电阻抗成像的数值模拟和分析
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摘要
电阻抗成像(Electrical Impedance Tomography,简称EIT)是一种新兴的生物医学成像技术,它的具体做法是在人体表面设置大量电极,注入安全电流,测量人体表皮上的电势分布,把这些数据传输给计算机,通过计算,得到人体内部的电阻率分布,并将其转化为灰度或彩色图像.由于人体各部分电阻率不同,因此这实际上是一个人体内部的图像.EIT在地球物理探矿,工业无伤检测,水下目标探测等领域也有着重要的应用前景([5,10,11,12,13]),从而引起了人们广泛的兴趣,其研究涉及理论,数值计算和实验等几个方面,横跨数学,物理学,电子信息,生物医学等多个学科.
电阻抗成像实际上是一类椭圆型偏微分方程反问题.反问题有别于传统的数学物理方程的定解问题(通常称为正问题),反问题研究由解的部分已知信息来求定解问题中的某些未知量,正问题通常都是适定的,而反问题往往是不适定的,该特点也是反问题的难点所在.基于实际应用问题的推动,20世纪60年代中期苏联科学院院士Tikhonov([2])提出了处理不适定问题的正则化方法,为不适定问题和反问题的研究开辟了道路.在中国,反问题的研究始于20世纪80年代冯康院士的大力倡导.
正是由于非线性和严重不适定性,EIT问题的数值求解是非常复杂和困难的.目前求解EIT问题的算法大体可分为三类:
1.基于全局线性化的非迭代算法,
2.直接法,
3.解完全非线性问题的迭代法.
其中基于全局线性化的非迭代法可看作迭代法的一步迭代即终止而来,如Calderon算法,等势线反投影法和一步牛顿法(NOSER),其局限性在于只能求解和均匀分布较为接近的电阻率分布,而且求解精度较差;直接法是指刻画在均匀分布介质上嵌入的分解法或实现唯一性证明的直接方法,由于其需要在目标的整个边界上施加电流,测量电压,用数学的语言来讲,就是要观测Neumann-to-Dirichlet映射,不适用于有限电极系统,因而在实际应用上受到限制.基于有限元的迭代法目前是求解EIT问题的一类比较有效的方法,但其收敛性分析一直没有得到解决.除了该问题的严重不适定性所带来的数值解的稳定性差这一难题外,由于计算量大,难以实时成像是其另一难点.为解决这一问题,Cheney([6])等人提出了著名的一步牛顿法(NOSER).在该方法中,把均匀分布作为初始猜测解,这样,均匀分布下的Jacobi矩阵可预先计算好并存储起来,实际计算时只进行一步迭代,从而有助于快速求解.但这样的方法也限制了其适用范围和求解精度.
本文作者在袁益让教授的精心指导下,在大量的数值试验的基础上,选择Levenberg-Marquardt方法与偏微分方程数值解法相结合,对电阻抗成像进行数值模拟研究.L-M方法是一种信赖域方法,也是一种正则化的迭代算法,因而可以用来稳定地求解非线性反问题([18,19]).本文主要考虑两种电阻抗成像模型:连续模型(continuum model)和电极模型(electrode model).分别应用有限元、体积元和块中心有限体积方法在规则和不规则区域,对二维和三维EIT问题进行求解.其中体积元和块中心有限体积方法是首次用于求解电阻抗成像问题.对于连续模型,我们用精确解来进行数值模拟,验证模型的正确性和算法的可靠性及可行性,然后应用到电极模型上去,并在电极模型下给出了适用于以上三种数值方法(对称格式)的一类电流模式,将计算量最大的Jacobi矩阵的计算彻底简化.
全文共分三章.
第一章介绍基于有限元方法的电阻抗成像的数值模拟和分析.§1.2-1.5研究了基于有限元的二维EIT问题的数值模拟的方法和技术.采用了两种不同的网格:三角形网格和四边形网,给出了伪单元刚度矩阵的计算公式,进行分析比较.在矩形区域上,采用连续模型,对边界条件施以不同幅度的扰动,分别用不同空间步长的数值解和精确解进行对比,验证了连续模型的正确性和算法的可靠性及可行性.数值实验表明,当步长缩小时,反问题的数值解能逼近精确解;数值解关于Neumann边界条件的稳定性优于Dirichlet边界条件,四边形网格优于三角形网格.§1.5在电极模型上的应用,研究两种网格上EIT问题的各种数值表现.特别比较研究了固定发射电极模式和轮换发射电极模式下的数值表现,结果表明轮换发射电极模式优于固定电极模式.在§1.6-1.9研究了三维EIT问题的数值模拟的方法和技术.针对基于有限元方法的迭代算法计算量大的特点,给出了基于伪单元刚度矩阵不变性的计算技巧,可大大减少有限元方法所需的数值积分的计算.提出了一类特殊的电流施加模式,可充分利用每一步迭代中已有的计算结果来简单计算目标函数Jacobi矩阵,而不再需要另外求解任何大规模线性方程组.最后给出三维不规则区域上对电阻抗成像的实际数值模拟的结果.§1.2-1.5的主要内容已投稿《Acta Mathematica Scientia》,§1.6-1.9的主要内容已投稿《Computers and Mathematics with Applications》,并已投修改稿.
第一章的创新之处有:(1)首次采用精确解对EIT问题进行数值模拟,尤其是三维EIT问题.而前人所做的都是二维的,而且是用有限元方法求得的近似解来代替边界测量(边界条件)来进行数值模拟.(2)首次提出伪单元刚度矩阵方法,简化了有限元方法中数值积分的计算.(3)本文提出的Jacobi矩阵的计算方法适合多次迭代,并使每次迭代中求解正问题的次数降到最低.
第二章介绍基于有限体积元方法的电阻抗成像的数值模拟和分析.本章§2.2-2.5为二维区域上的数值模拟和分析.将体积元方法用于二维电阻抗成像的数值模拟,给出了Jacobi矩阵的计算公式.并在正方形区域上,用精确解进行数值模拟,结果验证了连续模型的正确性和算法的可靠性和可行性.在2.5应用于电极模型,给出了在扇形区域上图像重建的仿真结果.本章§2.6-2.9为三维区域上的数值模拟和分析.对于三维椭圆方程Neumann边值问题,提出了四面体单元上的一类对称体积元格式,并证明了格式的半正定性及解的存在性;引入单元形状矩阵的概念,简化了系数矩阵的计算;提出了对电阻率进行拼接逼近的方法来降低反问题求解规模,使之与正问题的求解规模相匹配;导出了相应的误差泛函的Jacobi矩阵的计算公式.在正方体上进行了一系列数值实验,结果验证三维情形下EIT连续模型的正确性和算法的可靠性及可行性.§2.9论述了利用体积元格式的对称性和特殊的电流基向量同样可以简化Jacobi矩阵的计算,将每次迭代中需要求解的正问题的个数降到最低.并将这些方法成功应用于三维不规则区域上电阻抗成像的实际数值模拟.§2.2-2.5内容已投稿《APPLIEDMATHEMATICAL MODELLING》,并已提交修改稿.§2.6-2.9内容已在《计算数学》上发表.
