求解几类不适定问题的非经典正则化方法研究
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摘要
本篇论文用几种新的非经典正则化方法研究了三类数学物理反问题——热方程中的未知源识别问题、椭圆方程Cauchy问题,以及数值微分问题.
热方程的未知源识别问题具有极广泛的物理背景,目前是反问题研究中的重要问题之一.本文对该问题的几种非经典正则化方法做了较系统地研究.分析了未知源识别问题的不适定性,在最优性分析的框架下推导出该问题正则化方法的最优误差界.而后,分别在无界和有界区域上用Fourier方法、小波Galerkin方法、小波对偶最小二乘方法及拟逆方法等多种非经典正则化方法对该问题进行了数值求解.
椭圆方程Cauchy问题是严重不适定的.文中用高维小波方法求解了几类椭圆方程Cauchy问题,包括Laplace方程Cauchy问题、Helmholtz方程Cauchy问题和修正的Helmholtz方程Cauchy问题.
本文还利用小波Galerkin方法对数值微分问题进行了进一步地深入研究,得到了较好的理论和数值结果.
文中对用上述方法求解的三类数学物理反问题都进行了严格的理论分析,给出了正则逼近解和精确解之间的误差估计.此外,本文还给出大量的数值例子并做了相应的数值试验.数值结果验证了这些方法都是非常有效的.
In the present thesis, we study three kinds of ill-posed problems in mathematical physics, including identification of unknown source in the heat equation, the Cauchy problems for the elliptic equation, numerical differentiation and etc., by some new non-classical regularization methods.
Identification of unknown source term in the heat equation is one of the focal points in the research area of inverse problem for its wide physical application. In this thesis we work over some non-classical regularization methods for this problem. We analyze the ill-posedness for the problem. Under an a priori condition we answer the question concerning the best possible accuracy. Then we study the problem in the unbounded and bounded domain by the Fourier method, wavelet-Galerkin method, wavelet dual least square method and quasi-reversibility method, respectively.
The Cauchy problems of elliptic equation are severely ill-posed. We solve several Cauchy problems of elliptic equation, including the Cauchy problem of the Laplace equation, the Cauchy problem of the Helmholtz equation and the Cauchy problem of the modified Helmholtz equation (Yukawa) by multidimensional Meyer wavelet method.
We also study numerical differentiation further by the wavelet-Galerkin method, and obtain some preferable results in both theory and numerical aspects.
We discuss the stability of all the above regularization methods for solving these ill-posed problems in mathematical physics, and prove the convergence estimates for the exact solutions and their regularized approximation. Moreover, we give lots of examples and make numerical tests. The numerical results show the efficiency and accuracy of the proposed methods.
热方程的未知源识别问题具有极广泛的物理背景,目前是反问题研究中的重要问题之一.本文对该问题的几种非经典正则化方法做了较系统地研究.分析了未知源识别问题的不适定性,在最优性分析的框架下推导出该问题正则化方法的最优误差界.而后,分别在无界和有界区域上用Fourier方法、小波Galerkin方法、小波对偶最小二乘方法及拟逆方法等多种非经典正则化方法对该问题进行了数值求解.
椭圆方程Cauchy问题是严重不适定的.文中用高维小波方法求解了几类椭圆方程Cauchy问题,包括Laplace方程Cauchy问题、Helmholtz方程Cauchy问题和修正的Helmholtz方程Cauchy问题.
本文还利用小波Galerkin方法对数值微分问题进行了进一步地深入研究,得到了较好的理论和数值结果.
文中对用上述方法求解的三类数学物理反问题都进行了严格的理论分析,给出了正则逼近解和精确解之间的误差估计.此外,本文还给出大量的数值例子并做了相应的数值试验.数值结果验证了这些方法都是非常有效的.
In the present thesis, we study three kinds of ill-posed problems in mathematical physics, including identification of unknown source in the heat equation, the Cauchy problems for the elliptic equation, numerical differentiation and etc., by some new non-classical regularization methods.
Identification of unknown source term in the heat equation is one of the focal points in the research area of inverse problem for its wide physical application. In this thesis we work over some non-classical regularization methods for this problem. We analyze the ill-posedness for the problem. Under an a priori condition we answer the question concerning the best possible accuracy. Then we study the problem in the unbounded and bounded domain by the Fourier method, wavelet-Galerkin method, wavelet dual least square method and quasi-reversibility method, respectively.
