数学物理中反问题与边值问题的积分方程方法
|
摘要
本文主要围绕数学物理中的若干类反问题与边值问题进行研究，包括Hausdorff矩问题、弹性接触反问题、热传导方程反问题、Laplace方程Cauchy问题和二阶椭圆型方程组非线形边值问题等。随着科学技术的纵深发展和社会与经济的全面进步，越来越多的问题可归结为上述问题，如地质勘察、无损探伤、CT技术、军事侦察、环境治理、遥感遥测、信号处理、控制论、经济学等。本文创造性地应用积分方程方法，借助现代数学手段，着重研究这些问题的可解性条件、条件适定性(特别是条件稳定性)，构造稳定化算法，并进行数值模拟。论文的主要成果包括：
利用积分方程方法讨论通过有限个矩量求解Hausdorff矩问题，我们先将矩问题化为第一类Fredholm积分方程，分析该问题的不适定性。由于问题的不适定性，求解变得非常困难。为克服困难，我们先讨论矩问题的条件稳定性(包括整体估计和局部估计)，创造性地获得了对数型稳定性结果。并且基于稳定性，结合Tikhonov正则化方法，构造稳定化算法，成功地获得了矩问题的近似解——正则化解与精确解的整体误差估计和局部误差估计。利用有限元方法(有限维空间逼近)进行离散和数值模拟，数值实验结果很好地验证了算法的稳定性和有效性。本文提出的矩问题的稳定化算法可用之于Laplace方程Cauchy问题和一些反问题的数值求解。
在弹性理论中，接触反问题是一类重要的问题，属于一
2001年上梅大学博士学位论文
类无损探伤间题.我们首先给出了接触反问题的一种合理提
法，即通过接触域外部的位移测量值决定不可直接测量的接
触域和接触域上的应力.从数学上证明了接触反问题提法的
合理性，第一次提出了接触反问题，并证明了其不适定性.为
决定接触域和接触域上的应力，必须克服不适定性，构造稳
定化算法，我们的基本思路是:先通过Fourier变换将接触反
问题等价地转化为第一类Redhofm积分方程，再讨论积分方
程解的唯一性条件和条件稳定性，从而通过位势理论创造性
地获得了接触反问题的唯一性和对数型条件稳定性，包括整
体和局部条件稳定性.最后构造反问题的Ti比onov正则化解
(近似解)，利用稳定化估计证明了近似解的对数型整体误差
估计和局部误差估计，我们的方法独具创新，为其他接触反
问题的求解指明了方向.
本文讨论了高维情形下热传导方程中初始源项和中间某时
刻温度分布的决定问题，证明了热传导方程反间题的合理提法，
即通过部分边界或部分内部区域上、在有限时间内的温度分布
值决定温度的初始分布和中间某一时刻温度分布.分别利用
Carlemann估计、精确控制理论获得了初始温度分布决定的对数
性稳定性估计、中间时刻温度分布的Lipschitz型稳定性.基于
稳定性估计和正则化参数的先验选取，构造了正则化算法，获得
了稳定化解，证明了该稳定化解的收敛速率.
利用小波分析工具讨论了助place方程Cauchy问题的稳定
化求解，在小波多尺度分析中构造稳定化算法，获得了近似解
的H说der型误差估计.
利用积分算子理论、奇异积分方程理论，讨论了二阶椭圆型
方程组非线形边值问题(E2类和非瓦类)，建立了与间题等价的
数学物理中反问题与边值问题的积分方程方法
奇异积分方程，并进而得到了问题的可解性条件及存在性证明.
为便于阅读，在本文开始还提纲掣领地叙述了反问题与边值
问题的基本思想与方法，本文结束时，对本文进行了总结和评论，
并对将来数学物理反问题和边值问题的研究进行了展望.
The dissertation discusses several classes of inverse problems and boundary value problems in mathematical physics, including Hausdorff moment problems, inverse contact problem in the theory of elasticity, inverse heat conductivity problem, Cauchy problem for Laplace equations, and non-linear boundary value problems for second-order elliptic differential systems. With the in-depth development in science and technology and all-around progress in society and economy, more and more practical problems, such as geological prospecting, non-destructive testing, CT technology, military reconnaissance, environmental disposal, remote sensing, signal processing, cybernetics, economics and so on, have been formulated into inverse problems and boundary value problems for partial differential equations. By means of integral equation methods creatively along with other modern mathematical theories, this paper focuses on finding solvability conditions and conditional well-posedness (especially conditional stability), constr
ucting stabilized algorithms, and carrying through numerical simulation. The main achievements are as follows:
Using integral equation methods to solve the Hausdorff moment problems (HMP) by finite moments, we first transform the HMP into the first kind of Fredholm integral equation equivalently and analyze the ill-posedness of the HMP, for which it is very difficult to solve. In order to cure the ill-posedness, we first discuss the conditional stability estimates in moment problems including global and local estimates, and have obtained the stability results of logarithmic rate creatively. Basing on the stability estimates and the famous Tikhonov regularization method, we present some stabilized algorithms and have successfully derived the error estimates
between the approximate solution (regularized solution) and the exact solution for the HMP, including the global and local error estimates. We discretize the algorithms by means of the finite element method and do some numerical simulations. The numerical results show the nice stability and efficiency of the algorithms. The presented algorithms here for the HMP are applicable to numerically solving Cauchy problems for Laplace equations and several inverse problems.