第二章的创新之处有:(1)对于三维椭圆方程Neumann边值问题,提出了四面体单元上的一类对称体积元格式,并证明了格式的半正定性及解的存在性.引入单元形状矩阵的概念,使得体积元方法可以象有限元一样,通过单元分析、整体合成、代数解算,实现程序的标准化.(2)首次提出用体积元方法求解EIT问题,该方法可保持局部电量守恒.并在二维、三维,规则及不规则区域上进行了数值模拟.(3)给出了在电极模型下,相应于体积元方法的计算Jacobi矩阵的简单方法,可把每次迭代中需要求解正问题的个数降到最低.
第三章介绍基于块中心有限体积方法的电阻抗成像的数值模拟和分析.本章§3.2-3.5为二维区域上的数值模拟和分析.提出了块中心有限体积格式,并证明了格式的半正定性及解的存在性,导出了误差泛函的Jacobi矩阵的计算公式.通过用精确解进行数值模拟,验证了连续模型的正确性和算法的可靠性和可行性.在电极模型下的模拟也取得了成功.本章§3.6-3.9为三维区域上的数值模拟和分析.块中心有限体积格式对三维EIT的模拟也取得了类似的结果.在§3.9论述了利用块中心有限体积格式的对称性和特殊的电流基向量,同样可以简化Jacobi矩阵的计算将每次迭代中需要求解的正问题的个数降到最低.这些方法也成功地应用于电极模式下三维阶梯形区域上电阻抗成像的实际数值模拟.本章§3.6-3.9已投稿《Applied Mathematics and Computation》
第三章的创新之处在于:首次将块中心有限体积方法用于的EIT求解,并进行了二维以及三维的数值模拟.该格式简单对称,可实现Jacobi矩阵的简单计算并保持电量守恒,是求解EIT的这三种算法中工作量最小的.
Electrical impedance tomography(EIT) is a developing biomedical imaging technology. By means of injecting safe electric currents into a person through electrodes attached to the skin, it measures the resulting electric voltages on the surface, and sends data to the computer. By computing, the impedance or resistivity distribution is obtained and revealed as a gray or color image. It is the one's picture of the interior, because different tissues have different impedances or resistances. EIT has caused a wide attention for its safety, low cost and excellent application prospect in geophysical exploration, dam detecting, tracing of contaminants, and the search for underwater objects, etc., see [5,10, 11,12, 13]. The research covers the subjects of mathematics, physics, electronics and biomedical engineering.
EIT is an inverse problem of elliptic partial differential equation. Different from the traditional problems of mathematical physics(direct problems) which are well-posed, inverse problems is to solve something unknown starting from only part of information and are usually ill-posed. It is the difficulty of inverse problems. On account of the promotion of practical problems, in 1960s, the former Soviet Union mathematician Tikhonov([2]) proposed the famous regularization method for ill-posed problems and led the research on ill-posed and inverse problems to a new stage. In China the research of inverse problems began from 80s of twenty century on the initiative of mathematician Feng Kang.
Because of nonlinearity and severe ill-posedness, it is very complicated and difficult to solve EIT numerically. Algorithms known to me can be categorized as
1. noniterative algorithms based on global linearization,
2. direct methods,
3. iterative solvers tackling the nonlinear problem.
Noniterative algorithms based on global linearization can be built by stopping any iterative algorithm after the first step, a prominent example is the NOSER algorithm. The class of direct methods splits into two subclasses: factorization methods use special singular testfunctions to characterize inclusions in a homogeneous background medium and direct methods that implement a constructive existence and uniqueness proof. As far as I know both direct methods are not able to deal with finite electrode models but need to apply currents and measure the voltages along the whole boundary of the object(in mathematical terms: they need to observe the Neumann-to-Dirichlet mapping). Their use for a realistic setting is therefore limited. Iterative algorithms are appropriate methods for EIT, but the convergence proof has not been completed so far, and in addition that stability of numerical solution is bad for the ill-posedness of inverse problem, realtime reconstruction is difficult for the high computational cost. Cheney([6]) proposed the famous NOSER algorithm to reduce the cost, which takes homogeneous distribution as the initial guess, saving the Jacobian in advance and iterates only one step. But it loses the accuracy and is only appropriate for the medium approximating to be homogeneous.
Under the supervision of Prof. Yuan Yirang, the author use Levenberg-Marquardt iterative algorithm combined with various numerical method for partial differential equations to simulate EIT problem. L-M method is a trust region method, also a regularization algorithm. It can be used to obtain stable numerical solution for nonlinear inverse problem([18,19]). Two models are considered: continuum model and electrode model(shunt model). Finite element methods, finite volume element methods and cell-centered finite volume methods are used to solve EIT. It is the first time that finite volume element methods and cell-centered finite volume methods are applied in EIT . We verify the correctness of the continuum model, reliability and feasibility of the iterative algorithm by numerical simulation with exact solution of inverse problem and apply the methods to the electrode model. A class of current patterns are proposed to simplify completely the computation of Jacobian matrix and reduce the number of direct problems solved per iteration to the least.
The thesis consists of three chapters.
In Chapter 1, the numerical simulation and analysis based on finite element method for EIT is addressed. The method and technique of numerical simulation on two- and three-dimensional domains are studied. At first, continuum model of EIT is considered. The computational technique for an inverse boundary value problem is proposed based on invariance of pseudo element stiffness matrix. This offers a significant reduction in computation of numerical integration arising from finite element methods. The results of numerical simulation on a 3-dimensional domain show that numerical solutions approach those exact solutions gradually with space step length becoming smaller, which verifies the correctness of its continuum model, the reliability and feasibility of the algorithm. A method to reduce the number of direct problems to be solved at each iteration step is proposed as well. These methods have been applied practically in simulation of electrical impedance tomography.
In Chapter 2, the numerical simulation and analysis based on finite volume element method for EIT is addressed, numerical simulations and analysis for it on two- and three-dimensional domains are presented. In this chapter a modified symmetric finite volume element method is proposed, semi-positive definiteness and existence of solution for this scheme are proved; element geometry matrix is introduced, which is helpful for simplifying the calculation of coefficient matrix; patch approximation for electrical resistivity is present to lower the scale of this inverse problem; the computational formula of Jacobian matrix of error functional is obtained, a class of electrical current patterns is proposed, under which the number of direct problems to solve in each iteration can be reduced to the least. A series of numerical experiments verify the correctness of the continuum model, the reliability and feasibility of the algorithm. These methods have been applied successfully in practical simulation of electrical impedance tomography.
In Chapter 3, the numerical simulation and analysis based on cell-centered finite volume element method for EIT is addressed. Numerical simulations and analysis on two- and three-dimensional domains are presented in this paper. Cell-centered finite volume scheme for Neumann boundary value problem is proposed and proved to be semi-positive definite; the formula of Jacobian matrix of error functional is derived, fast algorithm of which is presented. The results of numerical experiments verify the correctness of continuum model, the reliability and feasibility of this algorithm. These methods in this chapter have been successful in numerical simulation of three-dimensional practical imaging.
电阻抗成像实际上是一类椭圆型偏微分方程反问题.反问题有别于传统的数学物理方程的定解问题(通常称为正问题),反问题研究由解的部分已知信息来求定解问题中的某些未知量,正问题通常都是适定的,而反问题往往是不适定的,该特点也是反问题的难点所在.基于实际应用问题的推动,20世纪60年代中期苏联科学院院士Tikhonov([2])提出了处理不适定问题的正则化方法,为不适定问题和反问题的研究开辟了道路.在中国,反问题的研究始于20世纪80年代冯康院士的大力倡导.