The Cauchy problems of elliptic equation are severely ill-posed. We solve several Cauchy problems of elliptic equation, including the Cauchy problem of the Laplace equation, the Cauchy problem of the Helmholtz equation and the Cauchy problem of the modified Helmholtz equation (Yukawa) by multidimensional Meyer wavelet method.
We also study numerical differentiation further by the wavelet-Galerkin method, and obtain some preferable results in both theory and numerical aspects.
We discuss the stability of all the above regularization methods for solving these ill-posed problems in mathematical physics, and prove the convergence estimates for the exact solutions and their regularized approximation. Moreover, we give lots of examples and make numerical tests. The numerical results show the efficiency and accuracy of the proposed methods.
引文
[1]G.Alessandrini,Stable determination of a crack from boundary measurements,Proc.Roy.Soc,Edinburgh Sect.A,123(1993) 497-516.
[2]R.S.Anderssen and M.Hegland,For numerical differentiation,dimensionality can be a blessing,Math,comput.,68(227)(1999) 1121-1141.
[3]D.D.Ang,N.H.Nghia and N.C.Tarn,Regularized solutions of a Cauchy problem for the Laplace equation in an irregular layer:A three-dimensional model,Acta Math.Vietnam.,23(1998) 65-74.
[4]N.Bleistein,J.Cohen,J.Jr.Stockwell,Mathematics of Multidimensional Seismic Imaging,Migration,and Inversion,Springer,Berlin,2001.
[5]G.Beylkin,On the representation of operators in bases of compactly supported wavelets,SIAM J.Numer.Anal.,6(1992) 1716-1740.
[6]L.Bourgeois,A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation,Inverse Problems,21(2005) 1087-1104.
[7]A.L.Bukhgeim,J.Cheng,M.Yamamoto,Stability for an inverse boundary problem of determining a part of a boundary,Inverse Problems,15(1999) 1021-1032.
[8]A.L.Bukhgeim,J.Cheng,M.Yamamoto,On a sharp estimate in a nondestructive testing:determination of unknown boundaries.In:K.Miya,M.Yamamoto,Nguyen Xuan Hung(eds.):Applied electromagnetism mechanics.JSAEM 1998,64-75.
[9]A.A.Burykin,A.M.Denisov,Determination of the unknown sources in the heat-conduction equation,Comput.Math.Model.,8(1997) 309-313.
[10]A.Calder(?)n,Uniqueness in the Cauchy problem for partial differential equations,Amer.J.Math.,80(1958) 16-36.
[11]J.R.Cannon,One dimensional heat equation,Addison-Wesley Publishing Company,California,1984.
[12]J.R.Cannon and P.DuChateau,Structural identification of an unknown source term in a heat equation,Inverse Problems,14(1998) 535-551.
[13]J.R.Cannon and S.P.Estevz,An inverse problem for the heat equation,Inverse Problems,2(1986) 395-403.
[14]H.W.Cheng,J.F.Huang and T.J.Leiterman,An adaptive fast solver for the modified Helmholtz equation in two dimensions,J.Comput.Phys.,211(2006)616-637.
[15]J.Cheng,Y.C.Hon and Y.B.Wang,A numerical method for the discontinuous solutions of Abel integral equations,In inverse problems and spectral theory,Contemp.Math.,Amer,Math.Soc,Providence,RI,348(2004) 23-3-243.
[16]J.Cheng,Y.C.Hon,T.Wei and M.Yamamoto,Numerical computation of a Cauchy problem for Laplace's equation,ZAMM Z.Angew.Math.Mech.,81(2001),665-674.
[17]J.Cheng,S.Prossdorf,M.Yamamoto,Local estimation for an integral equation of first kind with analytic kernel,J.Inverse Ill-Posed Probl,6(1998) 115-126.
[18]J.Cheng,M.Yamamoto,Unique continuation on a line for harmonic functions,Inverse Problems,14(1998) 869-882.
[19]J.Cheng,M.Yamamoto,Local stability of a linearized inverse problem in detecting steel reinforcement bars,Proc.Int.Conf.Inverse Problems and Applications,Quezon City,1998.Matimylas Mat.21(1998) Special issue,18-33.
[20]M.Choulli and M.Yamamoto,Conditional stability in determining a heat source,J.Inv.Ill-posed Probl.,12(2004) 233-243.