Inverse contact problems, as a class of non-destructive problems, are important issues in the theory of elasticity. Firstly we give an appropriate presentation for an inverse contact problem, that is, we determine the inaccessible contact domain and the stress on the domain from the displacement measurements outside of the contact domain. We have formulated the inverse contact problem for the first time, and proved mathematically its rationality and its ill-posedness. In order to determine the contact domain and the stress on the domain, we must cure the ill-posedness and construct stabilized algorithms. Our main ideas are as follows: firstly we transform equivalently the inverse contact problem into the Fredholm integral equation of the first kind by the Fourier transforms, and discuss the uniqueness and conditional stability for the integral equation, and then by the potential theory we derive the uniqueness and stability estimates creatively, including the global and local stability estimates, finally we construct the Tikhonov regularized solution for the inverse contact problem and prove the global and local error estimates of logarithmic rate. Our method shows the originality and gives some instructive ideas for solving other inverse contact problems.
The dissertation also discusses multi-dimensional inverse heat conductivity problems(IHCP), that is, the determination of initial source terms and heat distribution at any intermediate moment. We have proved the reasonable presentation of the IHCP, i.e., we can uniquely determine the
heat distribution at the initial moment and at any moments from the heat distribution measurements on the part of the boundary or in arbitrary accessible sub-domain at any time-interval. Utilizing the Carlemann estimates and exact controllability theory res
利用积分方程方法讨论通过有限个矩量求解Hausdorff矩问题，我们先将矩问题化为第一类Fredholm积分方程，分析该问题的不适定性。由于问题的不适定性，求解变得非常困难。为克服困难，我们先讨论矩问题的条件稳定性(包括整体估计和局部估计)，创造性地获得了对数型稳定性结果。并且基于稳定性，结合Tikhonov正则化方法，构造稳定化算法，成功地获得了矩问题的近似解——正则化解与精确解的整体误差估计和局部误差估计。利用有限元方法(有限维空间逼近)进行离散和数值模拟，数值实验结果很好地验证了算法的稳定性和有效性。本文提出的矩问题的稳定化算法可用之于Laplace方程Cauchy问题和一些反问题的数值求解。
在弹性理论中，接触反问题是一类重要的问题，属于一
2001年上梅大学博士学位论文
类无损探伤间题.我们首先给出了接触反问题的一种合理提
法，即通过接触域外部的位移测量值决定不可直接测量的接
触域和接触域上的应力.从数学上证明了接触反问题提法的
合理性，第一次提出了接触反问题，并证明了其不适定性.为
决定接触域和接触域上的应力，必须克服不适定性，构造稳
定化算法，我们的基本思路是:先通过Fourier变换将接触反
问题等价地转化为第一类Redhofm积分方程，再讨论积分方
程解的唯一性条件和条件稳定性，从而通过位势理论创造性
地获得了接触反问题的唯一性和对数型条件稳定性，包括整
体和局部条件稳定性.最后构造反问题的Ti比onov正则化解
(近似解)，利用稳定化估计证明了近似解的对数型整体误差
估计和局部误差估计，我们的方法独具创新，为其他接触反
问题的求解指明了方向.
本文讨论了高维情形下热传导方程中初始源项和中间某时
刻温度分布的决定问题，证明了热传导方程反间题的合理提法，
即通过部分边界或部分内部区域上、在有限时间内的温度分布
值决定温度的初始分布和中间某一时刻温度分布.分别利用
Carlemann估计、精确控制理论获得了初始温度分布决定的对数
性稳定性估计、中间时刻温度分布的Lipschitz型稳定性.基于
稳定性估计和正则化参数的先验选取，构造了正则化算法，获得
了稳定化解，证明了该稳定化解的收敛速率.
利用小波分析工具讨论了助place方程Cauchy问题的稳定
化求解，在小波多尺度分析中构造稳定化算法，获得了近似解
的H说der型误差估计.
利用积分算子理论、奇异积分方程理论，讨论了二阶椭圆型
方程组非线形边值问题(E2类和非瓦类)，建立了与间题等价的
数学物理中反问题与边值问题的积分方程方法
奇异积分方程，并进而得到了问题的可解性条件及存在性证明.
为便于阅读，在本文开始还提纲掣领地叙述了反问题与边值
问题的基本思想与方法，本文结束时，对本文进行了总结和评论，
并对将来数学物理反问题和边值问题的研究进行了展望.
The dissertation discusses several classes of inverse problems and boundary value problems in mathematical physics, including Hausdorff moment problems, inverse contact problem in the theory of elasticity, inverse heat conductivity problem, Cauchy problem for Laplace equations, and non-linear boundary value problems for second-order elliptic differential systems. With the in-depth development in science and technology and all-around progress in society and economy, more and more practical problems, such as geological prospecting, non-destructive testing, CT technology, military reconnaissance, environmental disposal, remote sensing, signal processing, cybernetics, economics and so on, have been formulated into inverse problems and boundary value problems for partial differential equations. By means of integral equation methods creatively along with other modern mathematical theories, this paper focuses on finding solvability conditions and conditional well-posedness (especially conditional stability), constr
ucting stabilized algorithms, and carrying through numerical simulation. The main achievements are as follows:
Using integral equation methods to solve the Hausdorff moment problems (HMP) by finite moments, we first transform the HMP into the first kind of Fredholm integral equation equivalently and analyze the ill-posedness of the HMP, for which it is very difficult to solve. In order to cure the ill-posedness, we first discuss the conditional stability estimates in moment problems including global and local estimates, and have obtained the stability results of logarithmic rate creatively. Basing on the stability estimates and the famous Tikhonov regularization method, we present some stabilized algorithms and have successfully derived the error estimates
between the approximate solution (regularized solution) and the exact solution for the HMP, including the global and local error estimates. We discretize the algorithms by means of the finite element method and do some numerical simulations. The numerical results show the nice stability and efficiency of the algorithms. The presented algorithms here for the HMP are applicable to numerically solving Cauchy problems for Laplace equations and several inverse problems.
