抛物型偏微分方程中未知区域重构的反问题及其算法
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摘要
在论文中,我们主要讨论介质的无损测试,在实际中,我们可以通过声波,电磁波,电流和热流来确定介质中的裂缝,空腔以及内含物。在本文中,我们在介质的边界上施加热流,并测量相应边界的温度,由此来重构热导体介质中的内含物。
第一章,我们介绍了所研究的问题的背景,主要来自于介质的无损测试,并且,我们给出了相关的主要结论。
第二章,我们给出了基本的空间的定义,以及其相关的性质。并且我们还介绍了正则化方法,它将应用于我们的数值计算中。
第三章,在本章中,我们给出了热方程的数学模型,以及热方程相关的一些基本的性质。最后我们利用有限元方法,给出了热方程正问题的数值解法。
第四章,我们将应用探针法,从Neumann-to-Dirichlet映射来确定同向同性介质中的内含物。简单的讲,探针法就是基于探针来构造一个指示函数I,当探针接触到未知内含物的边界时,指示函数将会爆破。利用这种性质,我们可以重构得到内部未知内含物的边界。对于一维和二维的热方程,我们将证明这种爆破的性质,并且我们给出了探测法的数值实现。
第五章,我们将介绍在热成像中的内含物的大小的估计方法,在热成像中,我们可以在边界上施加热流,然后再测的边界上的温度,利用这些边界测量的数据,我们将给出完全包含在Q中内含物D的大小的估计。并且,通过选择两种不同的边界条件,我们进行数值测试,数值的结果表明我们的方法准确而有效。
In this thesis, we discuss a mathematical study of non-destructive testing of media. Usually, acoustic waves, electromagnetic waves, electric current and heat are used for the non-destructive testing to find cracks, cavities and inclusions inside the medium. We will mainly focus on the non-destructive testing of an unknown inclusion of heat conductor by ejecting heat from the boundary of the medium and measuring the temperature on the boundary.
In Chapter 1, we present the background of our problems, which originated from the study of non-destructive testing of media. Finally, we also introduce the some main inclusions about this.
In Chapter 2, we introduce some definitions of basic space and their relational properties. And, we also introduce the regularization method, which will be used in our numerical realization.
In Chapter 3, in this part, we introduce the mathematic model for heat equation, and list some basic properties for heat equation. Finally, we give the algorithm for the numerical solution of heat equation, by using finite element method.
In Chapter 4, we consider an inverse problem for identifying an inclusion inside an isotropic homogeneous heat conductive medium from Neumann-to-Dirichlet map, by using probe method. Roughly speaking, this method is to construct a function I called indicator function based on a needle, when the needle approaches boundary of inclusion, the indicator function will blow up, using this property, we can reconstruct the boundary of the inclusion. For one and two dimensional space case, we will show the blow-up properties and carry out numerical implementation of this method.
In Chapter 5, we present the size estimate approach in the thermal imaging problem, which consists of applying a heat flux to the surface of a medium and observing the temperature over time. We estimate the Lebesgue measure |D| of inclusion in term of the boundary data provided D is compactly contained inΩ, and also perform our numerical tests with two different choice of boundary data to check how this effects the accuracy of our results.
第一章,我们介绍了所研究的问题的背景,主要来自于介质的无损测试,并且,我们给出了相关的主要结论。
第二章,我们给出了基本的空间的定义,以及其相关的性质。并且我们还介绍了正则化方法,它将应用于我们的数值计算中。
第三章,在本章中,我们给出了热方程的数学模型,以及热方程相关的一些基本的性质。最后我们利用有限元方法,给出了热方程正问题的数值解法。
第四章,我们将应用探针法,从Neumann-to-Dirichlet映射来确定同向同性介质中的内含物。简单的讲,探针法就是基于探针来构造一个指示函数I,当探针接触到未知内含物的边界时,指示函数将会爆破。利用这种性质,我们可以重构得到内部未知内含物的边界。对于一维和二维的热方程,我们将证明这种爆破的性质,并且我们给出了探测法的数值实现。
第五章,我们将介绍在热成像中的内含物的大小的估计方法,在热成像中,我们可以在边界上施加热流,然后再测的边界上的温度,利用这些边界测量的数据,我们将给出完全包含在Q中内含物D的大小的估计。并且,通过选择两种不同的边界条件,我们进行数值测试,数值的结果表明我们的方法准确而有效。
In this thesis, we discuss a mathematical study of non-destructive testing of media. Usually, acoustic waves, electromagnetic waves, electric current and heat are used for the non-destructive testing to find cracks, cavities and inclusions inside the medium. We will mainly focus on the non-destructive testing of an unknown inclusion of heat conductor by ejecting heat from the boundary of the medium and measuring the temperature on the boundary.
