非饱和土中污染物传输参数反演方法研究
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摘要
以非饱和土中污染物非平衡传输参数反演问题为应用背景和出发点，对求解线性／非线性反问题的正则化方法进行了深入研究。论文主要层次如下：一，针对非饱和土中污染物非平衡传输参数反演问题，以增大求解的收敛范围为主要目标，将同伦方法引入该问题的求解，同时对同伦参数的意义进行分析，在此基础上提出了几种有效的同伦参数修正方法；二，在第一部分工作的基础上，将研究范围拓展至一般非线性反问题，以正则化泛函为主要研究目标，将D-函数作为正则化泛函引入非线性反问题的求解，拓展了正则化泛函选取范围，提出了一类新的求解非线性反问题的正则化方法；三，在前两部分工作的基础上，将研究范围进一步拓展至一般线性／非线性反问题，以极大极小问题替代传统的最小二乘问题表达反问题，并采用凝聚函数对目标函数进行光滑化，提出一类求解一般反问题的新方法；四，将前面三部分的工作从数学表达和程序实现两个方面进行总结，一方面将本文所提出的几种方法在数学表达上进行统一，给出一般的表达形式，另一方面将本文提出的几种方法在程序上进行模块化，开发出一个基于Matlab平台的软件包，即非线性反演工具箱；五，基于本文提出的方法，使用自行开发的软件包依据实际的实验数据对镉离子在不同非饱和土样中的非平衡传输参数进行反演计算，并与现有方法进行对比，通过求解实际问题验证了本文提出方法的有效性及本文所开发软件包的可靠性。本文具体章节安排如下：
第一章首先简要介绍本文的应用背景，提出本文研究的出发点和意义。之后对目前求解线性／非线性反问题的主要方法进行了归纳和总结，在此基础上确立本文的主要研究基点，即正则化方法。最后简要地介绍本文的主要内容和论文层次结构。
第二章对本文的应用背景，即土壤中污染物传输规律的相关知识进行详细介绍，明确界定了本文所主要针对的具体问题，即非饱和土中污染物非平衡传输参数反演问题。
第三章以增大求解的收敛域为主要目标，将同伦方法引入非饱和土中污染物非平衡传输参数反演问题的求解。同时针对反问题自身的特点，对同伦参数的意义进行分析，在此基础上提出了几种同伦参数修正方法，并分析比较了各自的特点。数值算例表明，同伦方法较一般的梯度正则化方法具有更大的收敛范围，并避免了后者正则化参数初值较难选取的难点。
非饱和土中污染物传输参数反演方法研究
第四章以正则化泛函为主要研究目标，通过对正则化泛函的意义进行分
析，将D一函数作为正则化泛函引入非线性反问题的求解，提出一类求解非线
性反问题的新方法，即基于D一函数的迭代正则化方法，有效地拓展了正则化
泛函的选取范围。同时基于三种经常使用的D一函数，给出了该方法的三种具
体形式，并将其与求解线性反问题的几种正则化方法相比较，明确了该方法的
意义。数值算例验证了该方法的可行性及对其进行深入研究的价值。
第五章将研究领域进一步拓展至一般线性/非线性反问题，以Minmax问题
替代最小二乘问题表达反问题，并使用凝聚函数对目标函数进行光滑化，提出
一类求解一般反问题的新方法，即基于Minmax一正则化方法。几个经典算例的
计算结果表明，较几种现有的方法而言，应用该方法可以得到更准确的反演结
果。
在第六章中，首先将本文提出的几种方法在数学表达上进行总结，归结为
一个一般的表达形式。其次介绍在本文方法研究的基础上，基于Matlab平台
所开发的非线性反演工具箱NIP。最后利用该工具箱，分别使用本文提出的几
种方法依据14组实际的实验数据，对锡离子在非饱和土中的非平衡传输参数
进行反演计算，并与梯度正则化方法所得的结果相比较，验证了本文提出方法
的有效性及非线性反演工具箱的可靠性。
第七章对全文进行总结，并在此基础上提出了几个可以依据本文内容进一
步开展的工作方向。
附录A给出了本文使用的实验结果。
附录B对非线性反演工具箱NIP中的主要函数进行了说明。
附录C以实例的形式对非线性反演工具箱的使用方法进行了简要说明。
Nonlinear inverse problem arises from many areas of scientific and engineering applications. Up to date, various methods have been applied to solving this problem, of which iterative regularization method is the frequently used one. This dissertation takes the parameter inversion problem of contaminant transport through unsaturated soils as its platform and is mainly devoted to improve iterative regularization methods for solving nonlinear inverse problems. The emphases are focused on the following aspects: widen the convergence region, efficiently update the homotopy parameter, choice the appropriate regularization functional as well as test the other error function. Consequently, some efficient methods are proposed and a toolbox for solving general inverse problems is developed, in particular it has been successfully applied to solving the inverse problem of parameter inversion of contaminant transport through unsaturated soils.
Main contents of this dissertation are as follows:
In chapter 1, the background of this dissertation is introduced and some existing methods for solving linear/nonlinear inverse problems are surveyed. Main research areas and structure of this dissertation are discussed.
The background of this dissertation, named the non-equilibrium process of contaminant transport through unsaturated soils, is introduced in chapter 2. Taking the transport process of Cadmium through unsaturated soils as the example, corresponding mathematical models and laboratory soil-column testing method is introduced as well.
In chapter 3, the aim is targeted at widening the convergence region of the method. To this end, a homotopy method is employed for solving the inverse problem of parameter inversion of contaminant transport. At the same time, the effect of the homotopy parameter is studied and some efficient methods for updating the homotopy parameter are presented.