Inverse contact problems, as a class of non-destructive problems, are important issues in the theory of elasticity. Firstly we give an appropriate presentation for an inverse contact problem, that is, we determine the inaccessible contact domain and the stress on the domain from the displacement measurements outside of the contact domain. We have formulated the inverse contact problem for the first time, and proved mathematically its rationality and its ill-posedness. In order to determine the contact domain and the stress on the domain, we must cure the ill-posedness and construct stabilized algorithms. Our main ideas are as follows: firstly we transform equivalently the inverse contact problem into the Fredholm integral equation of the first kind by the Fourier transforms, and discuss the uniqueness and conditional stability for the integral equation, and then by the potential theory we derive the uniqueness and stability estimates creatively, including the global and local stability estimates, finally we construct the Tikhonov regularized solution for the inverse contact problem and prove the global and local error estimates of logarithmic rate. Our method shows the originality and gives some instructive ideas for solving other inverse contact problems.
The dissertation also discusses multi-dimensional inverse heat conductivity problems(IHCP), that is, the determination of initial source terms and heat distribution at any intermediate moment. We have proved the reasonable presentation of the IHCP, i.e., we can uniquely determine the
heat distribution at the initial moment and at any moments from the heat distribution measurements on the part of the boundary or in arbitrary accessible sub-domain at any time-interval. Utilizing the Carlemann estimates and exact controllability theory res
引文
1 栾文贵．地球物理中的反问题．北京：科学出版社，1987
2 冯康．数值计算方法．北京：科学出版社，1979
3 苏超伟．偏微分方程逆问题的数值方法及其应用．西安：西北工业大学出版社，1995
4 李明忠，侯宗义，徐振远．椭圆型方程组理论和边值问题．上海：复旦大学出版社，1990
5 Tikhonov A.N., Arsenin N.Y. Solutions of Ill-Posed Problems. Washington D.C.: V.H.Winston & Sons, 1977
6 Groestch C.W. Inverse Problems in the Mathematical Sciences. Braun-schweig: Vieweg, 1993
7 Kitsch A. An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag New York Inc., 1996.
8 Kress R. Linear Integral Equations. Berlin: Springer-Vedag, 1989
9 Isakov V. Inverse Problems for Partial Differential Equations. Berlin: Springer, 1998
10 Engl H.W., Hanke M., Neubauer A. Regularization of Inverse Problems. Dordrecht/Boston/London: Kluwer Academic Publishers, 1996
11 Murio D.A. The Mollification Method and the Numerical Solution of Ill-Posed Problems. NewYork: John Wiley & Sons Inc., 1993
12 Lavrent'ev M.M., Romanov V.G., Shishatskii S.P. Ill-Posed Problems of Mathematical Physics and Analysis, Providence RI: American Mathematical Society, 1986
13 Colton D., Kress R. Integral Equation Methods in Scattering Theory. New
York: John Wiley & Sons, 1983
14 Ramm A.G. Scattering by Obstacles. Dordrecht: D. Reldel Publishing Company, 1986
15 Fursikov A.V., Imanuvilov O.Yu. Controllability of Evolution Equations. Notes of the Series of Lectures held at the Seoul National University, 1996
16 赵彦．逆问题研究关联科学创新．科学时报(中国科学院主办，科学时报出版社出版)，1999；(1494)
17 Agmon S. Lectures on Elliptic Boundary Value Problems. Princeton, N. J.: Van Nostrand, 1965
18 Ladyzhenskaya O.A., Ural'tseva N.N. Linear and Quasi-linear Elliptic Equations. New York: Academic Press, (English Translation Edition), 1968
19 李大潜，秦铁虎．物理学与偏微分方程．北京：高等教育出版社，1997
20 Quarteroni A., Valli A. Numerical Approximation for Partial Differential Equations. Berlin: Springer-Verlag, 1997
21 夏道行，严绍宗，吾作人，舒五昌．泛函分析．北京：人民教育出版社，1979
22 Chui C.K. An Introduction to Wavelets. NewYork: Academic Press, Inc.,1992．(有中译本：程正兴译，自居宪审校．小波分析导论．西安：西安交通大学出版社，1994)
23 Prasad L., Iyengar S.S. Wavelet Analysis with Applications to Image Processing, Boca Raton: CRC Press, 1997
24 Adams R. A.(叶其孝等译)．Sobolev空间．北京：人民教育出版社，1979
25 Courant R., Hilbert D. Methods of Mathematical Physics. InterScience Publishers, 1962
26 Muskhelishvili N.I. Singular Integral Equations, Boundary Value Problems of Function Theory and Their Application to Mathematical Physics. New York: Dover Publications Inc., 1992
27 Gilbarg D., Trudinger N.S. Elliptic Partial Differential Equations of
Second Order. (2nd edition), Berlin, NewYork: Springer, 1983
28 Engl H.W., Isakov V. On the identifiability of steel reinforcement bars in concrete from magnetostatic measurements. Europe J. Appl. Math., 1992, 13:255-162
29 Bukhgeim A.L., Cheng J., Yamamoto M. Local stability of a lineaized inverse problem in determining steel reinforcement bars. Proceedings of International Conference on Inverse Problems and Applications, Philippin,1998
30 Hadamard J. Lectures on the Cauchy Problem in Linear Partial Differential Equations. New Haven: Yale University Press, 1923
31 Kaltenbacher B. Some Newton-type methods for the regularization of nonlinear ill-posed problems. Inverse Problems, 1997; 13:729-753
32 Cheng J., Yamamoto M. One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization. Inverse Problems, 2000; 16:L31-38
