一些反问题的数值解法研究
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摘要
自20世纪60年代以来,在地球物理,生命科学,材料科学,模式识别,遥感技术,信号图像处理,工业控制,乃至经济决策等众多的科学技术领域中,都提出了“由效果,表现(输出)反求原因,原像(输入)”的反问题,我们把它们通称为“数学物理反问题”.由于绝大多数的反问题无法解析求解,数值方法对反问题的研究起着根本性的作用.由于此类问题的理论和算法具有极大的挑战性,又有着广泛而重要的应用背景,因此吸引了许多学者从事该项研究.近三十年来,数学物理反问题是应用数学中发展和成长最快的领域之一.
反问题的研究内容非常广泛,本文将着重研究三种反问题的数值解法. 1)拉普拉斯变换的数值反演.对于这种情况,我们关心的是光滑的解. 2)第一类Fredholm积分方程的分段常数解的数值解法,这是一个线性的不适定问题.3) Robin反问题的数值解法,我们考虑Robin系数为分段常数的情形,这是一个非线性的不适定问题.
本文分为四章,主要内容如下:
第一章将介绍反问题的一些基础知识,包括问题的适定与不适定等基本概念,并对第一类算子方程(特别是第一类积分方程)的求解方法作简要的叙述.介绍了正则化方法的原理,正则算子的构造,正则解的误差估计和正则化参数的确定方法.本章最后将介绍一些关于Robin反问题的背景知识和基本理论.
在第二章,我们提出了拉普拉斯变换反演的一种基于高阶数值积分的算法.通过选择适当的求积方法和样本点,得到误差足够小的离散方程组.再对该线性方程组引进正则化算子以控制解的光滑性,并对正则化后的问题选定适当的数值解法,我们得到一个高精度的数值方法.数值结果表明,利用该章提出的算法来对拉普拉斯变换反演进行近似,可以得到非常精确的数值结果.
在第三章,我们考虑第一类Freldholm积分方程的分段常数解的数值方法.我们假定解取k不同的值,并分别就解的取值给定和待定两种情况建立了修正的Tikhonov–TV正则化方法.我们对Tikhonov–TV泛函用二次泛函进行逼近,从而得到修正牛顿法.数值结果表明:在分段值的数目和分段的数量比较小时,通过本章提出的数值方法可以取得比较好的结果,可以比较准确的找出断点位置和各段的取值.
在第四章,我们将讨论Robin系数为分段常数的Robin反问题的数值解法,这是一个非线性的反问题.我们的方法建立在[ F. Lin and W. Fang,Inverse Problems, 21 (2005), pp. 1757–1772]的基础上:先把该问题转化为边界积分方程问题,再引进一个新的函数把问题线性化.与第三章相似,我们考虑两种情况.第一种情况是已知解有k个不同的取值{c1,c2,...,ck}且ci (i =1,2,...,k)已给定.对于这种情况,我们把这个信息加入到Tikhonov泛函中,建立变形-Modica–Mortola-泛函(简称变形-MM-泛函),然后考虑该泛函的正定的二次逼近,从而得到修正牛顿法.第二个情况是只知道Robin系数是分段常数及其上界.对于这种情况,我们采用Tikhonov–TV正则化方法.类似地,我们考虑Tikhonov–TV泛函的正定的二次逼近,然后用系列二次规划方法求解该问题.数值结果表明,应用这一章提出的算法,我们可以较好地估计分段常数Robin系数.
Since the 1960s, inverse problems for“tracing the reason or preimage (in-put) from the e?ect or representation (output)”have appeared in many fieldsin science and technology, such as geophysics, life sciences, material science,remote sensing technique, signal and image processing, industrial control, andeven economic decision. We call such inverse problems mathematical physicsinverse problems. Since most inverse problems cannot be solved analytically,numerical solution methods play a fundamental role. Many scholars have beenattracted to the field of inverse problems because the theories and algorithmsof such problems are of great challenging, and the applications are widespread.In the last few decades, the field of researches in mathematical physics inverseproblems has been one of the fastest developing field.
