国著名的数学家Jacques Salomon Hadamard(1865年12月8日-1963年10月17日)在文献[38]里首次提出不适定性的概念,即如果解不满足存在性、唯一性、稳定性这三种性质中的任意一个就称所对应的问题是不适定的。
21789年8月21日-1857年5月23日
2该方法在Matlab中是直接用反斜杠来实现。
3本论文中我们用浮点的操作(Floating point operations)个数来计算一个算法的执行时间。用flop来表示一次运算[25].
4这里S*表示矩阵S的共轭转置.
5正弦变换可以用FFT算法来操作。
6这种思想类似与文献[91,P274]中处理二维问题所提出的一个直接算法[91,P274]
7对于每个矩阵A=[aij]∈Mm,n(F),定义vec A≡ [a11, …,am1,a12,…,am2,…,a1n,…,amn]T,则向量vec(A)∈Fmn参见文献[50]
8用tridiag表示三对角矩阵
9对于一个向量v,我们通过对其加一个随机扰动来构造噪音数据,即,vδ=v+ε randn(size(v)).
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