33 Kirsch A., Schomberg B., Berendt G. The Backus-Gilbert method. Inverse Problems, 1988; 10:771-783
34 吴乃龙，袁素云著．最大熵方法．长沙：湖南科学技术出版社，1991
35 Hettlich F., Rundell W. Iterative methods for the reconstruction of an inverse potential problem. Inverse Problems, 1999;12:251-266
36 Hansen P.C. The Backus-Gilbert method: SVD analysis and fast implementation. Inverse Problems, 1994; 10:895-904
37 Chen Y.M. Generalized pulse-spectrum technique. Geophysics, 1985; 50: 1664-1675
38 Kormonik V. Exact Controllability and Stabilization. Chichester: John Wiley & Sons, 1994
39 Beretta E., Vessella S. Stable determination of boundaries from Cauchy data. SIAM J. Math. AnaL, 1998; 30:220-232
40 Alessandrini G., Isakov V., Powell J. Local uniqueness in the inverse conductivity problem with one measurement. Trans. of American Mathematical Society, 1995; 347:3031-3041
41 Cedio-Fengya D.J., Moskow S., Vogelius M.S. Identification of conductivity imperfection of small diameter by boundary measyrements: continuous dependence and computational reconstruction. Inverse Problems, 1998; 14:553-595
42 Su C.W. An iterative algorithm for determining time-dependent coefficient of two-dimensional linear wave equation. Annals of Differential Equations, 1995; 11:213 -223
43 Bekya H．广义解析函数．北京：人民教育出版社，1960
44 Groetsch C.W. The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Boston: Pitman, 1984
45 Gorenflo R., Vessella S. Abel integral equations:analysis and applications. Lectures Notes in Mathematics, 1461, Berlin: Springer, 1991
46 Engl H.W., Landl G. Maximum entropy regularization of nonlinear ill-posed problems. In Lakshmikantham V(eds.): Proceedings of the First World Congress of Nonlinear Analysis, Vol.I: 513-525, Berlin: de Gruyter, 1996
47 Neubauer A. Tikhonov regularization for nonlinear ill-posed problems: optimal convergence and finite-dimensional approximation. Inverse Problems, 1989, 5:541-557
48 Ang D.D., Vy L.K., Gorerflo R. A regularization method for the moment problem, in: Inverse Problems: Principles and Applications in Geophysics, Technology and Medicine, Math. Research, Berlin: Akademic Verlag, 1993, 74:37-45
49 Backus G.E., Gilbert J.F. The resolving power of gross earth data. Geophysical Journal of the Royal Astronomical Society, 1968; 16:169-205
50 Inglese G. Recent results in the study of the moment problem. In:
Geophysical Data Interpretation by Inverse Modeling, Proceeding 9th International Seminar on Model Optimization in Exploration Geophysics, Berlin, 1991
51 Engl H.W., Lions A.K., Rundell W. Problems in Medical Imaging and Nondetructive Testing. Wien, New York: Springer, 1996
52 Lagendijk R.L., Biemond J. Iterative Identification and Restoration of Images. Boston, Dordrecht, London: Kluwer, 1991
53 Landau H.J. eds. Moments in Mathematics. Providence, Rhode Island: AMS, 1987
54 Xia X.G., Nashed M.Z. The Backus-Gilbert method for signals in reproducing kernel Hilbert spaces and wavelet subspaces. Inverse Problems, 1994; 10:785-804
55 Ang D.D., Gorenflo R., Trong D.D. A multidimensional Hausdorff moment problem: Regularization by finite moments and error estimates. Zeitschrift fur Anwendungen(Journal for Analysis and Applications), 1999, 18:13-25
56 Askey G., Schoenberg I.J., Sharma A. Hausdorff moment problem and expansion in Legendre polynomials. J. Math. Anal. Appl., 1982; 86
57 Talenti G. Recovering a function from a finite number of moments. Inverse Problems, 1987; 3:501-517
58 Shohat J.A., Tamarkin J.D. The Problem of Moment. In: Math. Surveys, Providence, RI: Am. Math. Soc., 1943
59 Wang L.A. modified method for linear moment problem. Mathematica Numerica Sinica, 1999; 21(3): 303-308(in Chinese)
60 Cheng J., Hon Y.C.; Wei T., Yamamoto M. Numerical computation of Cauchy problem for Laplace's equation. ZAMM, 2001; 81(10): 665-674
61 Cheng J., Xu D.H., Yamamoto M. An inverse contact problem in the theory of elasticity. Mathematical Methods in Applied Sciences, 1999; 22: 1001-1015
62 Rudin W. Real and Complex Analysis. New York: McGraw Hill, 1966
63 Bruckner G., Cheng J. Tikhonov regularization for an integral equation of the first kind with logarithmic kernel. Preprint (Weierstrass Institute of Angewandte Analysis und Stochastik), 1998
64 Xu D.H., Li M.Z. Stability estimation and numerical analysis for Hausdorff moment problems．CSIAM'2000中国工业与应用数学学会第六次大会论文集，北京大学出版社，2000：307-312
65 Xu D.H., Huang S.X.,Li M.Z.Local condidonal stability and numerical analysis for Hausdorff moment problems.,to appear in Inverse Problems
66 钱伟长．弹性力学．北京：科学出版社，1980
67 Cheng J., Yamamoto M. Conditional stabilizing estimation for an integral equation of the first kind with analytic kernel. Journal of Inverse and Ill- Posed Problems, 1998; 6:115-126
68 Cheng J., Prosdorf S., Yamamoto M. Local estimation for an integral equation of the first kind with analytic kernel. Journal of Integral Equations and Applications, 2000; 12:39-61
69 Kolakowski H. An integral equation of elasticity. Mathematical Methods in Applied Sciences, 1985; 7:261-268
70 Payne L.E. Bounds in Cauchy problems for the Laplace equation. Arch. Rational Mech. Anal., 1960; 5:35-45
