基于积分方程的逆时热传导问题数值方法研究
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摘要
这篇论文研究在一般二维区域D(?)R~2上求解二维逆时热传导问题.这些问题产生在许多工程领域中,如考古学和反应扩散进程等.它的物理描述是从在时间T＞0时温度场的测量值来确定初始温度场分布,数学问题归属于抛物方程的逆问题.本文研究了一般二维区域上的逆时热传导问题的数值方法,首先基于位势理论把逆时热传导方程的求解转化为等价的积分方程的求解,然后利用一种改进的Tikhonov正则化方法求解此问题.最后通过数值实验验证了所给方法的有效性.
本文分为四个章节.在第一章中给出了热传导问题及逆热传导问题的有关知识;第二章介绍了不适定问题的基本概念,Tikhonov正则化方法,改进的Tikhonov正则化方法以及正则化参数的选取;第三章建立了二维热传导问题的数学模型,并给出了基于积分方程的数值方法;第四章对二维逆时热传导问题进行了不适定性分析,基于积分方程的正则化方法对该方程进行了求解,同时对该算法进行了数值模拟.
The aim of this paper is to present an inversion scheme for 2-D backward heat problem in general 2-D domain D (?) R~2. Such problems arise in many engineering areas such as archaeology and reaction-diffusion process. The physical description is to determine the initial field distribution from its final measurement given at some time T > 0 .Mathematically, this problem belongs to the category of inverse problems for parabolic equations. This paper study the numerical method of backward heat problem in general 2-D domain. Firstly based on the potential theory, we transform the solution of backward heat equations into that of equivalent integral equations, then an improved Tikhonov regularization method can be used to solve the problem. Finally, numerical performances are given to show the validity of this regularizing scheme.
This paper is organized for four chapters. In chapter 1 we introduce some knowledge of heat conduction problem and inverse heat conduction problem; In chapter 2 we present the fundamental definitions for ill-posed problem, Tikhonov regularization method, improved Tikhonov regularization method and regularizing parameter selection; The mathematical model of 2-D heat conduction problem is established in chapter 3, and propose the numerical method based on the integral equation; In chapter 4 we analyze the ill-posed property for 2-D backward heat conduction problem , based on the integral equation's regularizing method, we solute the equation and give the numerical simulation for the algorithm.
本文分为四个章节.在第一章中给出了热传导问题及逆热传导问题的有关知识;第二章介绍了不适定问题的基本概念,Tikhonov正则化方法,改进的Tikhonov正则化方法以及正则化参数的选取;第三章建立了二维热传导问题的数学模型,并给出了基于积分方程的数值方法;第四章对二维逆时热传导问题进行了不适定性分析,基于积分方程的正则化方法对该方程进行了求解,同时对该算法进行了数值模拟.
The aim of this paper is to present an inversion scheme for 2-D backward heat problem in general 2-D domain D (?) R~2. Such problems arise in many engineering areas such as archaeology and reaction-diffusion process. The physical description is to determine the initial field distribution from its final measurement given at some time T > 0 .Mathematically, this problem belongs to the category of inverse problems for parabolic equations. This paper study the numerical method of backward heat problem in general 2-D domain. Firstly based on the potential theory, we transform the solution of backward heat equations into that of equivalent integral equations, then an improved Tikhonov regularization method can be used to solve the problem. Finally, numerical performances are given to show the validity of this regularizing scheme.
This paper is organized for four chapters. In chapter 1 we introduce some knowledge of heat conduction problem and inverse heat conduction problem; In chapter 2 we present the fundamental definitions for ill-posed problem, Tikhonov regularization method, improved Tikhonov regularization method and regularizing parameter selection; The mathematical model of 2-D heat conduction problem is established in chapter 3, and propose the numerical method based on the integral equation; In chapter 4 we analyze the ill-posed property for 2-D backward heat conduction problem , based on the integral equation's regularizing method, we solute the equation and give the numerical simulation for the algorithm.
引文
[1]杨文采.非线性地球物理反演方法回顾:回顾和展望.地球物理学进展,2002,17(2):255-261.
[2]杨辉,戴世坤,宋海斌,黄临乎.综合地球物理联合反演综述.地球物理学进展,2000,17(2):262-271.
[3]杨晓春,李小凡,张美根.地震波反演方法研究的某些进展及其数学基础.地球物理学进展.2001,16(4):96-109.