[21]P.Colli-Franzone,L.Guerri,S.Tentoni,C.Viganotti,S.Baruffi,S.Spaggiari,B.Taccardi,A mathematical procedure for solving the inverse potential problem of electrocardiography,Analysis of the time-space accuracy from in vitro experimental data,Math.Biosci.,77(1985) 353-396.
[22]P.Colli-Franzone and E.Magenes,On the inverse potential problem of electrocardiology,Calcolo,16(1979) 459-538.
[23]J.Cullum,Numerical differentiation and regularization,SIAM J.Numer.Anal.,8(1971) 254-265.
[24]I.Daubechies,Ten lectures on wavelets,CBMS-NSF Regional Conf.Series in Applied Mathematics,SIAM,Philadelphia,1992.
[25]S.R.Deans,The Radon transform and some of its applications,A Wiley- Interscience Publication,John Wiley & Sons Inc.,New York,1983.
[26]J.R.Dorroh and X.P.Ru,The application of the method of quasi-reversibility to the sideways heat equation,J.Math.Anal.Appl.,236(1999) 503-519.
[27]F.F.Dou,C.L.Fu,Determining unknown source in the heat equation by a wavelet dual least squares method,Applied Mathematics Letters,22(2009)661-667.
[28]F.F.Dou,C.L.Fu,F.L.Yang,Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation,Journal of Computational and Applied Mathematics,Doi:10.1016/j.ca m.2009.01.008.
[29]F.F.Dou,C.L.Fu,F.Yang,Identifying an unknown source term in a heat equation,Inverse Problems in Science and Engineering,accepted.
[30]L.Elden,Approximations for a Cauchy problem for the heat equation,Inverse Problems,3(1987) 263-273.
[31]L.Eld(?)n,Solving an inverse heat conduction problem by 'Method of lines',Trans.ASME J.Heat Transfer,119(1997) 913-923.
[32]L.Elden,F.Berntsson and T.Reginska,Wavelet and Fourier methods for solving the sideways heat equation,J.Sci.Comput.,21(2000) 2187-2205.
[33]H.W.Engl,M.Hanke and A.Newbauer,Regularization of inverse problems,Kluwer Academic,Boston,MA 1996.
[34]A.Farcas and D.Lesnic,The boundary-element method for the determination of a heat source dependent on one variable,J.Engin.Math.,54(2006) 375-388.
[35]A.Farcas and D.Lesnic,The boundary-element method for the determination of a heat source dependent on one variable,J.Eng.Math.,54(2006) 375-388.
[36]C.L.Fu,Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation,J.Comput.Appl.Math.,167(2004) 449-463.
[37]C.L.Fu,F.F.Dou,X.L.Feng and Z.Qian,A simple regularization method for stable analytic continuation,Inverse problems,24(2008) 065003.
[38]C.L.Fu,X.L.Feng and Z.Qian,The Fourier regularization for solving the Cauchy problem for the Helmholtz equation,Appl.Numer.Math.,to appear.
[39]C.L.Fu,H.F.Li ,Z.Qian and X.T.Xiong,Fourier regularization method for solving a Cauchy problem for the Laplace equation,J.Inv.Ill-posed Probl.,16(2008) 159-169.
[40]C.L.Fu,X.T.Xiong and Z.Qian,Fourier Regularization for a Backward Heat Equation,J.Math.Anal.Appl.,331(2007) 472-480.
[41]G.Golubev,R.Khasminskii,A statistical approach to the Cauchy problem for the Laplace equation,IMS Lecture Notes-Monograph Series,Volume 36 Beachwood,OH:Institute of Mathematical Statistics,616(2001) 419-433.
[42]R.Gorenflo,Funktionentheoretische Bestimmung des Aussenfeldes zu einer zweidimensionalen magnetohydrostatischen Konfiguration,Z.Angew.Math.Phys.16(1965) 279-290.
[43]R.Gorenflo and S.Vessella,Abel integral equations,volume 1461 of Lecture Notes in Mathematics,Springer-Verlag,Berlin,1991.
[44]C.W.Groetsch,Differentiation of approximately specified functions,Amer.Math.Monthly,98(9)(1991) 847-850.
[45]J.Hadamard,Lectures on Cauchy problem in linear partial differential equations,University Press,London,1923.
[46]M.Hanke and O.Scherzer,Error analysis of an equation error method for the identification of of the diffusion coefficient in a quasi-linear parabolic differentical equation,SIAM J.Appl,Math.,59(1999) 1012-1027.(eletronic)
[47]M.Hanke and O.Scherzer,Inverse problems light:numerical differentiation,Amer.Math.Monthly,108(6)(2001) 512-521.