71 Stein E.M. Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press, New Jersey, 1970
72 Verchota G. Layer potential and regularity for the Dirichlet problem for Laplace's equation in the Lipschitz domains. J. Funct. Anal., 1984; 59: 572-611
73 Protter M.H.,Weinberger H.F. Maximum Principles in Differential Equations. NJ: Prentice Hall, Englewood Cliffs, 1967
74 Fainburd V.M. Contact Problems of the Theory of Elasticity. Moskow: MIR, 1975(in Russian)
75 Isakov V. Stability estimates for obstacles in inverse scattering. Journal of Computational and Applied Mathematics, 1992; 42:79-88
76 Friedman A., Vogelius M. Determining cracks by boundary measurements. Indiana Univ.Math.J., 1989; 38:527-556
77 Kellog O.D. Foundations of Potential Theory. New York: Dover, 1953
78 Cannon J.R. The One-Dimensional Heat Equation. MA: Addison-Wesley, Reading, 1984
79 路见可．平面弹性复变方法．武汉：武汉大学出版社，1986
80 Beck J., Blackwell B., Clair C.S. Inverse Heat Conduction. Sussex: Wiley,1965
81 Binder A., Engl H.W., Vessella S. Some inverse problems for a nonlinear parabolic equation connected with continuous casting of steel: stability estimates and regularization. Numer. Funct. Anal. Optim, 1990; 11:643-672
82 Landel G., Engl H.W., Chert J., Zeman K. Optimal strategies for the cooling of steel strips in hot strip mills. Inverse Problems in Engineering, 1995; 2:103-118
83 Choulli M., Yamamoto M. An inverse parabolic problem with non-zero initial condition. Inverse Problems, 1997; 13:19-27
84 Isakov V. A class of inverse problems for parabolic equations. Dokl. Akad. Nauk SSSR, 1982; 263:1296-1299
85 Saitoh S., Yamamoto M. Stability of Lipschitz type in determination of initial heat distribution. Journal of Inequalities and Applications, 1997; 1:73-83
86 Yamamoto M. Conditional stability in determination of force terms in a rectangle. Math. Comput. Modelling, 1993; 18:79-88
87 Imanuvilov O. Yu., Yamamoto M. Lipschitz stability in inverse parabolic problems by Carleman estimate. Inverse Problems, 1998; 14:1229-1245
88 Imanuvilov O.Yu. Controllability of parabolic equations. Sbornik Math., 1995; 186:879-900
89 Yamamoto M. Conditional stability in determination of densities of heat sources in a bounded domain. International Series of Numerical Mathematics, Birkhauser Verlag Basel, 1994; 118:359-370
90 Payne L.E. Improperly Posed Problems in Partial Differential Equations. Philadelphia: SIAM, 1975
91 Isakov V. Inverse Source Problems. Providence, Rhode Island: American Mathematical Society, 1990
92 Lions J.L. Exact controllability, stabilization and perturbations for distributed systems. SIAM Review, 1988; 30:1-68
93 Friedman A.(夏宗伟译，姜礼尚校)抛物型偏微分方程．北京：科学出版社，1984
94 Hua L., Lin W., Wu Z. Second Order Systems of Equations on the Plane. Pitman Publishing Ltd., 1985
95 栾文贵．位场解析延拓的稳定性估计．地球物理学报，1983；26(3)：263-273
96 栾文贵．一般线性偏微分方程 Cauchy问题的唯一性和连续依赖性．中国科学(A辑)，1983；(10)：871-882
97 Xu Dinghua,Cheng Jin, Li Mingzhong. On the stability estimation of analytic continuation for potential fields. Applied Mathematics and Mechanics, 1998; 19:563-572
98 Dinh Nho H?o, Schneider A,, Reinhardt H.J. Regularization of a non-characteristic Cauchy problem for a parabolic equation. Inverse Problems, 1995, 11:1247-1263
99 Daubechies I. Orthonormal bases of compactly supported wavelets. Comm.
Pure Appl. Math., 1988, 41:909-996
100 Daubechies I.Ten Lectures on Wavelets. Providence:SIAM,1992
101 Meyer Y. Wavelets and Operators. Cambridge University Press, 1992
102 Beylkin G., Coifmann, Rokhlin V. Fast wavelet transforms and numerical algorithm I. Comm. Pure Appl. Math., 1991; 44:141-183
103 Donoho D.L. Non-linear solution of linear inverse problems by wavelet-vaguelette decomposition. Applied and Computational Harmonic Analysis, 1995; 2:101-126
104 Freden W., Schneider F.Regularization wavelets and multiresolution. Inverse Problems, 1998; 14:225-243
105 Lin W., Liu M. On semi-orthogonal wavelet bases of periodic splines and their duals.Proceeding of the International Conference on Differential Equations and Control Theory, (Wuhan May 1994), Marcel Dekker, 1995
106 Wolfersdorf L.V. A class of nonlinear Riemann-Hilbert problems for holomorphic functions. Math. Nachr., 1984; 116: 89-107
107 Wolfersdorf L.V. A class of nonlinear Riemann-Hilbert problems with monotone nonlinearity. Math. Nachr., 1987; 130:111-119
108 Kencht S., Mashimba A.S. On the existence of a solution to the Riemann-Hilbert problems for a partial differential equation. Complex Variables, 1986; 6:240-263
109 Gilbert R. P., Li M.Z. A class of nonlinear Riemann-Hilbert problems for the second order elliptic system. Complex Variables, 1997; 32:105-129
110 李明忠．一阶椭圆型方程组一般形式的非线性边值问题．非线性力学国际会议论文集，上海，1998
111 Begehr H., Hsiao G.C. Nonlinear Boundary Value Problem for a Class of Elliptic Systems. Lecture Notes in Math., Berlin: Springer-Vrelag, 1980
112 Li M.Z., Xu D.H. A class of nonlinear boundary value problems for a second order elliptic system in general form. Applicable Analysis, 1999,
73(1-2): 137-152
113 Li M.Z., Xu D.H. Non-E2 class of strongly nonlinear generalized R-H boundary value problems for second order elliptic system. Journal of Shanghai University, 1998; 2(3): 173-181