[4]C F Weber.Analysis and solution of the ill-posed inverse heat conduction problem [J].Int J Heat Mass Transfer,1981,24(11):1783-1792.
[5]P.A Terrola.Method to determine the thermal conductivity from measured temperature[J].International Journal of Heat and Mass Transfer,1989,32:1452-1430.
[6]刘家琦.数学物理反问题的分类及不适定问题的求解.应用数学与计算数学,1983.4:43-65.
[7]J V Beck,B Blackwell,A Haji-Sheikh.Comparison of inverse heat conduction methods using experimental data[J].Int J Heat Mass Transfer,1996,39(17):3649-3657.
[8]A lifanov O M.Solution of an inverse problem of heat conduction by iteration methods[J].Jounal of Engineering Physics,1972,26:471-476.
[9]Chen Han-Taw,Lin Sheng-Yih.Estimation of two-sided boundary conditions for two-dimensional inverse conduction problems[J].International Journal of Heat and Mass Transfer,2002,45:15-23.
[10]C H Huang,B H Chao.An inverse geometry problem in identifying irregular boundary configrations[J].Int J Heat Mass Transfer,1997,40:2045-2053.
[11]J U Beck,B Blackwell,C R Clair.Inverse Heat Conduction:Ill-posed Problem.New York:Wiley,1985.
[12]A Carrasso.Determining surface temperature from interior observations.SIAM J Appl Math,1982,42:558-574.
[13]F Scarpa,G Milano.Kalman smoothing technique applied to the inverse heat [J].Numerical Heat Transfer Part B,1995,28:79-96.
[14]周富军,傅初黎.确定表面热流的非标准逆热传导问题的Fourier正则化方法.中山大学学报,2003,Vol.42,No.4.
[15]J.R.Cannon.The One-dimensional Heat Equation.Addsion-Wesley,Menlo Park,CA,1984.
[16]P.Jonas,A.K.Louis.Approximate inverse for a one-dimensional inverse heat conduction problem.Inverse Problems,2000,Vol.16,175-185.
[17]D.Lesnic,L.Elliott.The decomposition approach to inverse heat conduction.J.Math.Anal.Appl.,1999,Vol.232,No.1,82-98.
[18]J V Beck,B Blackwell,C R S Jr Clair.Inverse Heat Corr ductiorr ill-posed Problems[M].New York:John Wiley & Sons,1985.
[19]J.J.Liu,D.J.Luo.On stability and regularization for backward heat equation.Chinese Annals of Mathematics,2003,24B:1,35-44.
[20]钱炜脐,蔡金狮,任斌.二维稳态热传导问题初步研究[J].计算物理,2002,19(6):512-516.
[21]L.Guo,D.A.Murio.A mollified space-marching finite-difference algorithm for the two-dimensional inverse heat conducion problem with slab symmetry.Inverse Problems,1991,Vol.7,No.2,247-259.
[22]A.Shidfar,A.Neisy.A two-dimensional inverse heat conduction problem for estimating heat flux.Far East J.Appl.,2003 Vol.10,No.2,145-150.
[23]刘继军.二维热传导方程的三层显式差分格式.应用数学和力学,2003,第24卷第5期.
[24]J.J.Liu.On Ill-posedness and Inversion Scheme for 2-D Backward Heat Conduction.2004.
[25]G D Smith.Numerical Solutions of Partial Differential Eqtions:Finite Difference Methods[M].New York:Oxford Uiniversity Press,1985.
[26]J.J.Liu.Numerical solution of forward and backward problem for 2-D heat conduction problem.J.Comput.Appl.Maths.,2002,Vol.145,No.2,459-482.
[27]黄光远,刘小军.数学物理反问题[M].济南:山东科学技术出版社,1993.
[28]王元明.数学是什么.南京:东南大学出版社,2003.
[29]J Baumeister.Stable Solution of Inverse Problems.Vieweg-Verlag,Brauschweig,1987.
[30]V K Ivanov.On Linear Ill-posed Problems.Dokl.Acad.Nauk USSR,1962,145(2),270-272.
[31]J Hadamard.Lectures on the Cauchy Problems in Linear Partial Differential Equations.New Haven:Yale University Press,1923.
[32]刘继军.不适定问题的正则化方法及应用.北京:科学出版社,2005.2-3.
[33]韩波,付友和.求解非线性不适定问题的正则化同伦方法.黑龙江大学自然科学学报,2005(5).
[34]C W Groetsch.The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind.Boston:Pitman,1984.