[48]D.N.Hao and P.M.Hien,Stability results for the Cauchy problem for the Laplace equation in a strip,Inverse Problems,19(2003) 833-844.
[49]D.N.Hao and D.Lesnic,The Cauchy for Laplace's equation via the conjugate gradient method,IMA J.Appl.Math.,65(2000) 199-217.
[50]D.N.Hao,A.Schneiderx and H.J.Reinhardt,Regularization of a noncharacteristic Cauchy problem for a parabolic equation,Inverse Problems,11(1995) 124-1263.
[51]Y.C.Hon and T.Wei,Backus-Gilbert algorithm for the Cauchy problem of Laplace equation,Inverse Problems,17(2001) 261-271.
[52]Y.C.Hon and T.Wei,Solving Cauchy problems of elliptic equations by the method of fundamental solutions,In Boundary elements ⅩⅩⅦ,volume 39 of WIT Trans.Model.Simul,57-65.WIT Press,Southampton,2005.
[53]T.Hrycak and V.Isakov,Increased stability in the continuation of solutions to the Helmholtz equation,Inverse Problems,20(2004) 697-712.
[54]V.Isakov,Inverse problems for partial differential equations,Springer-Verlag,New York,1998.
[55]M.I.Ivanchov,The inverse problem of determining the heat source power for a parabolic equation under arbitrary boundary conditions,J.Math.Sci.,88(1998) 432-436.
[56]C.R.Johnson,Computational and numerical methods for bioelectric field problems,Crit.Rev.Biomed.Eng.,25(1997) 1-81.
[57]T.Johansson and D.Lesnic,Determining a spacewise dependent heat source,J.Comput.Appl.Math.,209(2007) 66-80.
[58]T.Johansson and D.Lesnic,A variational method for identifying a spacewisedependent heat source,IMA J.Appl.Math.,72(2007) 748-760.
[59]D.Q.Kern and A.D.Kraus,Extended surface heat transfer,McGraw-Hill,New York,1972.
[60]A.Kirsch,An introduction to the mathematical theory of inverse problems,Springer,Berlin,1999.
[61]M.V.Klibanov and F.Santosa,A computational quasi-reversibility method for Cauchy problems for Laplace's equation,SIAM J.Appl.Math.,51(1991),1653-1675.
[62]Y.Knosowski,E.Lieres and A.Schneider,Regularization of a non-characteristic Cauchy problem for a parabolic equation in multiple dimensions,Inverse Problems,15(1999) 731-743.
[63]E.D.Kolaczyk,Wavelet methods for the inversion of certain homogeneous linear operators in thepresence of noisy data,PhD Thesis,Stanford University,Stanford,CA,1994.
[64]A.V.Kononov,CD.Riyanti,S.W.de Leeuw,C.W.Oosterlee and C.Vuik,Numerical performance of a parallel solution method for a heterogeneous 2D Helmholtz equation,Comput.Visual Sci.,11(2008) 139-146.
[65]L.Landweber,An iteration formula for Fredholm integral equation of the first kind,Am.J.Math.,73(1951),615-624.
[66]R.Lattes and J.L.Lions,The method of quasi-reversibility:applications to partial differential equations,Elsevier,New York,1969.
[67]M.M.Lavrent'ev,V.G.Romanov and S.P.Shishatskii,Ill-Posed Problems of Mathematical Physics and Analysis,Translations of Mathematical Monographs vol 64,Providence,RI:American Mathematical Society,1986.
[68]G.S.Li,Data compatibility and conditional stability for an inverse source problem in the heat equation,Appl.Math.Comput.,173(2006) 566-581.
[69]G.S.Li and M.Yamamoto,Stability analysis for determining a source term in a 1-D advection-dispersion equation ,J.Inv.Ill-posed Probl.,14(2006) 147-155.
[70]李洪芳,几类逆边值问题的正则化方法及最优性分析,博士学位论文,兰州大学,2007.
[71]李振平,几类椭圆型方程Cauchy问题的磨光化求解方法,硕士学位论文,兰州大学,2008.
[72]刘继军,不适定问题的正则化方法和应用,科学出版社,2005.
[73]L.Ling,M.Yamamoto,Y.C.Hon and T.Takeuchi,Indentification of source locations in two-demensional heat equations.Inverse Problems,22(2006) 1289-1305.