In Chapter 1, we present the background of our problems, which originated from the study of non-destructive testing of media. Finally, we also introduce the some main inclusions about this.
In Chapter 2, we introduce some definitions of basic space and their relational properties. And, we also introduce the regularization method, which will be used in our numerical realization.
In Chapter 3, in this part, we introduce the mathematic model for heat equation, and list some basic properties for heat equation. Finally, we give the algorithm for the numerical solution of heat equation, by using finite element method.
In Chapter 4, we consider an inverse problem for identifying an inclusion inside an isotropic homogeneous heat conductive medium from Neumann-to-Dirichlet map, by using probe method. Roughly speaking, this method is to construct a function I called indicator function based on a needle, when the needle approaches boundary of inclusion, the indicator function will blow up, using this property, we can reconstruct the boundary of the inclusion. For one and two dimensional space case, we will show the blow-up properties and carry out numerical implementation of this method.
In Chapter 5, we present the size estimate approach in the thermal imaging problem, which consists of applying a heat flux to the surface of a medium and observing the temperature over time. We estimate the Lebesgue measure |D| of inclusion in term of the boundary data provided D is compactly contained inΩ, and also perform our numerical tests with two different choice of boundary data to check how this effects the accuracy of our results.
引文
[1] S. R. Arridge, Optical tomography in medical imaging[J]. Inverse Problem, 1999,15: 41-93.
[2] G. Alessandrini, A. Bilotta, G. Formica, A. Morassi, E. Rosset, E Turco. Computing volume bounds on inclusions by EIT measurements [J]. J. Sci. Comput, 2007, 33: 293-312.
[3] G. Alesandrini, A. Morassi, E. Rosset. Size estimates [J]. Inverse problems: theory and applications (Cortona/Pisa, 2002), 1-33.
[4] G. Alesandrini, E. Rosset. The inverse conductivity problem with one measurement: bounds on the size of the unknown object [J]. SIAM J. Appl. Math, 1998, 58: 1060-1071.
[5] G. Alesandrini, E. Rosset. Volume bounds of inclusions from physical EIT measurements [J]. Inverse Problems, 2004, 20: 575-588.
[6] G. Alesandrini, A. Rosset, J.K. Seo. Optimal size estimates for the inverse conductivity problem with one measurement [J]. Proc. Amer. Math. Soc, 2000, 128: 53-64.
[7] G. Alessandrini, V. Isakov. Analyticity and uniqueness for the inverse conductivity problem [J]. Rend. Istit. Mat. Univ. Trieste, 1996, 28: 351-369.
[8] G. Alessandrini, M. Di Cristo. Stable determination of an inclusion by boundary measurements [J]. SIAM J. Math. Anal., 2005, 37: 200-217.
[9] K. Bryan. Single measurement detection of a discontinuous conductivity [J]. Comm. Partial Differential Equations, 1990, 15: 503-514.
[10] M. Di Cristo, L. Rondi, S. Vessella. Stability properties of an inverse parabolic problem with unknown boundaries [J]. Ann. Mat. Pura Appl, 2006, 185: 223-255.
[11] J. R. Cannon. The one-dimensional heat equation [M]. Addison-Wesley Publishing Company, 1984.
[12]陈恕行.现代偏微分方程导论[M].北京：科学出版社,2005.
[13] J Cheng, J J Liu, G Nakamyra. The numerical realization of probe method for the inverse scattering problems from the near-field data [J]. Inverse Problem, 2005, 21: 839-855.
[14] Y. Daido, H. Kang, G. Nakamura. A probe method for the inverse boundary value problem of non-stationary heat equations[J]. Inverse Problem, 2007,23: 1787-1800.
[15] K. Erhard, R. Potthast. A numerical study of the probe method [J]. Society for Industrial and applied Mathematics, 2006, 28: 1597-1612.
[16] A. Friedman. Partial Differential Equations of Parabolic Type[M]. INC: Prentice-Hall, 1964.
[17]A.Friedman.Detection of mines by electric measurements.SIAM J.Appl.Math,1987,47:201-212.
[18]P.C.Hansen.Regularization tools:a Matlab package for analysis and solution of discrete ill-posed problems[J].Numer.Algorithms,1994,6:1-35.
[19]M.Ikehata.Reconstruction of the shape of the inclusion by boundary measurements[J].Commun.PDE,1998,23:1459-1474.