[35]G Bruckner,J Cheng.Tikhonov regularization for an intergral equation of the first kind with logarithmic kernel.J.Inveser and Ill-posed Problems,2000,8(6):665-675.
[36]A Kitsch.An Introduction to the Mathematical Theory of Inverst Problems.New York:Springer Verlag,1996.
[37]R Kress.Linear Integral Equations.New York:Springer-Verlag,1989.
[38]肖庭延,于慎根,王彦飞.反问题数值解法.北京:科学出版社,2003
[39]A N Tikhonov,V Y Arsenin.Solutions of Ill-Posed Problems[m].New York:Winston Wiley,1997.136-163.
[40]A Kirscn.An introduction to the mathematical theory of inverse peoblems[M].Springer-Verlag,1999.
[41]H W Engl,M Hanke,A Neubauer.Regularization of inverse peoblems[Ml.Kluwer Academic Puhlishers,1996.
[42]R.Ramlau.Morozov's Discrepancy Principle for Tikhonov Regularization of Nonlinear Operators.Numer.Funct.Anal.and Optimiz.2002,23(1&2):147-172.
[43]O.Scherzer.The Use of Morozov's Discrepancy Principle for Tikhonov Regularization for Solving Nonlinear Ill-Posed Problems.Computing,1993,(51):45-60.
[44]G.Chavent,K.Kunisch.Convergence of Tikhonov Regularization for Constraines Ill-Posed Inverse Problems.Inverse Problems,1994,(10):63-76.
[45]李荷农,候宗义.算子和右端都近似给定的第一类算子方程的Tikhonov正则解的渐进阶的估计.数学年刊,1993,14A(4):458-463。
[46]Z.Y.Hou,Q.N.Jin.Tikhonov Regularization for Nonliear Ill-posed Problems.Nonlinear Analyses,1997,28(11):1799-1809.
[47]A.Neubauer,O.scherzer.Finite-dimtensional Approximation of Tikhonov Regularized Solutions of Nonlinear Ill-posed Problems.Numer.Anal.and Optimiz,1990,(11):85-89.
[48]A.Neubauer.Tikhonov Regularization for Nonlinear Ill-posed Problems:Optimal Convergence Rates and Finite-dimensional Approximation.Inverse problems,1989,(5):541-557.
[49]H.W.Engl.Discrepancy Principles for Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates.J.Optim.Theory appl,1987,(52):523-540.
[50]H.W.Engl,K.Kunisch,A.Neubarer.Convergence Rates for Tikhonov Regularization of Nonlinear Ill-Posed Problems.Inverse Problems,1989(5):523-540.
[51]O.Scherzer,H.W.Engle and K.Kunisch.Optimal a Psteriori Parameter Choice for Tikhonov Regularization for Solving Nonlinear Ill-Posed Problems.SIAM J.Numer.Anal,1992,30(6):1796-1838,
[52]黄向鼎,曾钟钢.非线性数值分析的理论与方法.武汉大学出版社,2004.
[53]O.Scherzer.The Use of Morozov's Discrepancy Principle for Tikhonov Regularization for Solving Nonlinear Ill-Posed Problems.Computing,1993,(51):45-60.
[54]H W Engl.Discrepancy principles for Tikhonov regularization of ill-posed problems.J.Optim.Th.and App.,1987,52(2):209-215.
[55]R.Kress.Linear integral equations,2nd,Springer-Velag,Berlin,Heidelberg,1999.
[56]宋丽红,陈仲英.解不适定问题的一种多尺度方法.高等学校计算数学学报,2007,29(1):63-73.
[57]Z Y Chen,B Wu,Y S Xu.Multilevel augmentation methods for solving operator equations.Numer.Math.,J.Chinese Univerdities,2005,14(1):31-55.
[58]Z Y Chen,Y S Xu,H Q Yang.Multilevel augmentation methods for solving ill-posed operator equations.Inverse Problems,2006,22:155-174.
[59]B.Han,Y.H.Fu.A regularization homotopy method for solving nonlinear illposed problems.&Natural science of Heilongjiang University,2005,Vol.22,No.5.
[60]Gang Bao,Jun liu.Numerical solution of invese scattering problems with multiexperimental limited aperture data.SIAM.J.Sci.Comput,2003,Vol.25,No.3,1102-1117.
[2]杨辉,戴世坤,宋海斌,黄临乎.综合地球物理联合反演综述.地球物理学进展,2000,17(2):262-271.