[74]L.Matin,L.Elliott,P.J.Heggs,D.B.Ingham,D.Lesnic and X.Wen,An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation.Comput.Methods Appl.Mech.Eng.,192(2003) 709-722.
[75]L.Matin,L.Elliott,P.J.Heggs,D.B.Ingham,D.Lesnic and X.Wen,Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations.Comput.Mech.,31(2003) 367-377.
[76]L.Marin,L.Elliott,P.J.Heggs,D.B.Ingham,D.Lesnic and X.Wen,BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method,Eng.Anal.Bound.Elem.,28(2004) 1025-1034.
[77]L.Marin and D.Lesnic,The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations,Computers and Structures,83(2005) 267-278.
[78]Y.Meyer,Wavelets and Operators,Cambridge University Press,Cambridge,1992.
[79]D.A.Murio,C.E.Mej(?)a,S.Zhan,Discrete mollification and automatic numerical differentiation,Computers Math.Appl.35(1998) 1-16.
[80]L.E.Payne,Bounds in the Cauchy problem for the Laplace' s equation,Arch.Rat.Mech.Anal.,5(1960) 35-45.
[81]A.Peterson,S.Ray and R.Mittra,Computational Methods for Electromagnet-ics.IEEE Press,New York,1998.
[82]A.I.Prilepko and V.V.Solov'ev,Solvability theorems and Rote's method for inverse problems for a parabolic equation,J.Diff.Equ.,23(1998) 1230-1237.
[83]钱志,数学物理反问题的正则化,博士学位论文,兰州大学,2008.
[84]Z.Qian,C.L.Fu and X.T.Xiong,Fourth-order modified method for the Cauchy problem for the Laplace equation,J.Comput.Appl.Math.,192(2006) 205-218.
[85]Z.Qian,C.L.Fu,X.L.Feng,A modified method for high order numerical derivatives,Appl.Math.Comput.,182(2006) 1191-1120.
[86]Z.Qian,C.L.Fu,Regularization strategies for a two-dimensional inverse heat conduction problem,Inverse Problems,23(2007) 1053-1068.
[87]H.H.Qin,T.Wei,Modified regularization method for the Cauchy problem of the Helmholtz equation,Applied Mathematical Modelling,33(5)(2009) 2334-2348.
[88]H.H.Qin,T.Wei,R.Shi,Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation,J.Comput.Appl.Math.,224(1)(2009) 39-53.
[89]H.H.Qin,D.W.Wen,Tikhonov type regularization method for the Cauchy problem of the modified Helmholtz equation,Appl.Math.Comput.,203(2)(2008) 617-628.
[90]C.Y.Qiu and C.L.Fu,Wavelets and regularization of the Cauchy Problem for the Laplace equation,J.Math.Anal.Appl.,338(2008) 1440-1447.
[91]R.Qu,A new approach to numerical differentiation and integration,Math.comput.,24(10)(1996) 55-68.
[92]A.G.Ramm and A.B.Smirnova,On stable numerical differentiation,Math.Comput.,70(2001) 1131-1153.(electronic)
[93]V.S.Ryaben'kii,S.V.Tsynov and S.V.Utyuzhnikov,Inverse source problem and active shielding for composite domains,Appl.Math.Lett.,20(2007) 511-515.
[94]T.Regi(?)ska,Sideways heat equation and wavelets,J.Comput.Appl.Math.,63(1995) 209-214.
[95]T.Regi(?)ska and L.Eld(?)n,Solving the sideways heat equation by a wavelet-Galerkin method,Inverse Problems,13(1997) 1093-1106.
[96]T.Regi(?)ska and L.Eld(?)n,Stability and convergence of a wavelet-Galerkin method for the sideways heat equation,J.Inv.Ill-posed Problems,8(2000)31-49.
[97]T.Regi(?)ska,Application of wavelet shrinkage to solving sideways heat equation,BIT,41(5)(2001) 1101-1110.
[98]T.Regi(?)ska and K.Regi(?)ski,A Cauchy problem for the Helmholtz equation:application to analysis of light propagation in solids,Inverse Problems,22(2006)975-989.
[99]T.Regi(?)ska and A.Wakulicz,Wavelet moment method for Cauchy problem for the Helmholtz equation,J.Comput.Appl.Math.,223(1)(2009) 218-229.
[100]V.S.Ryaben'kii,S.V.Tsynov and S.V.Utyuzhnikov,Inverse source problem and active shielding for composite domains,Appl.Math.Lett.,20(2007) 511-515.