The contents of the field in inverse problems are widespread. In this thesiswe will mainly discuss three inverse problems. 1) The numerical inversion ofLaplace transform. In this case, we are concerned with smooth solutions. 2) Nu-merical solution methods for piecewise constant solutions of Fredholm integralequations of the first kind. This is an ill-posed linear problem. 3) Numerical so-lution methods for the Robin inverse problem. We are concerned with piecewiseconstant Robin coe?cients. This is an ill-posed nonlinear problem.
This thesis consists of four chapters. The main contents are as follow:
In Chapter one, we give some introductory materials and some back groundabout inverse problems, including the well-posedness and ill-posedness. We alsointroduce some numerical methods for the Fredholm integral equation of the firstkind, including the idea and construction of regularization operator, the errorestimation of regularized solution and the determination of regularization pa-rameter. In the last section of this chapter, we will introduce some backgroundabout the Robin inverse problem.
In Chapter two, we propose numerical algorithms for inversion of Laplace transform based on high order numerical quadratures. It is shown that bychoosing suitable quadrature rules and sample points, small discretization er-rors can be guaranteed. Thus, by applying a suitable regularization to thelinear system, a numerical solution of high accuracy can be found. Numericalresults show that the approximate inverse Laplace transform obtained by thealgorithms proposed in this paper can be very accurate.
In Chapter three, we consider numerical solutions for Fredholm integralequations of the first kind with piecewise constant solutions. We assume thatthe true solution has k di?erent function values. We consider the cases whichthe function values are known or unknown respectively and propose a modifiedTikhonov-TV regularization method. The modified Tikhonov-TV functionalis then approximated by a positive quadratic functional and thus we derivean approximate modified Newton method. Numerical results show that if thenumber of function values and the number of discontinuous points are small, wecan obtain quite good numerical results. That is, we can recover the functionvalues and the discontinuous points accurately.
In Chapter four, we consider the Robin inverse problem where the Robin co-e?cient is piecewise constant. This is a nonlinear inverse problem. Our methodsare based on [F. Lin and W. Fang, Inverse Problems, 21 (2005), pp. 1757–1772]:transform the problem into a boundary value integral equation first and thenintroduce a new function to transform the nonlinear problem into a linear one.As in Chapter three, we also consider two cases. In the first case, we assumethat the Robin coe?cient takes k di?erent values {c1,c2,...,ck} with knownci (i = 1,2,...,k). In this case, we insert this information into the Tikhonovfunctional to build up a variant Modica-Mortola-functional (or variant MM-functional for short), and approximate the functional by a positive quadraticfunctional. In this way, we obtain a modified approximate Newton method. Inthe second case, we only know that the Robin coe?cient is piecewise constantand the bound of the coe?cient. For this case, we consider using Tikhonov-TVagain. Similarly, we approximate the functional a positive quadratic functional.Then we derive a sequence quadratic programming algorithm for the problem.Numerical results show that we can estimate the piecewise constant Robin co-e?cient quite well by using the algorithms proposed in this chapter.
反问题的研究内容非常广泛,本文将着重研究三种反问题的数值解法. 1)拉普拉斯变换的数值反演.对于这种情况,我们关心的是光滑的解. 2)第一类Fredholm积分方程的分段常数解的数值解法,这是一个线性的不适定问题.3) Robin反问题的数值解法,我们考虑Robin系数为分段常数的情形,这是一个非线性的不适定问题.
本文分为四章,主要内容如下:
第一章将介绍反问题的一些基础知识,包括问题的适定与不适定等基本概念,并对第一类算子方程(特别是第一类积分方程)的求解方法作简要的叙述.介绍了正则化方法的原理,正则算子的构造,正则解的误差估计和正则化参数的确定方法.本章最后将介绍一些关于Robin反问题的背景知识和基本理论.