114 黄思训，伍荣生编著．大气科学中的数学物理问题．北京：气象出版社，2001
115 Ciarlet P.G.(蒋尔雄等译)有限元素法的数值分析．上海：上海科学技术出版社，1978
2 冯康．数值计算方法．北京：科学出版社，1979
3 苏超伟．偏微分方程逆问题的数值方法及其应用．西安：西北工业大学出版社，1995
4 李明忠，侯宗义，徐振远．椭圆型方程组理论和边值问题．上海：复旦大学出版社，1990
5 Tikhonov A.N., Arsenin N.Y. Solutions of Ill-Posed Problems. Washington D.C.: V.H.Winston & Sons, 1977
6 Groestch C.W. Inverse Problems in the Mathematical Sciences. Braun-schweig: Vieweg, 1993
7 Kitsch A. An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag New York Inc., 1996.
8 Kress R. Linear Integral Equations. Berlin: Springer-Vedag, 1989
9 Isakov V. Inverse Problems for Partial Differential Equations. Berlin: Springer, 1998
10 Engl H.W., Hanke M., Neubauer A. Regularization of Inverse Problems. Dordrecht/Boston/London: Kluwer Academic Publishers, 1996
11 Murio D.A. The Mollification Method and the Numerical Solution of Ill-Posed Problems. NewYork: John Wiley & Sons Inc., 1993
12 Lavrent'ev M.M., Romanov V.G., Shishatskii S.P. Ill-Posed Problems of Mathematical Physics and Analysis, Providence RI: American Mathematical Society, 1986
13 Colton D., Kress R. Integral Equation Methods in Scattering Theory. New
York: John Wiley & Sons, 1983
14 Ramm A.G. Scattering by Obstacles. Dordrecht: D. Reldel Publishing Company, 1986
15 Fursikov A.V., Imanuvilov O.Yu. Controllability of Evolution Equations. Notes of the Series of Lectures held at the Seoul National University, 1996
16 赵彦．逆问题研究关联科学创新．科学时报(中国科学院主办，科学时报出版社出版)，1999；(1494)
17 Agmon S. Lectures on Elliptic Boundary Value Problems. Princeton, N. J.: Van Nostrand, 1965
18 Ladyzhenskaya O.A., Ural'tseva N.N. Linear and Quasi-linear Elliptic Equations. New York: Academic Press, (English Translation Edition), 1968
19 李大潜，秦铁虎．物理学与偏微分方程．北京：高等教育出版社，1997
20 Quarteroni A., Valli A. Numerical Approximation for Partial Differential Equations. Berlin: Springer-Verlag, 1997
21 夏道行，严绍宗，吾作人，舒五昌．泛函分析．北京：人民教育出版社，1979
22 Chui C.K. An Introduction to Wavelets. NewYork: Academic Press, Inc.,1992．(有中译本：程正兴译，自居宪审校．小波分析导论．西安：西安交通大学出版社，1994)
23 Prasad L., Iyengar S.S. Wavelet Analysis with Applications to Image Processing, Boca Raton: CRC Press, 1997
24 Adams R. A.(叶其孝等译)．Sobolev空间．北京：人民教育出版社，1979
25 Courant R., Hilbert D. Methods of Mathematical Physics. InterScience Publishers, 1962
26 Muskhelishvili N.I. Singular Integral Equations, Boundary Value Problems of Function Theory and Their Application to Mathematical Physics. New York: Dover Publications Inc., 1992
27 Gilbarg D., Trudinger N.S. Elliptic Partial Differential Equations of
Second Order. (2nd edition), Berlin, NewYork: Springer, 1983
28 Engl H.W., Isakov V. On the identifiability of steel reinforcement bars in concrete from magnetostatic measurements. Europe J. Appl. Math., 1992, 13:255-162
29 Bukhgeim A.L., Cheng J., Yamamoto M. Local stability of a lineaized inverse problem in determining steel reinforcement bars. Proceedings of International Conference on Inverse Problems and Applications, Philippin,1998
30 Hadamard J. Lectures on the Cauchy Problem in Linear Partial Differential Equations. New Haven: Yale University Press, 1923
31 Kaltenbacher B. Some Newton-type methods for the regularization of nonlinear ill-posed problems. Inverse Problems, 1997; 13:729-753
32 Cheng J., Yamamoto M. One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization. Inverse Problems, 2000; 16:L31-38
33 Kirsch A., Schomberg B., Berendt G. The Backus-Gilbert method. Inverse Problems, 1988; 10:771-783
34 吴乃龙，袁素云著．最大熵方法．长沙：湖南科学技术出版社，1991
35 Hettlich F., Rundell W. Iterative methods for the reconstruction of an inverse potential problem. Inverse Problems, 1999;12:251-266
36 Hansen P.C. The Backus-Gilbert method: SVD analysis and fast implementation. Inverse Problems, 1994; 10:895-904
37 Chen Y.M. Generalized pulse-spectrum technique. Geophysics, 1985; 50: 1664-1675
38 Kormonik V. Exact Controllability and Stabilization. Chichester: John Wiley & Sons, 1994
39 Beretta E., Vessella S. Stable determination of boundaries from Cauchy data. SIAM J. Math. AnaL, 1998; 30:220-232
40 Alessandrini G., Isakov V., Powell J. Local uniqueness in the inverse conductivity problem with one measurement. Trans. of American Mathematical Society, 1995; 347:3031-3041
41 Cedio-Fengya D.J., Moskow S., Vogelius M.S. Identification of conductivity imperfection of small diameter by boundary measyrements: continuous dependence and computational reconstruction. Inverse Problems, 1998; 14:553-595
42 Su C.W. An iterative algorithm for determining time-dependent coefficient of two-dimensional linear wave equation. Annals of Differential Equations, 1995; 11:213 -223