[101]P.C.Sabatier,Inverse problems:an introduction,Inverse problems,1(1985)1-4.
[102]E.G.Savateev,On problems of determining the source function in a parabolic equation,J.Inv.Ill-posed problems,3(1995) 83-102.
[103]A.Schneider,Wavelet-based mollication methods for some ill-posed problems,Dissertation Universit(a|¨)t Siegen,1996.
[104]T.Schrhter and U.Tautenhahn,On the 'optimal' regularization methods for solving linear ill-posed problems.Z.Anal.Anw.,13(1994) 697-710.
[105]V.V.Solov'ev,Solvability of the inverse problem of finding a source,using overdetermination on the upper base for a parabolic equation,Diff.Equ.,25(1990) 1114-1119.
[106]孙炯,王忠,线性算子的谱分析,科学出版社,北京,2005.
[107]U.Tautenhahn,Optimal stable solution of Cauchy problems for elliptic equations,Z.Anal.Anw.15(1996) 961-984.
[108]U.Tautenhahn,Optimal stable approximations for the sideways heat equation,J.Inv.Ill-posed Problems,5(1997) 287-307.
[109]U.Tautenhahn,Optimality for ill-posed problems under general source conditions,Numer.Funct.Anal.Optimiz.,19(1998) 377-398.
[110]U.Tautenhahn and R.Gorenflo,On Optimal regularization methods for fractional differentiation,J.Anal.Appl.,18(1999) 449-467.
[111]A.N.Tikhonov,V.Y.Arsenin,Solutions of ill-posed problems,Winston and Sons,Washington,1977.
[112]D.D.Trong,N.T.Long and P.N.D.Alain,Nonhomogeneous heat equation:Identification and regularization for the inhomogeneous term,J.Math.Anal.Appl.,312(2005) 93-104.
[113]G.Vainikko,On the optimality of methods for ill-posed problems,Z.Anal.Anw.,64(1987) 351-362.
[114]C.Vani and A.Avudainayagam,Regularized solution of the Cauchy problem for the Laplace equation using Meyer Wavelets,Math.Comput.Model.,36(2002)1151-1159.
[115]Y.B Wang,X.Z.Jia and J.Cheng,A numerical differentiation method and its application to reconstruction of discontinuity,Inverse Problems,18(2002)1461-1476.
[116]T.Wei,Y.C.Hon and L.Ling,Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators,Eng.Anal.Bound.Elem.,31(2007) 373-385.
[117]T.Wei,H.H.Qin and R.Shi,Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation,Inverse Problems,24(3)(2008)035003(18pp).
[118]A.S.Wood,G.E.Tupholme,M.I.H.Bhatti and P.J.Heggs,Steady-state heat transfer through extended plane surfaces,Int.Commun.Heat Mass Transfer 22(1995) 99-109.
[119]熊向团,抛物型方程反问题的正则化理论和算法,博士学位论文,兰州大学,2007.
[120]X.T.Xiong,C.L.Fu and Z.Qian,Two numerical methods for solving a backward heat conduction problem,Appl.Math.Comput.,179(2006) 370-377.
[121]X.T.Xiong and C.L.Fu,Central difference regularization method for the Cauchy problem of the Laplace's equation,Appl.Math.Comput.,181(2006)675-684.
[122]X.T.Xiong and C.L.Fu,Determining surface temperature and heat flux by a wavelet dual least squares method,J.Comput.Appl.Math.,201(2007) 198-207.
[123]X.T.Xiong and C.L.Fu,Error estimates on a backward heat equation by a wavelet dual least squares method,Int.J.Wavelets,Multiresolut.Inf.Process.,5(2007) 389-397.
[124]X.T.Xiong and C.L.Fu,Two approximate methods of a Cauchy problem for the Helmholtz equation,Comput.Appl.Math.,26(2007) 285-307.
[125]M.Yamamoto,Condtional stability in determation of force terms of heat equations in a rectangle,Math.Comput.Modelling,18(1993) 79-88.
[126]L.Yah,C.L.Fu,and F.L.Yang,The method of fundamental solutions for the inverse heat source problem,Eng.Anal.Boundary Elem.,32(3)(2008) 216-222.
[127]杨帆,几类未知源识别问题与正则化方法与算法,硕士学位论文,兰州大学,2007.
[128]Z.Yi and D.A.Murio,Source Term identification in 1-D IHCP,Comput.Math.Appl.,47(2004) 1921-1933.