正是由于非线性和严重不适定性,EIT问题的数值求解是非常复杂和困难的.目前求解EIT问题的算法大体可分为三类:
1.基于全局线性化的非迭代算法,
2.直接法,
3.解完全非线性问题的迭代法.
其中基于全局线性化的非迭代法可看作迭代法的一步迭代即终止而来,如Calderon算法,等势线反投影法和一步牛顿法(NOSER),其局限性在于只能求解和均匀分布较为接近的电阻率分布,而且求解精度较差;直接法是指刻画在均匀分布介质上嵌入的分解法或实现唯一性证明的直接方法,由于其需要在目标的整个边界上施加电流,测量电压,用数学的语言来讲,就是要观测Neumann-to-Dirichlet映射,不适用于有限电极系统,因而在实际应用上受到限制.基于有限元的迭代法目前是求解EIT问题的一类比较有效的方法,但其收敛性分析一直没有得到解决.除了该问题的严重不适定性所带来的数值解的稳定性差这一难题外,由于计算量大,难以实时成像是其另一难点.为解决这一问题,Cheney([6])等人提出了著名的一步牛顿法(NOSER).在该方法中,把均匀分布作为初始猜测解,这样,均匀分布下的Jacobi矩阵可预先计算好并存储起来,实际计算时只进行一步迭代,从而有助于快速求解.但这样的方法也限制了其适用范围和求解精度.
本文作者在袁益让教授的精心指导下,在大量的数值试验的基础上,选择Levenberg-Marquardt方法与偏微分方程数值解法相结合,对电阻抗成像进行数值模拟研究.L-M方法是一种信赖域方法,也是一种正则化的迭代算法,因而可以用来稳定地求解非线性反问题([18,19]).本文主要考虑两种电阻抗成像模型:连续模型(continuum model)和电极模型(electrode model).分别应用有限元、体积元和块中心有限体积方法在规则和不规则区域,对二维和三维EIT问题进行求解.其中体积元和块中心有限体积方法是首次用于求解电阻抗成像问题.对于连续模型,我们用精确解来进行数值模拟,验证模型的正确性和算法的可靠性及可行性,然后应用到电极模型上去,并在电极模型下给出了适用于以上三种数值方法(对称格式)的一类电流模式,将计算量最大的Jacobi矩阵的计算彻底简化.
全文共分三章.
第一章介绍基于有限元方法的电阻抗成像的数值模拟和分析.§1.2-1.5研究了基于有限元的二维EIT问题的数值模拟的方法和技术.采用了两种不同的网格:三角形网格和四边形网,给出了伪单元刚度矩阵的计算公式,进行分析比较.在矩形区域上,采用连续模型,对边界条件施以不同幅度的扰动,分别用不同空间步长的数值解和精确解进行对比,验证了连续模型的正确性和算法的可靠性及可行性.数值实验表明,当步长缩小时,反问题的数值解能逼近精确解;数值解关于Neumann边界条件的稳定性优于Dirichlet边界条件,四边形网格优于三角形网格.§1.5在电极模型上的应用,研究两种网格上EIT问题的各种数值表现.特别比较研究了固定发射电极模式和轮换发射电极模式下的数值表现,结果表明轮换发射电极模式优于固定电极模式.在§1.6-1.9研究了三维EIT问题的数值模拟的方法和技术.针对基于有限元方法的迭代算法计算量大的特点,给出了基于伪单元刚度矩阵不变性的计算技巧,可大大减少有限元方法所需的数值积分的计算.提出了一类特殊的电流施加模式,可充分利用每一步迭代中已有的计算结果来简单计算目标函数Jacobi矩阵,而不再需要另外求解任何大规模线性方程组.最后给出三维不规则区域上对电阻抗成像的实际数值模拟的结果.§1.2-1.5的主要内容已投稿《Acta Mathematica Scientia》,§1.6-1.9的主要内容已投稿《Computers and Mathematics with Applications》,并已投修改稿.
第一章的创新之处有:(1)首次采用精确解对EIT问题进行数值模拟,尤其是三维EIT问题.而前人所做的都是二维的,而且是用有限元方法求得的近似解来代替边界测量(边界条件)来进行数值模拟.(2)首次提出伪单元刚度矩阵方法,简化了有限元方法中数值积分的计算.(3)本文提出的Jacobi矩阵的计算方法适合多次迭代,并使每次迭代中求解正问题的次数降到最低.
第二章介绍基于有限体积元方法的电阻抗成像的数值模拟和分析.本章§2.2-2.5为二维区域上的数值模拟和分析.将体积元方法用于二维电阻抗成像的数值模拟,给出了Jacobi矩阵的计算公式.并在正方形区域上,用精确解进行数值模拟,结果验证了连续模型的正确性和算法的可靠性和可行性.在2.5应用于电极模型,给出了在扇形区域上图像重建的仿真结果.本章§2.6-2.9为三维区域上的数值模拟和分析.对于三维椭圆方程Neumann边值问题,提出了四面体单元上的一类对称体积元格式,并证明了格式的半正定性及解的存在性;引入单元形状矩阵的概念,简化了系数矩阵的计算;提出了对电阻率进行拼接逼近的方法来降低反问题求解规模,使之与正问题的求解规模相匹配;导出了相应的误差泛函的Jacobi矩阵的计算公式.在正方体上进行了一系列数值实验,结果验证三维情形下EIT连续模型的正确性和算法的可靠性及可行性.§2.9论述了利用体积元格式的对称性和特殊的电流基向量同样可以简化Jacobi矩阵的计算,将每次迭代中需要求解的正问题的个数降到最低.并将这些方法成功应用于三维不规则区域上电阻抗成像的实际数值模拟.§2.2-2.5内容已投稿《APPLIEDMATHEMATICAL MODELLING》,并已提交修改稿.§2.6-2.9内容已在《计算数学》上发表.
第二章的创新之处有:(1)对于三维椭圆方程Neumann边值问题,提出了四面体单元上的一类对称体积元格式,并证明了格式的半正定性及解的存在性.引入单元形状矩阵的概念,使得体积元方法可以象有限元一样,通过单元分析、整体合成、代数解算,实现程序的标准化.(2)首次提出用体积元方法求解EIT问题,该方法可保持局部电量守恒.并在二维、三维,规则及不规则区域上进行了数值模拟.(3)给出了在电极模型下,相应于体积元方法的计算Jacobi矩阵的简单方法,可把每次迭代中需要求解正问题的个数降到最低.
第三章介绍基于块中心有限体积方法的电阻抗成像的数值模拟和分析.本章§3.2-3.5为二维区域上的数值模拟和分析.提出了块中心有限体积格式,并证明了格式的半正定性及解的存在性,导出了误差泛函的Jacobi矩阵的计算公式.通过用精确解进行数值模拟,验证了连续模型的正确性和算法的可靠性和可行性.在电极模型下的模拟也取得了成功.本章§3.6-3.9为三维区域上的数值模拟和分析.块中心有限体积格式对三维EIT的模拟也取得了类似的结果.在§3.9论述了利用块中心有限体积格式的对称性和特殊的电流基向量,同样可以简化Jacobi矩阵的计算将每次迭代中需要求解的正问题的个数降到最低.这些方法也成功地应用于电极模式下三维阶梯形区域上电阻抗成像的实际数值模拟.本章§3.6-3.9已投稿《Applied Mathematics and Computation》
第三章的创新之处在于:首次将块中心有限体积方法用于的EIT求解,并进行了二维以及三维的数值模拟.该格式简单对称,可实现Jacobi矩阵的简单计算并保持电量守恒,是求解EIT的这三种算法中工作量最小的.