[20]M.Ikehata.Reconstruction of inclusion from boundary measurements[J].J.Inverse and Ill-Posed Problems,2002,10:37-65.
[21]V.Isakov,K.Kim,G.Nakamura.Reconstruction of an unknown inclusion by thermal imaging.preprint.
[22]V.Isakov.Inverse problems for partial differential equations.Second edition[M].New York:Springer,2006.
[23]V.Isakov.On uniqueness of recovery of a discontinuous conductivity[J].Pure Appl.Math.,1988,41:865-877.
[24]J.J.Liu,J.Cheng and G.Nakamura.Reconstruction and uniqueness of the inverse scattering problem with impedance boundary[J].Sci.China,2002,45:1408-1419.
[25]J.J.Liu.Determination of Dirichlet-to-Neumann map for mixed boundary value problem[J].Appl.Math.Comput,2005,161:843-864.
[26]刘继军.不适定问题的正则化方法及应用[M].北京:科学出版社,2005.
[27]D.Issacson Distinguishability of conductivities by electric current computed tomography[J].IEEE TransMed Imagin,1986,5(1):91-95.
[28]P.K.Lamm,Zhewei Dai.On local regularization methods for linear Volterra equations and nonlinear equations of Hammerstein type[J].Inverse Problem,2005,21:1773-1790.
[29]O.A.Ladyzenskaja,V.A.Solonnikov,N.N.Uralceva.Linear and Quasi-linear Equations of Parabolic Type[M].American Mathematical Society,1968.
[30]H.Kang,J..Seo,D.Sheen.The inverse conductivity problem with one measurement:stability and estimation of size[J].SIAM J.Math.Anal,1997,28:1389-1405.
[31]R.Kress.Linear Integral Equations[M].Berlin:Springer-Verlag,1989.
[32]R.Muthupillai,D.J.Lomas and P.J.Rossman et al.Magnetic resonance imaging of transverse acoustic strain waves[J].Science,1995,269:1854-1857.
[33]P.Reulet,D.Nortershauser and P.Millan.Inverse method using infrared thermography for surface temperature and heat flux measurements[J].Instrumentation in Aerospace Shnulation Facilities,ICIASF apos;03.20th International Congress 2003,25:118 - 126.
[34]P.M.Patel,S.K.Lau and D.P.Aldmond.A review of image analysis techniques applied in transient tomographic nondestructive testing[J].Nondestructive testing and eval,1992,6:343-364.
[35]R.Potthast.Stability estimates and reconstruction in inverse acoustic scattering using point sources [J].J.Appl.Comput.Maths,2000,114(2):247-274.
[36]J.Wloka.Partial differential equation[M].Cambridge Univ.press,1987.
[37]王国荣.矩阵与算子广义逆[M].北京:科学出版社,1994.
[38]肖庭延,于慎根,王彦飞等.反问题的数值解法[M].北京:科学出版社,2003.
[39]O.C.Zienkiewicz,R.L.Taylor.The Finite Element method,Fourth Ed[M].London:McGraw-Hill Book Company,1989.
[2] G. Alessandrini, A. Bilotta, G. Formica, A. Morassi, E. Rosset, E Turco. Computing volume bounds on inclusions by EIT measurements [J]. J. Sci. Comput, 2007, 33: 293-312.
[3] G. Alesandrini, A. Morassi, E. Rosset. Size estimates [J]. Inverse problems: theory and applications (Cortona/Pisa, 2002), 1-33.
[4] G. Alesandrini, E. Rosset. The inverse conductivity problem with one measurement: bounds on the size of the unknown object [J]. SIAM J. Appl. Math, 1998, 58: 1060-1071.
[5] G. Alesandrini, E. Rosset. Volume bounds of inclusions from physical EIT measurements [J]. Inverse Problems, 2004, 20: 575-588.
[6] G. Alesandrini, A. Rosset, J.K. Seo. Optimal size estimates for the inverse conductivity problem with one measurement [J]. Proc. Amer. Math. Soc, 2000, 128: 53-64.
[7] G. Alessandrini, V. Isakov. Analyticity and uniqueness for the inverse conductivity problem [J]. Rend. Istit. Mat. Univ. Trieste, 1996, 28: 351-369.
[8] G. Alessandrini, M. Di Cristo. Stable determination of an inclusion by boundary measurements [J]. SIAM J. Math. Anal., 2005, 37: 200-217.
[9] K. Bryan. Single measurement detection of a discontinuous conductivity [J]. Comm. Partial Differential Equations, 1990, 15: 503-514.