[2]R.S.Anderssen and M.Hegland,For numerical differentiation,dimensionality can be a blessing,Math,comput.,68(227)(1999) 1121-1141.
[3]D.D.Ang,N.H.Nghia and N.C.Tarn,Regularized solutions of a Cauchy problem for the Laplace equation in an irregular layer:A three-dimensional model,Acta Math.Vietnam.,23(1998) 65-74.
[4]N.Bleistein,J.Cohen,J.Jr.Stockwell,Mathematics of Multidimensional Seismic Imaging,Migration,and Inversion,Springer,Berlin,2001.
[5]G.Beylkin,On the representation of operators in bases of compactly supported wavelets,SIAM J.Numer.Anal.,6(1992) 1716-1740.
[6]L.Bourgeois,A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation,Inverse Problems,21(2005) 1087-1104.
[7]A.L.Bukhgeim,J.Cheng,M.Yamamoto,Stability for an inverse boundary problem of determining a part of a boundary,Inverse Problems,15(1999) 1021-1032.
[8]A.L.Bukhgeim,J.Cheng,M.Yamamoto,On a sharp estimate in a nondestructive testing:determination of unknown boundaries.In:K.Miya,M.Yamamoto,Nguyen Xuan Hung(eds.):Applied electromagnetism mechanics.JSAEM 1998,64-75.
[9]A.A.Burykin,A.M.Denisov,Determination of the unknown sources in the heat-conduction equation,Comput.Math.Model.,8(1997) 309-313.
[10]A.Calder(?)n,Uniqueness in the Cauchy problem for partial differential equations,Amer.J.Math.,80(1958) 16-36.
[11]J.R.Cannon,One dimensional heat equation,Addison-Wesley Publishing Company,California,1984.
[12]J.R.Cannon and P.DuChateau,Structural identification of an unknown source term in a heat equation,Inverse Problems,14(1998) 535-551.
[13]J.R.Cannon and S.P.Estevz,An inverse problem for the heat equation,Inverse Problems,2(1986) 395-403.
[14]H.W.Cheng,J.F.Huang and T.J.Leiterman,An adaptive fast solver for the modified Helmholtz equation in two dimensions,J.Comput.Phys.,211(2006)616-637.
[15]J.Cheng,Y.C.Hon and Y.B.Wang,A numerical method for the discontinuous solutions of Abel integral equations,In inverse problems and spectral theory,Contemp.Math.,Amer,Math.Soc,Providence,RI,348(2004) 23-3-243.
[16]J.Cheng,Y.C.Hon,T.Wei and M.Yamamoto,Numerical computation of a Cauchy problem for Laplace's equation,ZAMM Z.Angew.Math.Mech.,81(2001),665-674.
[17]J.Cheng,S.Prossdorf,M.Yamamoto,Local estimation for an integral equation of first kind with analytic kernel,J.Inverse Ill-Posed Probl,6(1998) 115-126.
[18]J.Cheng,M.Yamamoto,Unique continuation on a line for harmonic functions,Inverse Problems,14(1998) 869-882.
[19]J.Cheng,M.Yamamoto,Local stability of a linearized inverse problem in detecting steel reinforcement bars,Proc.Int.Conf.Inverse Problems and Applications,Quezon City,1998.Matimylas Mat.21(1998) Special issue,18-33.
[20]M.Choulli and M.Yamamoto,Conditional stability in determining a heat source,J.Inv.Ill-posed Probl.,12(2004) 233-243.
[21]P.Colli-Franzone,L.Guerri,S.Tentoni,C.Viganotti,S.Baruffi,S.Spaggiari,B.Taccardi,A mathematical procedure for solving the inverse potential problem of electrocardiography,Analysis of the time-space accuracy from in vitro experimental data,Math.Biosci.,77(1985) 353-396.
[22]P.Colli-Franzone and E.Magenes,On the inverse potential problem of electrocardiology,Calcolo,16(1979) 459-538.
[23]J.Cullum,Numerical differentiation and regularization,SIAM J.Numer.Anal.,8(1971) 254-265.
[24]I.Daubechies,Ten lectures on wavelets,CBMS-NSF Regional Conf.Series in Applied Mathematics,SIAM,Philadelphia,1992.
[25]S.R.Deans,The Radon transform and some of its applications,A Wiley- Interscience Publication,John Wiley & Sons Inc.,New York,1983.