43 Bekya H．广义解析函数．北京：人民教育出版社，1960
44 Groetsch C.W. The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Boston: Pitman, 1984
45 Gorenflo R., Vessella S. Abel integral equations:analysis and applications. Lectures Notes in Mathematics, 1461, Berlin: Springer, 1991
46 Engl H.W., Landl G. Maximum entropy regularization of nonlinear ill-posed problems. In Lakshmikantham V(eds.): Proceedings of the First World Congress of Nonlinear Analysis, Vol.I: 513-525, Berlin: de Gruyter, 1996
47 Neubauer A. Tikhonov regularization for nonlinear ill-posed problems: optimal convergence and finite-dimensional approximation. Inverse Problems, 1989, 5:541-557
48 Ang D.D., Vy L.K., Gorerflo R. A regularization method for the moment problem, in: Inverse Problems: Principles and Applications in Geophysics, Technology and Medicine, Math. Research, Berlin: Akademic Verlag, 1993, 74:37-45
49 Backus G.E., Gilbert J.F. The resolving power of gross earth data. Geophysical Journal of the Royal Astronomical Society, 1968; 16:169-205
50 Inglese G. Recent results in the study of the moment problem. In:
Geophysical Data Interpretation by Inverse Modeling, Proceeding 9th International Seminar on Model Optimization in Exploration Geophysics, Berlin, 1991
51 Engl H.W., Lions A.K., Rundell W. Problems in Medical Imaging and Nondetructive Testing. Wien, New York: Springer, 1996
52 Lagendijk R.L., Biemond J. Iterative Identification and Restoration of Images. Boston, Dordrecht, London: Kluwer, 1991
53 Landau H.J. eds. Moments in Mathematics. Providence, Rhode Island: AMS, 1987
54 Xia X.G., Nashed M.Z. The Backus-Gilbert method for signals in reproducing kernel Hilbert spaces and wavelet subspaces. Inverse Problems, 1994; 10:785-804
55 Ang D.D., Gorenflo R., Trong D.D. A multidimensional Hausdorff moment problem: Regularization by finite moments and error estimates. Zeitschrift fur Anwendungen(Journal for Analysis and Applications), 1999, 18:13-25
56 Askey G., Schoenberg I.J., Sharma A. Hausdorff moment problem and expansion in Legendre polynomials. J. Math. Anal. Appl., 1982; 86
57 Talenti G. Recovering a function from a finite number of moments. Inverse Problems, 1987; 3:501-517
58 Shohat J.A., Tamarkin J.D. The Problem of Moment. In: Math. Surveys, Providence, RI: Am. Math. Soc., 1943
59 Wang L.A. modified method for linear moment problem. Mathematica Numerica Sinica, 1999; 21(3): 303-308(in Chinese)
60 Cheng J., Hon Y.C.; Wei T., Yamamoto M. Numerical computation of Cauchy problem for Laplace's equation. ZAMM, 2001; 81(10): 665-674
61 Cheng J., Xu D.H., Yamamoto M. An inverse contact problem in the theory of elasticity. Mathematical Methods in Applied Sciences, 1999; 22: 1001-1015
62 Rudin W. Real and Complex Analysis. New York: McGraw Hill, 1966
63 Bruckner G., Cheng J. Tikhonov regularization for an integral equation of the first kind with logarithmic kernel. Preprint (Weierstrass Institute of Angewandte Analysis und Stochastik), 1998
64 Xu D.H., Li M.Z. Stability estimation and numerical analysis for Hausdorff moment problems．CSIAM'2000中国工业与应用数学学会第六次大会论文集，北京大学出版社，2000：307-312
65 Xu D.H., Huang S.X.,Li M.Z.Local condidonal stability and numerical analysis for Hausdorff moment problems.,to appear in Inverse Problems
66 钱伟长．弹性力学．北京：科学出版社，1980
67 Cheng J., Yamamoto M. Conditional stabilizing estimation for an integral equation of the first kind with analytic kernel. Journal of Inverse and Ill- Posed Problems, 1998; 6:115-126
68 Cheng J., Prosdorf S., Yamamoto M. Local estimation for an integral equation of the first kind with analytic kernel. Journal of Integral Equations and Applications, 2000; 12:39-61
69 Kolakowski H. An integral equation of elasticity. Mathematical Methods in Applied Sciences, 1985; 7:261-268
70 Payne L.E. Bounds in Cauchy problems for the Laplace equation. Arch. Rational Mech. Anal., 1960; 5:35-45
71 Stein E.M. Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press, New Jersey, 1970
72 Verchota G. Layer potential and regularity for the Dirichlet problem for Laplace's equation in the Lipschitz domains. J. Funct. Anal., 1984; 59: 572-611
73 Protter M.H.,Weinberger H.F. Maximum Principles in Differential Equations. NJ: Prentice Hall, Englewood Cliffs, 1967
74 Fainburd V.M. Contact Problems of the Theory of Elasticity. Moskow: MIR, 1975(in Russian)
75 Isakov V. Stability estimates for obstacles in inverse scattering. Journal of Computational and Applied Mathematics, 1992; 42:79-88
76 Friedman A., Vogelius M. Determining cracks by boundary measurements. Indiana Univ.Math.J., 1989; 38:527-556