[3]杨晓春,李小凡,张美根.地震波反演方法研究的某些进展及其数学基础.地球物理学进展.2001,16(4):96-109.
[4]C F Weber.Analysis and solution of the ill-posed inverse heat conduction problem [J].Int J Heat Mass Transfer,1981,24(11):1783-1792.
[5]P.A Terrola.Method to determine the thermal conductivity from measured temperature[J].International Journal of Heat and Mass Transfer,1989,32:1452-1430.
[6]刘家琦.数学物理反问题的分类及不适定问题的求解.应用数学与计算数学,1983.4:43-65.
[7]J V Beck,B Blackwell,A Haji-Sheikh.Comparison of inverse heat conduction methods using experimental data[J].Int J Heat Mass Transfer,1996,39(17):3649-3657.
[8]A lifanov O M.Solution of an inverse problem of heat conduction by iteration methods[J].Jounal of Engineering Physics,1972,26:471-476.
[9]Chen Han-Taw,Lin Sheng-Yih.Estimation of two-sided boundary conditions for two-dimensional inverse conduction problems[J].International Journal of Heat and Mass Transfer,2002,45:15-23.
[10]C H Huang,B H Chao.An inverse geometry problem in identifying irregular boundary configrations[J].Int J Heat Mass Transfer,1997,40:2045-2053.
[11]J U Beck,B Blackwell,C R Clair.Inverse Heat Conduction:Ill-posed Problem.New York:Wiley,1985.
[12]A Carrasso.Determining surface temperature from interior observations.SIAM J Appl Math,1982,42:558-574.
[13]F Scarpa,G Milano.Kalman smoothing technique applied to the inverse heat [J].Numerical Heat Transfer Part B,1995,28:79-96.
[14]周富军,傅初黎.确定表面热流的非标准逆热传导问题的Fourier正则化方法.中山大学学报,2003,Vol.42,No.4.
[15]J.R.Cannon.The One-dimensional Heat Equation.Addsion-Wesley,Menlo Park,CA,1984.
[16]P.Jonas,A.K.Louis.Approximate inverse for a one-dimensional inverse heat conduction problem.Inverse Problems,2000,Vol.16,175-185.
[17]D.Lesnic,L.Elliott.The decomposition approach to inverse heat conduction.J.Math.Anal.Appl.,1999,Vol.232,No.1,82-98.
[18]J V Beck,B Blackwell,C R S Jr Clair.Inverse Heat Corr ductiorr ill-posed Problems[M].New York:John Wiley & Sons,1985.
[19]J.J.Liu,D.J.Luo.On stability and regularization for backward heat equation.Chinese Annals of Mathematics,2003,24B:1,35-44.
[20]钱炜脐,蔡金狮,任斌.二维稳态热传导问题初步研究[J].计算物理,2002,19(6):512-516.
[21]L.Guo,D.A.Murio.A mollified space-marching finite-difference algorithm for the two-dimensional inverse heat conducion problem with slab symmetry.Inverse Problems,1991,Vol.7,No.2,247-259.
[22]A.Shidfar,A.Neisy.A two-dimensional inverse heat conduction problem for estimating heat flux.Far East J.Appl.,2003 Vol.10,No.2,145-150.
[23]刘继军.二维热传导方程的三层显式差分格式.应用数学和力学,2003,第24卷第5期.
[24]J.J.Liu.On Ill-posedness and Inversion Scheme for 2-D Backward Heat Conduction.2004.
[25]G D Smith.Numerical Solutions of Partial Differential Eqtions:Finite Difference Methods[M].New York:Oxford Uiniversity Press,1985.
[26]J.J.Liu.Numerical solution of forward and backward problem for 2-D heat conduction problem.J.Comput.Appl.Maths.,2002,Vol.145,No.2,459-482.
[27]黄光远,刘小军.数学物理反问题[M].济南:山东科学技术出版社,1993.
[28]王元明.数学是什么.南京:东南大学出版社,2003.
[29]J Baumeister.Stable Solution of Inverse Problems.Vieweg-Verlag,Brauschweig,1987.
[30]V K Ivanov.On Linear Ill-posed Problems.Dokl.Acad.Nauk USSR,1962,145(2),270-272.
[31]J Hadamard.Lectures on the Cauchy Problems in Linear Partial Differential Equations.New Haven:Yale University Press,1923.
[32]刘继军.不适定问题的正则化方法及应用.北京:科学出版社,2005.2-3.