Electrical impedance tomography(EIT) is a developing biomedical imaging technology. By means of injecting safe electric currents into a person through electrodes attached to the skin, it measures the resulting electric voltages on the surface, and sends data to the computer. By computing, the impedance or resistivity distribution is obtained and revealed as a gray or color image. It is the one's picture of the interior, because different tissues have different impedances or resistances. EIT has caused a wide attention for its safety, low cost and excellent application prospect in geophysical exploration, dam detecting, tracing of contaminants, and the search for underwater objects, etc., see [5,10, 11,12, 13]. The research covers the subjects of mathematics, physics, electronics and biomedical engineering.
EIT is an inverse problem of elliptic partial differential equation. Different from the traditional problems of mathematical physics(direct problems) which are well-posed, inverse problems is to solve something unknown starting from only part of information and are usually ill-posed. It is the difficulty of inverse problems. On account of the promotion of practical problems, in 1960s, the former Soviet Union mathematician Tikhonov([2]) proposed the famous regularization method for ill-posed problems and led the research on ill-posed and inverse problems to a new stage. In China the research of inverse problems began from 80s of twenty century on the initiative of mathematician Feng Kang.
Because of nonlinearity and severe ill-posedness, it is very complicated and difficult to solve EIT numerically. Algorithms known to me can be categorized as
1. noniterative algorithms based on global linearization,
2. direct methods,
3. iterative solvers tackling the nonlinear problem.
Noniterative algorithms based on global linearization can be built by stopping any iterative algorithm after the first step, a prominent example is the NOSER algorithm. The class of direct methods splits into two subclasses: factorization methods use special singular testfunctions to characterize inclusions in a homogeneous background medium and direct methods that implement a constructive existence and uniqueness proof. As far as I know both direct methods are not able to deal with finite electrode models but need to apply currents and measure the voltages along the whole boundary of the object(in mathematical terms: they need to observe the Neumann-to-Dirichlet mapping). Their use for a realistic setting is therefore limited. Iterative algorithms are appropriate methods for EIT, but the convergence proof has not been completed so far, and in addition that stability of numerical solution is bad for the ill-posedness of inverse problem, realtime reconstruction is difficult for the high computational cost. Cheney([6]) proposed the famous NOSER algorithm to reduce the cost, which takes homogeneous distribution as the initial guess, saving the Jacobian in advance and iterates only one step. But it loses the accuracy and is only appropriate for the medium approximating to be homogeneous.
Under the supervision of Prof. Yuan Yirang, the author use Levenberg-Marquardt iterative algorithm combined with various numerical method for partial differential equations to simulate EIT problem. L-M method is a trust region method, also a regularization algorithm. It can be used to obtain stable numerical solution for nonlinear inverse problem([18,19]). Two models are considered: continuum model and electrode model(shunt model). Finite element methods, finite volume element methods and cell-centered finite volume methods are used to solve EIT. It is the first time that finite volume element methods and cell-centered finite volume methods are applied in EIT . We verify the correctness of the continuum model, reliability and feasibility of the iterative algorithm by numerical simulation with exact solution of inverse problem and apply the methods to the electrode model. A class of current patterns are proposed to simplify completely the computation of Jacobian matrix and reduce the number of direct problems solved per iteration to the least.
The thesis consists of three chapters.
In Chapter 1, the numerical simulation and analysis based on finite element method for EIT is addressed. The method and technique of numerical simulation on two- and three-dimensional domains are studied. At first, continuum model of EIT is considered. The computational technique for an inverse boundary value problem is proposed based on invariance of pseudo element stiffness matrix. This offers a significant reduction in computation of numerical integration arising from finite element methods. The results of numerical simulation on a 3-dimensional domain show that numerical solutions approach those exact solutions gradually with space step length becoming smaller, which verifies the correctness of its continuum model, the reliability and feasibility of the algorithm. A method to reduce the number of direct problems to be solved at each iteration step is proposed as well. These methods have been applied practically in simulation of electrical impedance tomography.
In Chapter 2, the numerical simulation and analysis based on finite volume element method for EIT is addressed, numerical simulations and analysis for it on two- and three-dimensional domains are presented. In this chapter a modified symmetric finite volume element method is proposed, semi-positive definiteness and existence of solution for this scheme are proved; element geometry matrix is introduced, which is helpful for simplifying the calculation of coefficient matrix; patch approximation for electrical resistivity is present to lower the scale of this inverse problem; the computational formula of Jacobian matrix of error functional is obtained, a class of electrical current patterns is proposed, under which the number of direct problems to solve in each iteration can be reduced to the least. A series of numerical experiments verify the correctness of the continuum model, the reliability and feasibility of the algorithm. These methods have been applied successfully in practical simulation of electrical impedance tomography.
In Chapter 3, the numerical simulation and analysis based on cell-centered finite volume element method for EIT is addressed. Numerical simulations and analysis on two- and three-dimensional domains are presented in this paper. Cell-centered finite volume scheme for Neumann boundary value problem is proposed and proved to be semi-positive definite; the formula of Jacobian matrix of error functional is derived, fast algorithm of which is presented. The results of numerical experiments verify the correctness of continuum model, the reliability and feasibility of this algorithm. These methods in this chapter have been successful in numerical simulation of three-dimensional practical imaging.
引文
[1]Courant R.Variational methods for the solutions of problems of equilibrium and vibration[J].Bull.Amer.Math.Soc.,1943,49,1-23
[2]Tikhonov A N.On solving incorrectly posed problems and method of regularition[J].Dokl.Acad.Nauk USSR,151(3).1963
[3]Feng K.The difference scheme based on variational principle(Chinese).J.Appl.Comp.1965,2(4):238-262.
[4]Ciarlet P G.The Finite Element Method for Elliptic Problems.Amsterdam:North-Holland,1978.
[5]Cheney M,Isaacson D,Newell J C.Electrical Impedance Tomography[J].SIAM Rev.,1999,41(1):85-101.
[6]Cheney M,Isaacson D,Newell J C,Simske S and Gogle J.Nosor:an algorithm for solving the inverse conductivity problem.Int.J.Imag.Syst.Technol.2,66-75.
[7]Bruhl M.Explicit characterization of inclusions in electrical impedance tomography[J].SIAM J.Math.Anal.,2001,32:1327-1341.
[8]Eric T.Chung,Tony F.Chan and Xue-Cheng Tai.Eelectrical Impedance Tomography using level set representation and total variational regularization[J].J.Comput.Phys.,2005,205:357-372.
[9]Breckon W R,Pidcock M K.Mathematical aspects of impedance imaging[J].Clin.Phys.Physiol.Mess.,1987,8(Suppl):77-84.
[10]Borcea L,Gray G A,Zhang Y.Variationally constrained numerical solution of electrical impedance tomography[J].Inverse Problems,2003,19(5):1159-1184.
[11]Lechleiter A,Kieder A.Newton regularizations for impedance tomography:a numerical study[J].Inverse Problems,2006,22:1967-1987.
[12]Du Y,Cheng J,Liu Z.Combined variable metric method for electrical impedance tomography problem[J].Chin.J.Biomed.Eng.(in Chinese),1997,16(2):167-173.
[13]Tamburrino A,Rubinacci G.A new non-iterative inversion for electrical resistance tomography [J].Inverse Problems,2002,18(6):1809-1829.