在第二章,我们提出了拉普拉斯变换反演的一种基于高阶数值积分的算法.通过选择适当的求积方法和样本点,得到误差足够小的离散方程组.再对该线性方程组引进正则化算子以控制解的光滑性,并对正则化后的问题选定适当的数值解法,我们得到一个高精度的数值方法.数值结果表明,利用该章提出的算法来对拉普拉斯变换反演进行近似,可以得到非常精确的数值结果.
在第三章,我们考虑第一类Freldholm积分方程的分段常数解的数值方法.我们假定解取k不同的值,并分别就解的取值给定和待定两种情况建立了修正的Tikhonov–TV正则化方法.我们对Tikhonov–TV泛函用二次泛函进行逼近,从而得到修正牛顿法.数值结果表明:在分段值的数目和分段的数量比较小时,通过本章提出的数值方法可以取得比较好的结果,可以比较准确的找出断点位置和各段的取值.
在第四章,我们将讨论Robin系数为分段常数的Robin反问题的数值解法,这是一个非线性的反问题.我们的方法建立在[ F. Lin and W. Fang,Inverse Problems, 21 (2005), pp. 1757–1772]的基础上:先把该问题转化为边界积分方程问题,再引进一个新的函数把问题线性化.与第三章相似,我们考虑两种情况.第一种情况是已知解有k个不同的取值{c1,c2,...,ck}且ci (i =1,2,...,k)已给定.对于这种情况,我们把这个信息加入到Tikhonov泛函中,建立变形-Modica–Mortola-泛函(简称变形-MM-泛函),然后考虑该泛函的正定的二次逼近,从而得到修正牛顿法.第二个情况是只知道Robin系数是分段常数及其上界.对于这种情况,我们采用Tikhonov–TV正则化方法.类似地,我们考虑Tikhonov–TV泛函的正定的二次逼近,然后用系列二次规划方法求解该问题.数值结果表明,应用这一章提出的算法,我们可以较好地估计分段常数Robin系数.
Since the 1960s, inverse problems for“tracing the reason or preimage (in-put) from the e?ect or representation (output)”have appeared in many fieldsin science and technology, such as geophysics, life sciences, material science,remote sensing technique, signal and image processing, industrial control, andeven economic decision. We call such inverse problems mathematical physicsinverse problems. Since most inverse problems cannot be solved analytically,numerical solution methods play a fundamental role. Many scholars have beenattracted to the field of inverse problems because the theories and algorithmsof such problems are of great challenging, and the applications are widespread.In the last few decades, the field of researches in mathematical physics inverseproblems has been one of the fastest developing field.
The contents of the field in inverse problems are widespread. In this thesiswe will mainly discuss three inverse problems. 1) The numerical inversion ofLaplace transform. In this case, we are concerned with smooth solutions. 2) Nu-merical solution methods for piecewise constant solutions of Fredholm integralequations of the first kind. This is an ill-posed linear problem. 3) Numerical so-lution methods for the Robin inverse problem. We are concerned with piecewiseconstant Robin coe?cients. This is an ill-posed nonlinear problem.
This thesis consists of four chapters. The main contents are as follow:
In Chapter one, we give some introductory materials and some back groundabout inverse problems, including the well-posedness and ill-posedness. We alsointroduce some numerical methods for the Fredholm integral equation of the firstkind, including the idea and construction of regularization operator, the errorestimation of regularized solution and the determination of regularization pa-rameter. In the last section of this chapter, we will introduce some backgroundabout the Robin inverse problem.
In Chapter two, we propose numerical algorithms for inversion of Laplace transform based on high order numerical quadratures. It is shown that bychoosing suitable quadrature rules and sample points, small discretization er-rors can be guaranteed. Thus, by applying a suitable regularization to thelinear system, a numerical solution of high accuracy can be found. Numericalresults show that the approximate inverse Laplace transform obtained by thealgorithms proposed in this paper can be very accurate.
