椭圆型偏微分方程反问题的正则化理论及算法
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摘要
众所周知,椭圆型偏微分方程Cauchy问题在Hadamard意义下严重不适定,表现在Cauchy数据的微小扰动可导致Cauchy问题解的巨大误差.来源于科学和工程中的许多理论和应用问题可归结为椭圆型方程Cauchy问题,如工程无损探测,地球物理勘查,心脏病学等.Cauchy问题的不适定性给上述问题的研究带来了很大困难,表现在难以构造稳定,高效的算法.一般来说,椭圆型方程Cauchy问题不具有稳定性,但若对该问题的解作先验有界的假设,则可获得稳定性估计,如Holder稳定性,对数稳定性等.
本文研究了三类椭圆型方程柯西问题：Laplace方程柯西问题,Helmholtz方程柯西问题和变系数椭圆方程柯西问题.分析了这些问题的不适定性本质和不适定性程度,给出了几种正则化方法.
本文分为四部分.第一章简要介绍了反问题的概念,反问题的数学特征以及正则化方法.第二章用线方法和谱方法求解了Laplace方程柯西问题,得到了稳定的误差估计.第三章用拟逆方法和拟边界值方法求解了Helmholtz方程和修正的Helmholtz方程,并且得到了Helmholtz方程在一般源条件下的最优误差界.第四章用Fourier方法,修正的Tikhonov正则化方法和小波对偶最小二乘法求解了变系数椭圆方程柯西问题,得到了稳定的误差估计.
此外,我们讨论了所有这些方法的数值实现,给出大量的数值例子来测试所提出的正则化方法各方面的性质.这些测试显示我们的方法是数值可行和有效的.
It is well know that the Cauchy problem of elliptic partial equations are severely ill-posed in the sense of Hadamard:a small noise in the Cauchy data may cause dramatically large error in the solution. The Cauchy problem of elliptic equation arises from many science and engineering problems such as nondestructive testing techniques, geophysics and cardiology. The ill-posedness of Cauchy problem lead to enormous difficulty to research these problems, that is, it is difficult to construct stable, efficient algorithm. In general, the Cauchy problem of elliptic equation has not stability. Under an additional a-priori bound condition, a stable estimate can be obtained, such as Holder stability or logarithmic stability.
In this thesis, we study three kinds of Cauchy problem of elliptic equa-tion:Cauchy problem of Laplace equation; Cauchy problem of Helmholtz equation and Cauchy problem of elliptic equation with variable coefficients.
We analyze the ill-posedness of these inverse problems, and discuss their degree of ill-posedness. For stably computing these problems, we proposed several regularization methods.
This thesis is divided into four parts. The first chapter is preface, the definition of inverse problem, the mathematical characteristic of inverse problem and the regularization method are reported. In the second chap-ter, we use line method and spectral method solve the Cauchy problem of Laplace equation, the stable error estimate is obtained. In the third chap-ter, we apply the quasi-reversibility method and the quasi-boundary value method to Helmholtz equation. Moreover, we get the optimal error bound for the Cauchy problem of Helmholtz equation at general source condition. The fourth chapter is devoted to Fourier method, revised Tikhonov method and Wavelet dual least squares method for the Cauchy problem of elliptic equation with variable coefficients, we obtain stable error estimate.
In addition, we discuss the numerical implementation of all these meth-ods, and give large number of numerical examples to test various properties of the proposed regularization methods. These tests show that our methods are numerically feasible and effective.
本文研究了三类椭圆型方程柯西问题：Laplace方程柯西问题,Helmholtz方程柯西问题和变系数椭圆方程柯西问题.分析了这些问题的不适定性本质和不适定性程度,给出了几种正则化方法.
本文分为四部分.第一章简要介绍了反问题的概念,反问题的数学特征以及正则化方法.第二章用线方法和谱方法求解了Laplace方程柯西问题,得到了稳定的误差估计.第三章用拟逆方法和拟边界值方法求解了Helmholtz方程和修正的Helmholtz方程,并且得到了Helmholtz方程在一般源条件下的最优误差界.第四章用Fourier方法,修正的Tikhonov正则化方法和小波对偶最小二乘法求解了变系数椭圆方程柯西问题,得到了稳定的误差估计.
此外,我们讨论了所有这些方法的数值实现,给出大量的数值例子来测试所提出的正则化方法各方面的性质.这些测试显示我们的方法是数值可行和有效的.
It is well know that the Cauchy problem of elliptic partial equations are severely ill-posed in the sense of Hadamard:a small noise in the Cauchy data may cause dramatically large error in the solution. The Cauchy problem of elliptic equation arises from many science and engineering problems such as nondestructive testing techniques, geophysics and cardiology. The ill-posedness of Cauchy problem lead to enormous difficulty to research these problems, that is, it is difficult to construct stable, efficient algorithm. In general, the Cauchy problem of elliptic equation has not stability. Under an additional a-priori bound condition, a stable estimate can be obtained, such as Holder stability or logarithmic stability.
In this thesis, we study three kinds of Cauchy problem of elliptic equa-tion:Cauchy problem of Laplace equation; Cauchy problem of Helmholtz equation and Cauchy problem of elliptic equation with variable coefficients.
We analyze the ill-posedness of these inverse problems, and discuss their degree of ill-posedness. For stably computing these problems, we proposed several regularization methods.
This thesis is divided into four parts. The first chapter is preface, the definition of inverse problem, the mathematical characteristic of inverse problem and the regularization method are reported. In the second chap-ter, we use line method and spectral method solve the Cauchy problem of Laplace equation, the stable error estimate is obtained. In the third chap-ter, we apply the quasi-reversibility method and the quasi-boundary value method to Helmholtz equation. Moreover, we get the optimal error bound for the Cauchy problem of Helmholtz equation at general source condition. The fourth chapter is devoted to Fourier method, revised Tikhonov method and Wavelet dual least squares method for the Cauchy problem of elliptic equation with variable coefficients, we obtain stable error estimate.
In addition, we discuss the numerical implementation of all these meth-ods, and give large number of numerical examples to test various properties of the proposed regularization methods. These tests show that our methods are numerically feasible and effective.
引文
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