[14]Hanke M,Bruhl M.Recent progress in electrical impedance tomography[J].Inverse Problems,2003,19:S65-S90.
[15]Webster J G.Electrical Impedance Tomography[M].Adam Hilger,Bristol England,1990.
[16]Kirsch A.An introduction to the mathematical theory of inverse problems[M].Number 120 in Applied Mathematical Sciences.Springer-Verlag,New York,1996.
[17]Wang Yanfei.On the regularity of trust region-cg algorithm:With application to deconvolution problem.Science in Chian(Series A),2003,46(3):312-325.
[18]Xiao T,Yu S,Wang Y.Numerical solution for inverse problems(in Chinese)[M].Science Press,Beijing,2003.
[19]Wang Yanfei,Yuan Yaxiang.On the regularity of trust region algorithms for nonlinear ill-posed inverse problems,short communications at the International Congress of Mathematician,China,August22,2002.
[20]Mishev I D.Finite volume methods on Voronoi meshes[J].Numer.Methods P.D.E,1998,14:193-212.
[21]Yuan Y,Sun W.Theory and methods of Optimization(in Chinese)[M].Science Press,Beijing,1997.
[22]Hayes L J.Galerkin alternating-direction methods for nonrecutangular regions using patch approximations[J].SIAM J.Numer.Anal.,1981,18(4):627-643.
[23]Hayes L J.A modified backward time discretization for nonlinear parabolic equations using patch approximations[J].SIAM J.Numer.Anal.,1981,18(5):781-793.
[24]Li T,Qin T.Physics and partial differntial equations(in Chinese)[M].Advanced Education Press,Beijing,second edition,2005.
[25]Liu J.Regularization methods for ill-posed problems and its application(in Chinese)[M].Science Press,Beijing,2005.
[26]Jiang Lishang,Pang Zhihuan.The finite element method with its theoretical foundation[M].Beijing:People's Education Press,1979.
[27]S.Brenner and R.Scott,The Mathematical Theory of Finite element methods,Springer-Verlag,New York,1994.
[28]Calderon A P.On an inverse boundary value problem.Seminar on Numerical Analysis and Its Applications to Continuum Physics,Rio de Janeiro,1980,65-73.
[29]Sylvester J,Uhlmann G.A global uniqueness theorem for an inverse boundary value problem.Ann.Math.125(1987),153-169.
[30]Nachman A I.Global uniqueness for a two-dimensional inverse boundary problem.Ann.Math.1996,143:71-96
[31]Brown R M,Uhlmann G.Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions.Comm.Partial Differential Equations,1997,22(5-6):1009-1027.
[32]Hackbnsch W.:Elliptic Differential Equations:Theory and Numerical Treatment.Springer-Verlag,Berlin(1992).
[33]Isakov V.:Inverse problems for partial differential equations.Springer-Verlsg,New York(1998).
[34]Du Yah.A study on algorithms for electrical impedance tomography.Beijing:Doctorial Dis sertation of Beijing University of Aeronautics and Astronautics,1997.
[35]Quarteroni A.,Sacco R.and Saleri F.:Numerical Mathematics.Springer-Verlag,New York(2000).
[36]Stoer J.and Bulirsch R.:Introduction to Numerical Analysis.Springer-Verlag,New York(1993).
[37]Nocedal,J and Wright S J.Numerical Optimization.Springer Science+Business,Inc.1999
[38]R.D.Lazarov,I.D.Mishev and P.S.Vassilevski.Finite volume methods for convection-diffusion problems.SIAM J.Numer.Anal.33(1):31-55(1996).
[39]S.Larsson and V.Thomee,Partial Differential Equations with Numerical Methods.Springer-Veriag,Berlin,2003,pp.43-50.
[40]Russell T F,Wheeler M F.Finite element and finite difference methods for continuous flows in porous media.In:Ewing R E.The mathematics of reservoir simulation[M],SIAM Phialdelphia,1983
[41]Weiser A,Wheeler M F.On convergence of block-centered finite difference for elliptic problem[J].SIAM J Numer Anal,1988;25(2):351-375.
[42]MscNeal R H.An asymmetrical finite difference network[J].Quart.Appl.Math.1953,11:295-310.
[43]R.E.Ewing,M.F.Wheeler.Galerkin methods for miscible displacement problems in porous media[J].SIAM.J.Numer.Anal.,1980,17(2):351-365.
[44]R.E.Ewing and T.F.Russell.Efficient time-stepping methods for miacible displacement problem in Porous Media[J].SIAM.J.Numer.Anal.,1982,19(1):1-67.
[45]R.E.Ewing.The mathematics of reservoir simulation[M].Philadelphia:SIAM,1983.
[46]J.Douglas.Jr,R.E.Ewing and M.F.Wheeler.Approximation of the pressure by a mixed method in the simulation of miscible displacement[J].RAIRO.Anal.Numer.1983,17(1):17-33.
[47]J.Douglas.Jr,R.E.Ewing and M.F.Wheeler.A time-discretization procedure for a mixed method in the simulation of miscible displacement in porous media[J].RAIRO.Anal.Numer.1983,17(1):249-265.
[48]J.Douglas and T.F.Russell.Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures[J].SIAM.J.Numer.Anal.,1982,19(5):871-885.
[49]J.Douglas.JR.Finite difference methods for two-phase incompressible flow in Porous Media[J].SIAM.J.Numer.Anal.,1983,20(4):681-696.
[50]R.E.Ewing,T.F.Russell and M.F.Wheeler.Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics[J].Comp.Meth.Appl.Mech.Engrg.,1984,47:73-92.
[51]T.F.Russell.Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in Porous Media[J].SIAM.J.Numer.Anal.,1985,22(5):970-1013.
[52]C.N.Dawson,T.F.Russell and M.F.Wheeler.Some improved error estimates for the modified method of characteristics[J].SIAM.J.Numer.Anal.,1989,26:1487-1512.
[53]C.N.Dawson.Godunov-mixed methods for advection flow problems in one space dimension[J].SIAM.J.Numer.Anal.,1991,28:1282-1309.
[54]C.N.Dawson.Godunov-mixed methods for advection-diffusion equations in multidimensions[J].SIAM.J.Numer.Anal.,1993,30(5):1315-1332.
[55]M.A.Celia,T.F.Russell,I.Herrera,and R.E.Ewing.An Eulerian-Lagrangian local adjoint method for the advection-diffusion equation[J].Adv.Water Resour.,1990,13:187-206.
[56]R.W.Healy and T.F.Russell.A Finite-Volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation[J].Adv.Water Resour.,1993,29(7):2399-2413.
[57]J.Douglas.Jr,F.Pereira,and L.M.Yeh.A locally conservative Eulerian-Lagrangian numerical method and its application to nonlinear transport in porous media[J].Computational Geosciences,2000,4:1-40.
[58]Li R.Generalized difference methods for two-point boundary value problem.J.Jilin Univ.,1982,1:16-40
[59]Li R,Zhu P.Generalized difference methods for 2nd order elliptic partial differential equations (Ⅰ)[J].Numer.Math.J Chin.Univ.,1982,(2):140-151.
[60]Li R,Chen Z,and Wu W.The generalized difference method for differential equations-Numerical analysis of finite volume methods.Marcel Dikker Inc,New York,2000.35-60.
[61]Li R,Chen Z.Generalized difference methods for differential equations(in Chinese)[M].Jilin University Press,Changchun,1994.