In chapter 4, the effect of regularization functional is studied. By employing the D-Function as the regularization functional, a new method, named HLB method for solving nonlinear inverse problems is presented and applied to solving
the inverse problem of parameter inversion of contaminant transport through unsaturated soils. The feasibility of the HLB method is justified with numerical computations.
In chapter 5, the nonlinear least square problem is replaced by a minmax problem in the formulation of inverse problem, and the HMB method is presented in which the aggregate function is used to smoothen the infinity norm and the D-Function and homotopy methods are also incorporated.
In chapter 6, a toolbox for solving nonlinear inverse problems, named NIP is developed. Methods discussed above are all integrated in NIP toolbox.
The last chapter gives a summary of the dissertation and some possible extensions of the present work.
Appendix A contains the experimental results and fitting curves referenced by this dissertation.
Appendix B briefly introduces the functionality of NIP toolbox.
Appendix C is a tutorial of NIP toolbox.
第一章首先简要介绍本文的应用背景，提出本文研究的出发点和意义。之后对目前求解线性／非线性反问题的主要方法进行了归纳和总结，在此基础上确立本文的主要研究基点，即正则化方法。最后简要地介绍本文的主要内容和论文层次结构。
第二章对本文的应用背景，即土壤中污染物传输规律的相关知识进行详细介绍，明确界定了本文所主要针对的具体问题，即非饱和土中污染物非平衡传输参数反演问题。
第三章以增大求解的收敛域为主要目标，将同伦方法引入非饱和土中污染物非平衡传输参数反演问题的求解。同时针对反问题自身的特点，对同伦参数的意义进行分析，在此基础上提出了几种同伦参数修正方法，并分析比较了各自的特点。数值算例表明，同伦方法较一般的梯度正则化方法具有更大的收敛范围，并避免了后者正则化参数初值较难选取的难点。
非饱和土中污染物传输参数反演方法研究
第四章以正则化泛函为主要研究目标，通过对正则化泛函的意义进行分
析，将D一函数作为正则化泛函引入非线性反问题的求解，提出一类求解非线
性反问题的新方法，即基于D一函数的迭代正则化方法，有效地拓展了正则化
泛函的选取范围。同时基于三种经常使用的D一函数，给出了该方法的三种具
体形式，并将其与求解线性反问题的几种正则化方法相比较，明确了该方法的
意义。数值算例验证了该方法的可行性及对其进行深入研究的价值。
第五章将研究领域进一步拓展至一般线性/非线性反问题，以Minmax问题
替代最小二乘问题表达反问题，并使用凝聚函数对目标函数进行光滑化，提出
一类求解一般反问题的新方法，即基于Minmax一正则化方法。几个经典算例的
计算结果表明，较几种现有的方法而言，应用该方法可以得到更准确的反演结
果。
在第六章中，首先将本文提出的几种方法在数学表达上进行总结，归结为
一个一般的表达形式。其次介绍在本文方法研究的基础上，基于Matlab平台
所开发的非线性反演工具箱NIP。最后利用该工具箱，分别使用本文提出的几
种方法依据14组实际的实验数据，对锡离子在非饱和土中的非平衡传输参数
进行反演计算，并与梯度正则化方法所得的结果相比较，验证了本文提出方法
的有效性及非线性反演工具箱的可靠性。
第七章对全文进行总结，并在此基础上提出了几个可以依据本文内容进一
步开展的工作方向。
附录A给出了本文使用的实验结果。
附录B对非线性反演工具箱NIP中的主要函数进行了说明。
附录C以实例的形式对非线性反演工具箱的使用方法进行了简要说明。
Nonlinear inverse problem arises from many areas of scientific and engineering applications. Up to date, various methods have been applied to solving this problem, of which iterative regularization method is the frequently used one. This dissertation takes the parameter inversion problem of contaminant transport through unsaturated soils as its platform and is mainly devoted to improve iterative regularization methods for solving nonlinear inverse problems. The emphases are focused on the following aspects: widen the convergence region, efficiently update the homotopy parameter, choice the appropriate regularization functional as well as test the other error function. Consequently, some efficient methods are proposed and a toolbox for solving general inverse problems is developed, in particular it has been successfully applied to solving the inverse problem of parameter inversion of contaminant transport through unsaturated soils.
Main contents of this dissertation are as follows:
In chapter 1, the background of this dissertation is introduced and some existing methods for solving linear/nonlinear inverse problems are surveyed. Main research areas and structure of this dissertation are discussed.
The background of this dissertation, named the non-equilibrium process of contaminant transport through unsaturated soils, is introduced in chapter 2. Taking the transport process of Cadmium through unsaturated soils as the example, corresponding mathematical models and laboratory soil-column testing method is introduced as well.
In chapter 3, the aim is targeted at widening the convergence region of the method. To this end, a homotopy method is employed for solving the inverse problem of parameter inversion of contaminant transport. At the same time, the effect of the homotopy parameter is studied and some efficient methods for updating the homotopy parameter are presented.
In chapter 4, the effect of regularization functional is studied. By employing the D-Function as the regularization functional, a new method, named HLB method for solving nonlinear inverse problems is presented and applied to solving
the inverse problem of parameter inversion of contaminant transport through unsaturated soils. The feasibility of the HLB method is justified with numerical computations.
In chapter 5, the nonlinear least square problem is replaced by a minmax problem in the formulation of inverse problem, and the HMB method is presented in which the aggregate function is used to smoothen the infinity norm and the D-Function and homotopy methods are also incorporated.
In chapter 6, a toolbox for solving nonlinear inverse problems, named NIP is developed. Methods discussed above are all integrated in NIP toolbox.
The last chapter gives a summary of the dissertation and some possible extensions of the present work.
Appendix A contains the experimental results and fitting curves referenced by this dissertation.
Appendix B briefly introduces the functionality of NIP toolbox.
Appendix C is a tutorial of NIP toolbox.
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