In Chapter three, we consider numerical solutions for Fredholm integralequations of the first kind with piecewise constant solutions. We assume thatthe true solution has k di?erent function values. We consider the cases whichthe function values are known or unknown respectively and propose a modifiedTikhonov-TV regularization method. The modified Tikhonov-TV functionalis then approximated by a positive quadratic functional and thus we derivean approximate modified Newton method. Numerical results show that if thenumber of function values and the number of discontinuous points are small, wecan obtain quite good numerical results. That is, we can recover the functionvalues and the discontinuous points accurately.
In Chapter four, we consider the Robin inverse problem where the Robin co-e?cient is piecewise constant. This is a nonlinear inverse problem. Our methodsare based on [F. Lin and W. Fang, Inverse Problems, 21 (2005), pp. 1757–1772]:transform the problem into a boundary value integral equation first and thenintroduce a new function to transform the nonlinear problem into a linear one.As in Chapter three, we also consider two cases. In the first case, we assumethat the Robin coe?cient takes k di?erent values {c1,c2,...,ck} with knownci (i = 1,2,...,k). In this case, we insert this information into the Tikhonovfunctional to build up a variant Modica-Mortola-functional (or variant MM-functional for short), and approximate the functional by a positive quadraticfunctional. In this way, we obtain a modified approximate Newton method. Inthe second case, we only know that the Robin coe?cient is piecewise constantand the bound of the coe?cient. For this case, we consider using Tikhonov-TVagain. Similarly, we approximate the functional a positive quadratic functional.Then we derive a sequence quadratic programming algorithm for the problem.Numerical results show that we can estimate the piecewise constant Robin co-e?cient quite well by using the algorithms proposed in this chapter.
引文
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[2]苏超伟,偏微分方程逆问题的数值解法及其应用,西北工业大学出版社,1995.
[3]袁亚湘,非线性优化计算方法,科技出版社, 2008.
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[2]苏超伟,偏微分方程逆问题的数值解法及其应用,西北工业大学出版社,1995.
[3]袁亚湘,非线性优化计算方法,科技出版社, 2008.
[4]张光澄,黄世萤,侯泽华,最优化计算方法,成都科技大学出版社, 1989.
[5] G. Alessandrini, L. Del Piero, and L. Rondi, Stable determination of cor-rosion by single electrostatic boundary measurement, Inverse Problems, 19(2003), pp. 973–984.
[6] H. C. Andrews and B. R. Hunt, Digital Image Restoration, EnglewoodCli?s, NJ: Prentice Hall, 1977.
[7] J. Ahn, S. Kang, and Y. H. Kwon, A ?exible inverse Laplace transformalgorithm and its application, Computing 71 (2003), pp. 115–131.
[8] K. E. Atkinson, The Numerical Solution of Integral Equations of SecondKind, Cambridge University Press, Cambridge, 1997.
[9] C. T. H. Baker, The Numerical Treannent of Integral Equations, Oxford:Clarendon, 1977.
[10] A. Baumeister, Stable Solution of Invers Problems, Vieweg-verlag,Brauschweig, 1987.
[11] P. Brianzi, A criterion for the choice of a sampling parameter in the prob-lem of Laplace transform inversion, Inverse Problem, 10 (1994), pp. 55–61.
[12] M. Burger and S. J. Osher, A survey on level set methods for inverseproblems and optimal design, European J. Appl. Math., 16 (2005), pp. 263–301.
[13] R. Campagna, L. D’Amore, and A. Murli, An e?cient algorithm for regu-larization of Laplace transform inversion in real case, Journal of Compu-tational and Applied Mathematics, 210 (2007), pp. 84–98.
[14] S. Chaabane, I. Fellah, M. Jaoua, and J. Leblond, Logarithmic stabilityestimates for a Robin coe?cient in two-dimensional Laplace inverse prob-lems, Inverse Problems, 20 (2004), pp.47-59.
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