偏微分方程参数反演问题的算法与分析

发布时间：2018-03-08 11:04

本文选题：参数反演问题　切入点：正则化方法　出处：《浙江大学》2015年博士论文　论文类型：学位论文

【摘要】：微分方程反问题在自然科学与工程技术诸多领域之中有着广泛应用。其一个突出的特征就是不适定性,这使得反问题的求解比正问题困难的多。因而反问题的求解算法是科学计算领域的重要研究方向。微分方程的参数反演问题是反问题的重要分支之一。本文详细介绍了椭圆型微分方程参数反演问题的基本理论、各方面应用以及其数值算法。解偏微分方程参数反演问题,通常做法是构造相应的最小二乘泛函,求得其极小化问题的解。以椭圆型方程Robin系数反演问题的数值解法为例进行研究,我们得到了以下创新性的成果：一、对于分片常数型Robin系数的反演问题,我们提出直接反演Robin参数间断点,以重构Robin系数的方法(下文称为间断点重构法)。我们引入相应的最小二乘泛函,给出了方程解及目标泛函关于间断点一阶和二阶Frechet导数所满足的方程,从而应用Gauss-Newton方法求解泛函极小点,得到各间断点的位置,较准确的重构Robin系数。二、对于一般的情况,我们提出复、实边界条件耦合方法。在可接触边界上,构造新的Robin边界条件,将Neumann和Dirichlet型边界测量值耦合,从而构造新的边值问题及相应的最小二乘数据拟合项,求解相应的泛函优化问题,以重构Robin系数。对于复边界条件耦合法,我们构造一个复空间上的边值问题,通过极小化复边值问题解的虚部模,得到稳定的数值反演结果。对于实边界条件耦合法,我们构造依赖于不同正参数α1和α2的两个边值问题,通过极小化相应的Kohn-Vogelius型最小二乘泛函,重构Robin系数。该方法对正则化参数选取的依赖程度小,对于非常小的正则化参数,也可得到满意的结果。
[Abstract]:Inverse problems of differential equations are widely used in many fields of natural science and engineering technology. Therefore, inverse problem solving algorithm is an important research direction in the field of scientific computation. Parameter inversion problem of differential equation is one of the important branches of inverse problem. The basic theory of parameter inversion for elliptic differential equations is introduced. The parameter inversion problem for solving partial differential equations is usually done by constructing the corresponding least square functional. Taking the numerical solution of the Robin coefficient inversion problem for elliptic equations as an example, we have obtained the following innovative results: first, for the piecewise constant Robin coefficient inversion problem, We propose a direct inversion of the discontinuity points of Robin parameters to reconstruct the Robin coefficients (hereafter called the discontinuous point reconstruction method). The solution of the equation and the equation of the first and second order Frechet derivatives of the discontinuous point are given. The Gauss-Newton method is used to solve the minimal point of the functional, the position of the discontinuous point is obtained, and the Robin coefficient is reconstructed more accurately. We propose a complex and real boundary condition coupling method. A new Robin boundary condition is constructed on the reachable boundary, and the Neumann and Dirichlet boundary measurements are coupled to construct a new boundary value problem and the corresponding least-squares data fitting term. For the complex boundary condition coupling method, we construct a boundary value problem in complex space by minimizing the imaginary modulus of the solution of the complex boundary value problem. For the real boundary condition coupling method, we construct two boundary value problems dependent on different positive parameters 伪 1 and 伪 2, and minimize the corresponding Kohn-Vogelius type least squares functional. The method has less dependence on the selection of regularization parameters, and satisfactory results can be obtained for very small regularization parameters.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175.2

【共引文献】