77 Kellog O.D. Foundations of Potential Theory. New York: Dover, 1953
78 Cannon J.R. The One-Dimensional Heat Equation. MA: Addison-Wesley, Reading, 1984
79 路见可．平面弹性复变方法．武汉：武汉大学出版社，1986
80 Beck J., Blackwell B., Clair C.S. Inverse Heat Conduction. Sussex: Wiley,1965
81 Binder A., Engl H.W., Vessella S. Some inverse problems for a nonlinear parabolic equation connected with continuous casting of steel: stability estimates and regularization. Numer. Funct. Anal. Optim, 1990; 11:643-672
82 Landel G., Engl H.W., Chert J., Zeman K. Optimal strategies for the cooling of steel strips in hot strip mills. Inverse Problems in Engineering, 1995; 2:103-118
83 Choulli M., Yamamoto M. An inverse parabolic problem with non-zero initial condition. Inverse Problems, 1997; 13:19-27
84 Isakov V. A class of inverse problems for parabolic equations. Dokl. Akad. Nauk SSSR, 1982; 263:1296-1299
85 Saitoh S., Yamamoto M. Stability of Lipschitz type in determination of initial heat distribution. Journal of Inequalities and Applications, 1997; 1:73-83
86 Yamamoto M. Conditional stability in determination of force terms in a rectangle. Math. Comput. Modelling, 1993; 18:79-88
87 Imanuvilov O. Yu., Yamamoto M. Lipschitz stability in inverse parabolic problems by Carleman estimate. Inverse Problems, 1998; 14:1229-1245
88 Imanuvilov O.Yu. Controllability of parabolic equations. Sbornik Math., 1995; 186:879-900
89 Yamamoto M. Conditional stability in determination of densities of heat sources in a bounded domain. International Series of Numerical Mathematics, Birkhauser Verlag Basel, 1994; 118:359-370
90 Payne L.E. Improperly Posed Problems in Partial Differential Equations. Philadelphia: SIAM, 1975
91 Isakov V. Inverse Source Problems. Providence, Rhode Island: American Mathematical Society, 1990
92 Lions J.L. Exact controllability, stabilization and perturbations for distributed systems. SIAM Review, 1988; 30:1-68
93 Friedman A.(夏宗伟译，姜礼尚校)抛物型偏微分方程．北京：科学出版社，1984
94 Hua L., Lin W., Wu Z. Second Order Systems of Equations on the Plane. Pitman Publishing Ltd., 1985
95 栾文贵．位场解析延拓的稳定性估计．地球物理学报，1983；26(3)：263-273
96 栾文贵．一般线性偏微分方程 Cauchy问题的唯一性和连续依赖性．中国科学(A辑)，1983；(10)：871-882
97 Xu Dinghua,Cheng Jin, Li Mingzhong. On the stability estimation of analytic continuation for potential fields. Applied Mathematics and Mechanics, 1998; 19:563-572
98 Dinh Nho H?o, Schneider A,, Reinhardt H.J. Regularization of a non-characteristic Cauchy problem for a parabolic equation. Inverse Problems, 1995, 11:1247-1263
99 Daubechies I. Orthonormal bases of compactly supported wavelets. Comm.
Pure Appl. Math., 1988, 41:909-996
100 Daubechies I.Ten Lectures on Wavelets. Providence:SIAM,1992
101 Meyer Y. Wavelets and Operators. Cambridge University Press, 1992
102 Beylkin G., Coifmann, Rokhlin V. Fast wavelet transforms and numerical algorithm I. Comm. Pure Appl. Math., 1991; 44:141-183
103 Donoho D.L. Non-linear solution of linear inverse problems by wavelet-vaguelette decomposition. Applied and Computational Harmonic Analysis, 1995; 2:101-126
104 Freden W., Schneider F.Regularization wavelets and multiresolution. Inverse Problems, 1998; 14:225-243
105 Lin W., Liu M. On semi-orthogonal wavelet bases of periodic splines and their duals.Proceeding of the International Conference on Differential Equations and Control Theory, (Wuhan May 1994), Marcel Dekker, 1995
106 Wolfersdorf L.V. A class of nonlinear Riemann-Hilbert problems for holomorphic functions. Math. Nachr., 1984; 116: 89-107
107 Wolfersdorf L.V. A class of nonlinear Riemann-Hilbert problems with monotone nonlinearity. Math. Nachr., 1987; 130:111-119
108 Kencht S., Mashimba A.S. On the existence of a solution to the Riemann-Hilbert problems for a partial differential equation. Complex Variables, 1986; 6:240-263
109 Gilbert R. P., Li M.Z. A class of nonlinear Riemann-Hilbert problems for the second order elliptic system. Complex Variables, 1997; 32:105-129
110 李明忠．一阶椭圆型方程组一般形式的非线性边值问题．非线性力学国际会议论文集，上海，1998
111 Begehr H., Hsiao G.C. Nonlinear Boundary Value Problem for a Class of Elliptic Systems. Lecture Notes in Math., Berlin: Springer-Vrelag, 1980
112 Li M.Z., Xu D.H. A class of nonlinear boundary value problems for a second order elliptic system in general form. Applicable Analysis, 1999,
73(1-2): 137-152
113 Li M.Z., Xu D.H. Non-E2 class of strongly nonlinear generalized R-H boundary value problems for second order elliptic system. Journal of Shanghai University, 1998; 2(3): 173-181
114 黄思训，伍荣生编著．大气科学中的数学物理问题．北京：气象出版社，2001
115 Ciarlet P.G.(蒋尔雄等译)有限元素法的数值分析．上海：上海科学技术出版社，1978