[33]韩波,付友和.求解非线性不适定问题的正则化同伦方法.黑龙江大学自然科学学报,2005(5).
[34]C W Groetsch.The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind.Boston:Pitman,1984.
[35]G Bruckner,J Cheng.Tikhonov regularization for an intergral equation of the first kind with logarithmic kernel.J.Inveser and Ill-posed Problems,2000,8(6):665-675.
[36]A Kitsch.An Introduction to the Mathematical Theory of Inverst Problems.New York:Springer Verlag,1996.
[37]R Kress.Linear Integral Equations.New York:Springer-Verlag,1989.
[38]肖庭延,于慎根,王彦飞.反问题数值解法.北京:科学出版社,2003
[39]A N Tikhonov,V Y Arsenin.Solutions of Ill-Posed Problems[m].New York:Winston Wiley,1997.136-163.
[40]A Kirscn.An introduction to the mathematical theory of inverse peoblems[M].Springer-Verlag,1999.
[41]H W Engl,M Hanke,A Neubauer.Regularization of inverse peoblems[Ml.Kluwer Academic Puhlishers,1996.
[42]R.Ramlau.Morozov's Discrepancy Principle for Tikhonov Regularization of Nonlinear Operators.Numer.Funct.Anal.and Optimiz.2002,23(1&2):147-172.
[43]O.Scherzer.The Use of Morozov's Discrepancy Principle for Tikhonov Regularization for Solving Nonlinear Ill-Posed Problems.Computing,1993,(51):45-60.
[44]G.Chavent,K.Kunisch.Convergence of Tikhonov Regularization for Constraines Ill-Posed Inverse Problems.Inverse Problems,1994,(10):63-76.
[45]李荷农,候宗义.算子和右端都近似给定的第一类算子方程的Tikhonov正则解的渐进阶的估计.数学年刊,1993,14A(4):458-463。
[46]Z.Y.Hou,Q.N.Jin.Tikhonov Regularization for Nonliear Ill-posed Problems.Nonlinear Analyses,1997,28(11):1799-1809.
[47]A.Neubauer,O.scherzer.Finite-dimtensional Approximation of Tikhonov Regularized Solutions of Nonlinear Ill-posed Problems.Numer.Anal.and Optimiz,1990,(11):85-89.
[48]A.Neubauer.Tikhonov Regularization for Nonlinear Ill-posed Problems:Optimal Convergence Rates and Finite-dimensional Approximation.Inverse problems,1989,(5):541-557.
[49]H.W.Engl.Discrepancy Principles for Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates.J.Optim.Theory appl,1987,(52):523-540.
[50]H.W.Engl,K.Kunisch,A.Neubarer.Convergence Rates for Tikhonov Regularization of Nonlinear Ill-Posed Problems.Inverse Problems,1989(5):523-540.
[51]O.Scherzer,H.W.Engle and K.Kunisch.Optimal a Psteriori Parameter Choice for Tikhonov Regularization for Solving Nonlinear Ill-Posed Problems.SIAM J.Numer.Anal,1992,30(6):1796-1838,
[52]黄向鼎,曾钟钢.非线性数值分析的理论与方法.武汉大学出版社,2004.
[53]O.Scherzer.The Use of Morozov's Discrepancy Principle for Tikhonov Regularization for Solving Nonlinear Ill-Posed Problems.Computing,1993,(51):45-60.
[54]H W Engl.Discrepancy principles for Tikhonov regularization of ill-posed problems.J.Optim.Th.and App.,1987,52(2):209-215.
[55]R.Kress.Linear integral equations,2nd,Springer-Velag,Berlin,Heidelberg,1999.
[56]宋丽红,陈仲英.解不适定问题的一种多尺度方法.高等学校计算数学学报,2007,29(1):63-73.
[57]Z Y Chen,B Wu,Y S Xu.Multilevel augmentation methods for solving operator equations.Numer.Math.,J.Chinese Univerdities,2005,14(1):31-55.
[58]Z Y Chen,Y S Xu,H Q Yang.Multilevel augmentation methods for solving ill-posed operator equations.Inverse Problems,2006,22:155-174.
[59]B.Han,Y.H.Fu.A regularization homotopy method for solving nonlinear illposed problems.&Natural science of Heilongjiang University,2005,Vol.22,No.5.
[60]Gang Bao,Jun liu.Numerical solution of invese scattering problems with multiexperimental limited aperture data.SIAM.J.Sci.Comput,2003,Vol.25,No.3,1102-1117.