[62]Li R.The Galerkin finite element methods for boundary value problems[M].Beijing:Science Press,2005.
[63]Li Ronghua.Generalized difference methods for a nonlinear Dirichlet problem[J],SIAM.J.Numer.Anal.1987,24(1):77-88.
[64]Yuan Yirang.Characteristic finite difference methods for positive semidefmite problem of two phase miscible flow in porous media[J].Systems Science and Mathematical Sciences,1999,12(4):299-306.
[65]J.W.Barrett and P.Knabner.Finite element approximation of the transport of reactive solutes in porous media part Ⅰ:Error estimates for noneqnilibrium adsorption processes[J].SIAM.J.Numer.Anal.,1997,34(1):201-227.
[66]J.W.Barrett and P.Knabner.Finite element approximation of the transport of reactive solutes in porous media part Ⅱ:Error estimates for equilibrium adsorption processes[J].SIAM.J.Numer.Anal.,1997,34(2):455-479.
[67]C.N.Dawson,C.J.Van.Duijn and M.F.Wheeler.Characteristic-Galerkin methods for contaminant transport with noneqnilibrium adsorption kinetics[J].SIAM.J.Numer.Anal.,1994,31(4):982-999.
[68]Clint Dawson.Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations[J].SIAM.J.Numer.Anal.,1998,35(5):1709-1724.
[69]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅰ:A linear model problem[J].SIAM.J.Numer.Anal.,1991,28(1):43-77.
[70]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅳ:Nonlinear problems[J].SIAM.J.Numer.Anal.,1995,32(5):1729-1749.
[71]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅴ:Longtime integration[J].SIAM.J.Numer.Anal,1995,32(5):1750-1763.
[72]R.Bank and R.Santos,Analysis of some moving space-time finite element methods[J].SIAM.J.Numer.Anal.,1993,30(1):1-18.
[73]T.F.Dupont,Mesh modification for evolution equations[J].Math.Comp.,1982,39:85-108.
[74]K.Miller and R.N.Miller,Moving finite elements Ⅰ[J].SIAM.J.Numer.Anal.,1981,18:1019-1032.
[75]K.Miller,Moving finite elements Ⅱ[J].SIAM.J.Numer.Anal.,1981,18:1033-1057.
[76]J.Douglns.JR and T.Dupont.Alternating direction Galerkin methods on rectangles[C].Proc.Symposium on Numerical Solution of Partial Differential Equations.B.Hubbard,ed,Academic Press,New York,1971,pp:133-214.
[77]J.E.Dendy.Jr.and G.Fairweather.Alternating-direction Galerkin methods for parabolic and hyperbolic problems on rectangular polygons[J].SIAM.J.Numer.Anal.1975,12(2):144-163.
[78]P.A.Raviart.and J.M.Thomas,A mixed finite element method for second order elliptic problems,in Mathematical Aspects of Finite Element Methods,Ⅰ.Galligani and E.Magenes,eds.,Lecture Notes in Math.606,Springer-Verlag,Berlin,1977,pp:292-315.
[79]Sun Tongjun.An ADI Galerkin method with moving finite element spaces for a class of secondorder hyperbolic equations[J].Numerical Mathematics,J.C.U.(Euglish Series),2001,10(1):45-58.
[80]J.Douglas.Jr,C.H.Hnang,and F.Pereira.The modified method of characteristics with adjusted adverction[J].Numer.Math.,1999,83:353-369.
[81]R.E.Ewing,Yirang Ynan and Gang Li.Time-stepping along characteristics for a mixed finite element approximation for compressible flow of contamination from nuclear waste in porous media[J].SIAM.J.Numer.Anal.,1989,26(6):1513-1534.
[82]R.A.Adams.Sobolev Space[M].New York,Academic Press.1975.
[83]Z.Q.Cai,J.Mandel and S.Mccormick.The finite volume element method for diffusion equations on general triangulations[J].SIAM.J.Numer.Anal.,1991,28(2):392-402.
[84]J.Jim Douglas and J.E.Roberts.Global estimates for mixed methods for second order elliptic equations[J].Math.Comp.,1985,44(1):39-52.
[85]T.Arbogast,M.Wheeler and I.Yotov.Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences[J].SIAM.J.Numer.Anal.,1997,34(2):828-852.
[86]Milos Zlamal.Finite element multistep discretization of parabolic boundary value problems[J].Mathematics of Computation,1975,29(130):350-359.
[87]J.H.Bramble et.Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data[J].Mathematics of Computation,1989,52(186):339-367.
[88]J.H.Bramble,R.E.Ewiug and Gang Li.Alternating direction multistep methods for parabolic problems iterative stabilization[J].SIAM.J.Numer.Anal.,1989,26(4):904-919.
[89]S.K.Godunov.A difference scheme for numerical computation of discontinuous solution of equations in fluid dynamics[J].Math.Sbornik,1959,47:271-306.
[90]D.Q.Yang,Mixed methods with dynamic finite-element space for miscible displacement in porous media[J].J.Comput.Appl.Math.,1990,30:313-328.
[91]D.Q.Yang,A characteristic mixed method with dynamic finite-element space for convectiondominated diffusion problems[J].J.Comput.Appl.Math.,1992,43:343-353.
[92]C.N.Dawson and R.Kirby.Solution of parabolic equations by backward euler-mixed finite element methods on a dynamically changing mesh[J].SIAM.J.Numer.Anal.2000,37(2):423-442.
[93]Yuan Yiraug and Wang Hong.The discrete-time finite dement methods for nonlinear hyperbolic equations and their theoretical analysis[J].Journal of Computational mathematics,1988,6(3):193-204.
[94]Z.Q.Cai and S.Mccormick.On the accuracy of the finite volume element method for diffusion equations on composite grids[J].SIAM.J Numer Anal,1990,27(3):636-655.
[95]S.H.Chou,D.Y.Kwak and P.S.Vassilevski.Mixed upwinding covolume methods on rectangular grids for convection-diffusion problems[J].SIAM.J.Sci.Comput.1999,21(1):145-165.
[96]S.H.Chou,D.Y.Kwak and P.S.Vassilevski.Mixed covolume methods for elliptic problems on triangular grids[J].SIAM.J.Numer.Anal.1998,35:1850-1861.
[97]H.Wang.An optimal-order error estimate for MMOC and MMOCAA schemes for multidimensional advection-reaction equations[J].Num.Meth.PDEs.2002,Vol.18,No.l:(69-84).
[2]Tikhonov A N.On solving incorrectly posed problems and method of regularition[J].Dokl.Acad.Nauk USSR,151(3).1963
[3]Feng K.The difference scheme based on variational principle(Chinese).J.Appl.Comp.1965,2(4):238-262.
[4]Ciarlet P G.The Finite Element Method for Elliptic Problems.Amsterdam:North-Holland,1978.
[5]Cheney M,Isaacson D,Newell J C.Electrical Impedance Tomography[J].SIAM Rev.,1999,41(1):85-101.
[6]Cheney M,Isaacson D,Newell J C,Simske S and Gogle J.Nosor:an algorithm for solving the inverse conductivity problem.Int.J.Imag.Syst.Technol.2,66-75.
[7]Bruhl M.Explicit characterization of inclusions in electrical impedance tomography[J].SIAM J.Math.Anal.,2001,32:1327-1341.