[10] M. Di Cristo, L. Rondi, S. Vessella. Stability properties of an inverse parabolic problem with unknown boundaries [J]. Ann. Mat. Pura Appl, 2006, 185: 223-255.
[11] J. R. Cannon. The one-dimensional heat equation [M]. Addison-Wesley Publishing Company, 1984.
[12]陈恕行.现代偏微分方程导论[M].北京：科学出版社,2005.
[13] J Cheng, J J Liu, G Nakamyra. The numerical realization of probe method for the inverse scattering problems from the near-field data [J]. Inverse Problem, 2005, 21: 839-855.
[14] Y. Daido, H. Kang, G. Nakamura. A probe method for the inverse boundary value problem of non-stationary heat equations[J]. Inverse Problem, 2007,23: 1787-1800.
[15] K. Erhard, R. Potthast. A numerical study of the probe method [J]. Society for Industrial and applied Mathematics, 2006, 28: 1597-1612.
[16] A. Friedman. Partial Differential Equations of Parabolic Type[M]. INC: Prentice-Hall, 1964.
[17]A.Friedman.Detection of mines by electric measurements.SIAM J.Appl.Math,1987,47:201-212.
[18]P.C.Hansen.Regularization tools:a Matlab package for analysis and solution of discrete ill-posed problems[J].Numer.Algorithms,1994,6:1-35.
[19]M.Ikehata.Reconstruction of the shape of the inclusion by boundary measurements[J].Commun.PDE,1998,23:1459-1474.
[20]M.Ikehata.Reconstruction of inclusion from boundary measurements[J].J.Inverse and Ill-Posed Problems,2002,10:37-65.
[21]V.Isakov,K.Kim,G.Nakamura.Reconstruction of an unknown inclusion by thermal imaging.preprint.
[22]V.Isakov.Inverse problems for partial differential equations.Second edition[M].New York:Springer,2006.
[23]V.Isakov.On uniqueness of recovery of a discontinuous conductivity[J].Pure Appl.Math.,1988,41:865-877.
[24]J.J.Liu,J.Cheng and G.Nakamura.Reconstruction and uniqueness of the inverse scattering problem with impedance boundary[J].Sci.China,2002,45:1408-1419.
[25]J.J.Liu.Determination of Dirichlet-to-Neumann map for mixed boundary value problem[J].Appl.Math.Comput,2005,161:843-864.
[26]刘继军.不适定问题的正则化方法及应用[M].北京:科学出版社,2005.
[27]D.Issacson Distinguishability of conductivities by electric current computed tomography[J].IEEE TransMed Imagin,1986,5(1):91-95.
[28]P.K.Lamm,Zhewei Dai.On local regularization methods for linear Volterra equations and nonlinear equations of Hammerstein type[J].Inverse Problem,2005,21:1773-1790.
[29]O.A.Ladyzenskaja,V.A.Solonnikov,N.N.Uralceva.Linear and Quasi-linear Equations of Parabolic Type[M].American Mathematical Society,1968.
[30]H.Kang,J..Seo,D.Sheen.The inverse conductivity problem with one measurement:stability and estimation of size[J].SIAM J.Math.Anal,1997,28:1389-1405.
[31]R.Kress.Linear Integral Equations[M].Berlin:Springer-Verlag,1989.
[32]R.Muthupillai,D.J.Lomas and P.J.Rossman et al.Magnetic resonance imaging of transverse acoustic strain waves[J].Science,1995,269:1854-1857.
[33]P.Reulet,D.Nortershauser and P.Millan.Inverse method using infrared thermography for surface temperature and heat flux measurements[J].Instrumentation in Aerospace Shnulation Facilities,ICIASF apos;03.20th International Congress 2003,25:118 - 126.
[34]P.M.Patel,S.K.Lau and D.P.Aldmond.A review of image analysis techniques applied in transient tomographic nondestructive testing[J].Nondestructive testing and eval,1992,6:343-364.
[35]R.Potthast.Stability estimates and reconstruction in inverse acoustic scattering using point sources [J].J.Appl.Comput.Maths,2000,114(2):247-274.
[36]J.Wloka.Partial differential equation[M].Cambridge Univ.press,1987.
[37]王国荣.矩阵与算子广义逆[M].北京:科学出版社,1994.
[38]肖庭延,于慎根,王彦飞等.反问题的数值解法[M].北京:科学出版社,2003.
[39]O.C.Zienkiewicz,R.L.Taylor.The Finite Element method,Fourth Ed[M].London:McGraw-Hill Book Company,1989.