[8]Eric T.Chung,Tony F.Chan and Xue-Cheng Tai.Eelectrical Impedance Tomography using level set representation and total variational regularization[J].J.Comput.Phys.,2005,205:357-372.
[9]Breckon W R,Pidcock M K.Mathematical aspects of impedance imaging[J].Clin.Phys.Physiol.Mess.,1987,8(Suppl):77-84.
[10]Borcea L,Gray G A,Zhang Y.Variationally constrained numerical solution of electrical impedance tomography[J].Inverse Problems,2003,19(5):1159-1184.
[11]Lechleiter A,Kieder A.Newton regularizations for impedance tomography:a numerical study[J].Inverse Problems,2006,22:1967-1987.
[12]Du Y,Cheng J,Liu Z.Combined variable metric method for electrical impedance tomography problem[J].Chin.J.Biomed.Eng.(in Chinese),1997,16(2):167-173.
[13]Tamburrino A,Rubinacci G.A new non-iterative inversion for electrical resistance tomography [J].Inverse Problems,2002,18(6):1809-1829.
[14]Hanke M,Bruhl M.Recent progress in electrical impedance tomography[J].Inverse Problems,2003,19:S65-S90.
[15]Webster J G.Electrical Impedance Tomography[M].Adam Hilger,Bristol England,1990.
[16]Kirsch A.An introduction to the mathematical theory of inverse problems[M].Number 120 in Applied Mathematical Sciences.Springer-Verlag,New York,1996.
[17]Wang Yanfei.On the regularity of trust region-cg algorithm:With application to deconvolution problem.Science in Chian(Series A),2003,46(3):312-325.
[18]Xiao T,Yu S,Wang Y.Numerical solution for inverse problems(in Chinese)[M].Science Press,Beijing,2003.
[19]Wang Yanfei,Yuan Yaxiang.On the regularity of trust region algorithms for nonlinear ill-posed inverse problems,short communications at the International Congress of Mathematician,China,August22,2002.
[20]Mishev I D.Finite volume methods on Voronoi meshes[J].Numer.Methods P.D.E,1998,14:193-212.
[21]Yuan Y,Sun W.Theory and methods of Optimization(in Chinese)[M].Science Press,Beijing,1997.
[22]Hayes L J.Galerkin alternating-direction methods for nonrecutangular regions using patch approximations[J].SIAM J.Numer.Anal.,1981,18(4):627-643.
[23]Hayes L J.A modified backward time discretization for nonlinear parabolic equations using patch approximations[J].SIAM J.Numer.Anal.,1981,18(5):781-793.
[24]Li T,Qin T.Physics and partial differntial equations(in Chinese)[M].Advanced Education Press,Beijing,second edition,2005.
[25]Liu J.Regularization methods for ill-posed problems and its application(in Chinese)[M].Science Press,Beijing,2005.
[26]Jiang Lishang,Pang Zhihuan.The finite element method with its theoretical foundation[M].Beijing:People's Education Press,1979.
[27]S.Brenner and R.Scott,The Mathematical Theory of Finite element methods,Springer-Verlag,New York,1994.
[28]Calderon A P.On an inverse boundary value problem.Seminar on Numerical Analysis and Its Applications to Continuum Physics,Rio de Janeiro,1980,65-73.
[29]Sylvester J,Uhlmann G.A global uniqueness theorem for an inverse boundary value problem.Ann.Math.125(1987),153-169.
[30]Nachman A I.Global uniqueness for a two-dimensional inverse boundary problem.Ann.Math.1996,143:71-96
[31]Brown R M,Uhlmann G.Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions.Comm.Partial Differential Equations,1997,22(5-6):1009-1027.
[32]Hackbnsch W.:Elliptic Differential Equations:Theory and Numerical Treatment.Springer-Verlag,Berlin(1992).
[33]Isakov V.:Inverse problems for partial differential equations.Springer-Verlsg,New York(1998).
[34]Du Yah.A study on algorithms for electrical impedance tomography.Beijing:Doctorial Dis sertation of Beijing University of Aeronautics and Astronautics,1997.
[35]Quarteroni A.,Sacco R.and Saleri F.:Numerical Mathematics.Springer-Verlag,New York(2000).
[36]Stoer J.and Bulirsch R.:Introduction to Numerical Analysis.Springer-Verlag,New York(1993).
[37]Nocedal,J and Wright S J.Numerical Optimization.Springer Science+Business,Inc.1999
[38]R.D.Lazarov,I.D.Mishev and P.S.Vassilevski.Finite volume methods for convection-diffusion problems.SIAM J.Numer.Anal.33(1):31-55(1996).
[39]S.Larsson and V.Thomee,Partial Differential Equations with Numerical Methods.Springer-Veriag,Berlin,2003,pp.43-50.
[40]Russell T F,Wheeler M F.Finite element and finite difference methods for continuous flows in porous media.In:Ewing R E.The mathematics of reservoir simulation[M],SIAM Phialdelphia,1983
[41]Weiser A,Wheeler M F.On convergence of block-centered finite difference for elliptic problem[J].SIAM J Numer Anal,1988;25(2):351-375.
[42]MscNeal R H.An asymmetrical finite difference network[J].Quart.Appl.Math.1953,11:295-310.
[43]R.E.Ewing,M.F.Wheeler.Galerkin methods for miscible displacement problems in porous media[J].SIAM.J.Numer.Anal.,1980,17(2):351-365.
[44]R.E.Ewing and T.F.Russell.Efficient time-stepping methods for miacible displacement problem in Porous Media[J].SIAM.J.Numer.Anal.,1982,19(1):1-67.
[45]R.E.Ewing.The mathematics of reservoir simulation[M].Philadelphia:SIAM,1983.
[46]J.Douglas.Jr,R.E.Ewing and M.F.Wheeler.Approximation of the pressure by a mixed method in the simulation of miscible displacement[J].RAIRO.Anal.Numer.1983,17(1):17-33.
[47]J.Douglas.Jr,R.E.Ewing and M.F.Wheeler.A time-discretization procedure for a mixed method in the simulation of miscible displacement in porous media[J].RAIRO.Anal.Numer.1983,17(1):249-265.
[48]J.Douglas and T.F.Russell.Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures[J].SIAM.J.Numer.Anal.,1982,19(5):871-885.
[49]J.Douglas.JR.Finite difference methods for two-phase incompressible flow in Porous Media[J].SIAM.J.Numer.Anal.,1983,20(4):681-696.
[50]R.E.Ewing,T.F.Russell and M.F.Wheeler.Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics[J].Comp.Meth.Appl.Mech.Engrg.,1984,47:73-92.
[51]T.F.Russell.Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in Porous Media[J].SIAM.J.Numer.Anal.,1985,22(5):970-1013.
[52]C.N.Dawson,T.F.Russell and M.F.Wheeler.Some improved error estimates for the modified method of characteristics[J].SIAM.J.Numer.Anal.,1989,26:1487-1512.
[53]C.N.Dawson.Godunov-mixed methods for advection flow problems in one space dimension[J].SIAM.J.Numer.Anal.,1991,28:1282-1309.
[54]C.N.Dawson.Godunov-mixed methods for advection-diffusion equations in multidimensions[J].SIAM.J.Numer.Anal.,1993,30(5):1315-1332.
[55]M.A.Celia,T.F.Russell,I.Herrera,and R.E.Ewing.An Eulerian-Lagrangian local adjoint method for the advection-diffusion equation[J].Adv.Water Resour.,1990,13:187-206.
[56]R.W.Healy and T.F.Russell.A Finite-Volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation[J].Adv.Water Resour.,1993,29(7):2399-2413.
[57]J.Douglas.Jr,F.Pereira,and L.M.Yeh.A locally conservative Eulerian-Lagrangian numerical method and its application to nonlinear transport in porous media[J].Computational Geosciences,2000,4:1-40.
[58]Li R.Generalized difference methods for two-point boundary value problem.J.Jilin Univ.,1982,1:16-40
[59]Li R,Zhu P.Generalized difference methods for 2nd order elliptic partial differential equations (Ⅰ)[J].Numer.Math.J Chin.Univ.,1982,(2):140-151.
[60]Li R,Chen Z,and Wu W.The generalized difference method for differential equations-Numerical analysis of finite volume methods.Marcel Dikker Inc,New York,2000.35-60.
[61]Li R,Chen Z.Generalized difference methods for differential equations(in Chinese)[M].Jilin University Press,Changchun,1994.
[62]Li R.The Galerkin finite element methods for boundary value problems[M].Beijing:Science Press,2005.
[63]Li Ronghua.Generalized difference methods for a nonlinear Dirichlet problem[J],SIAM.J.Numer.Anal.1987,24(1):77-88.
[64]Yuan Yirang.Characteristic finite difference methods for positive semidefmite problem of two phase miscible flow in porous media[J].Systems Science and Mathematical Sciences,1999,12(4):299-306.
[65]J.W.Barrett and P.Knabner.Finite element approximation of the transport of reactive solutes in porous media part Ⅰ:Error estimates for noneqnilibrium adsorption processes[J].SIAM.J.Numer.Anal.,1997,34(1):201-227.
[66]J.W.Barrett and P.Knabner.Finite element approximation of the transport of reactive solutes in porous media part Ⅱ:Error estimates for equilibrium adsorption processes[J].SIAM.J.Numer.Anal.,1997,34(2):455-479.
[67]C.N.Dawson,C.J.Van.Duijn and M.F.Wheeler.Characteristic-Galerkin methods for contaminant transport with noneqnilibrium adsorption kinetics[J].SIAM.J.Numer.Anal.,1994,31(4):982-999.
[68]Clint Dawson.Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations[J].SIAM.J.Numer.Anal.,1998,35(5):1709-1724.
[69]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅰ:A linear model problem[J].SIAM.J.Numer.Anal.,1991,28(1):43-77.
[70]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅳ:Nonlinear problems[J].SIAM.J.Numer.Anal.,1995,32(5):1729-1749.
[71]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅴ:Longtime integration[J].SIAM.J.Numer.Anal,1995,32(5):1750-1763.
[72]R.Bank and R.Santos,Analysis of some moving space-time finite element methods[J].SIAM.J.Numer.Anal.,1993,30(1):1-18.
[73]T.F.Dupont,Mesh modification for evolution equations[J].Math.Comp.,1982,39:85-108.
[74]K.Miller and R.N.Miller,Moving finite elements Ⅰ[J].SIAM.J.Numer.Anal.,1981,18:1019-1032.
[75]K.Miller,Moving finite elements Ⅱ[J].SIAM.J.Numer.Anal.,1981,18:1033-1057.
[76]J.Douglns.JR and T.Dupont.Alternating direction Galerkin methods on rectangles[C].Proc.Symposium on Numerical Solution of Partial Differential Equations.B.Hubbard,ed,Academic Press,New York,1971,pp:133-214.
[77]J.E.Dendy.Jr.and G.Fairweather.Alternating-direction Galerkin methods for parabolic and hyperbolic problems on rectangular polygons[J].SIAM.J.Numer.Anal.1975,12(2):144-163.
[78]P.A.Raviart.and J.M.Thomas,A mixed finite element method for second order elliptic problems,in Mathematical Aspects of Finite Element Methods,Ⅰ.Galligani and E.Magenes,eds.,Lecture Notes in Math.606,Springer-Verlag,Berlin,1977,pp:292-315.
[79]Sun Tongjun.An ADI Galerkin method with moving finite element spaces for a class of secondorder hyperbolic equations[J].Numerical Mathematics,J.C.U.(Euglish Series),2001,10(1):45-58.
[80]J.Douglas.Jr,C.H.Hnang,and F.Pereira.The modified method of characteristics with adjusted adverction[J].Numer.Math.,1999,83:353-369.
[81]R.E.Ewing,Yirang Ynan and Gang Li.Time-stepping along characteristics for a mixed finite element approximation for compressible flow of contamination from nuclear waste in porous media[J].SIAM.J.Numer.Anal.,1989,26(6):1513-1534.
[82]R.A.Adams.Sobolev Space[M].New York,Academic Press.1975.
[83]Z.Q.Cai,J.Mandel and S.Mccormick.The finite volume element method for diffusion equations on general triangulations[J].SIAM.J.Numer.Anal.,1991,28(2):392-402.
[84]J.Jim Douglas and J.E.Roberts.Global estimates for mixed methods for second order elliptic equations[J].Math.Comp.,1985,44(1):39-52.
[85]T.Arbogast,M.Wheeler and I.Yotov.Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences[J].SIAM.J.Numer.Anal.,1997,34(2):828-852.
[86]Milos Zlamal.Finite element multistep discretization of parabolic boundary value problems[J].Mathematics of Computation,1975,29(130):350-359.
[87]J.H.Bramble et.Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data[J].Mathematics of Computation,1989,52(186):339-367.
[88]J.H.Bramble,R.E.Ewiug and Gang Li.Alternating direction multistep methods for parabolic problems iterative stabilization[J].SIAM.J.Numer.Anal.,1989,26(4):904-919.
[89]S.K.Godunov.A difference scheme for numerical computation of discontinuous solution of equations in fluid dynamics[J].Math.Sbornik,1959,47:271-306.
[90]D.Q.Yang,Mixed methods with dynamic finite-element space for miscible displacement in porous media[J].J.Comput.Appl.Math.,1990,30:313-328.
[91]D.Q.Yang,A characteristic mixed method with dynamic finite-element space for convectiondominated diffusion problems[J].J.Comput.Appl.Math.,1992,43:343-353.
[92]C.N.Dawson and R.Kirby.Solution of parabolic equations by backward euler-mixed finite element methods on a dynamically changing mesh[J].SIAM.J.Numer.Anal.2000,37(2):423-442.
[93]Yuan Yiraug and Wang Hong.The discrete-time finite dement methods for nonlinear hyperbolic equations and their theoretical analysis[J].Journal of Computational mathematics,1988,6(3):193-204.
[94]Z.Q.Cai and S.Mccormick.On the accuracy of the finite volume element method for diffusion equations on composite grids[J].SIAM.J Numer Anal,1990,27(3):636-655.
[95]S.H.Chou,D.Y.Kwak and P.S.Vassilevski.Mixed upwinding covolume methods on rectangular grids for convection-diffusion problems[J].SIAM.J.Sci.Comput.1999,21(1):145-165.
[96]S.H.Chou,D.Y.Kwak and P.S.Vassilevski.Mixed covolume methods for elliptic problems on triangular grids[J].SIAM.J.Numer.Anal.1998,35:1850-1861.
[97]H.Wang.An optimal-order error estimate for MMOC and MMOCAA schemes for multidimensional advection-reaction equations[J].Num.Meth.PDEs.2002,Vol.18,No.l:(69-84).