[26]J.R.Dorroh and X.P.Ru,The application of the method of quasi-reversibility to the sideways heat equation,J.Math.Anal.Appl.,236(1999) 503-519.
[27]F.F.Dou,C.L.Fu,Determining unknown source in the heat equation by a wavelet dual least squares method,Applied Mathematics Letters,22(2009)661-667.
[28]F.F.Dou,C.L.Fu,F.L.Yang,Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation,Journal of Computational and Applied Mathematics,Doi:10.1016/j.ca m.2009.01.008.
[29]F.F.Dou,C.L.Fu,F.Yang,Identifying an unknown source term in a heat equation,Inverse Problems in Science and Engineering,accepted.
[30]L.Elden,Approximations for a Cauchy problem for the heat equation,Inverse Problems,3(1987) 263-273.
[31]L.Eld(?)n,Solving an inverse heat conduction problem by 'Method of lines',Trans.ASME J.Heat Transfer,119(1997) 913-923.
[32]L.Elden,F.Berntsson and T.Reginska,Wavelet and Fourier methods for solving the sideways heat equation,J.Sci.Comput.,21(2000) 2187-2205.
[33]H.W.Engl,M.Hanke and A.Newbauer,Regularization of inverse problems,Kluwer Academic,Boston,MA 1996.
[34]A.Farcas and D.Lesnic,The boundary-element method for the determination of a heat source dependent on one variable,J.Engin.Math.,54(2006) 375-388.
[35]A.Farcas and D.Lesnic,The boundary-element method for the determination of a heat source dependent on one variable,J.Eng.Math.,54(2006) 375-388.
[36]C.L.Fu,Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation,J.Comput.Appl.Math.,167(2004) 449-463.
[37]C.L.Fu,F.F.Dou,X.L.Feng and Z.Qian,A simple regularization method for stable analytic continuation,Inverse problems,24(2008) 065003.
[38]C.L.Fu,X.L.Feng and Z.Qian,The Fourier regularization for solving the Cauchy problem for the Helmholtz equation,Appl.Numer.Math.,to appear.
[39]C.L.Fu,H.F.Li ,Z.Qian and X.T.Xiong,Fourier regularization method for solving a Cauchy problem for the Laplace equation,J.Inv.Ill-posed Probl.,16(2008) 159-169.
[40]C.L.Fu,X.T.Xiong and Z.Qian,Fourier Regularization for a Backward Heat Equation,J.Math.Anal.Appl.,331(2007) 472-480.
[41]G.Golubev,R.Khasminskii,A statistical approach to the Cauchy problem for the Laplace equation,IMS Lecture Notes-Monograph Series,Volume 36 Beachwood,OH:Institute of Mathematical Statistics,616(2001) 419-433.
[42]R.Gorenflo,Funktionentheoretische Bestimmung des Aussenfeldes zu einer zweidimensionalen magnetohydrostatischen Konfiguration,Z.Angew.Math.Phys.16(1965) 279-290.
[43]R.Gorenflo and S.Vessella,Abel integral equations,volume 1461 of Lecture Notes in Mathematics,Springer-Verlag,Berlin,1991.
[44]C.W.Groetsch,Differentiation of approximately specified functions,Amer.Math.Monthly,98(9)(1991) 847-850.
[45]J.Hadamard,Lectures on Cauchy problem in linear partial differential equations,University Press,London,1923.
[46]M.Hanke and O.Scherzer,Error analysis of an equation error method for the identification of of the diffusion coefficient in a quasi-linear parabolic differentical equation,SIAM J.Appl,Math.,59(1999) 1012-1027.(eletronic)
[47]M.Hanke and O.Scherzer,Inverse problems light:numerical differentiation,Amer.Math.Monthly,108(6)(2001) 512-521.
[48]D.N.Hao and P.M.Hien,Stability results for the Cauchy problem for the Laplace equation in a strip,Inverse Problems,19(2003) 833-844.
[49]D.N.Hao and D.Lesnic,The Cauchy for Laplace's equation via the conjugate gradient method,IMA J.Appl.Math.,65(2000) 199-217.
[50]D.N.Hao,A.Schneiderx and H.J.Reinhardt,Regularization of a noncharacteristic Cauchy problem for a parabolic equation,Inverse Problems,11(1995) 124-1263.
[51]Y.C.Hon and T.Wei,Backus-Gilbert algorithm for the Cauchy problem of Laplace equation,Inverse Problems,17(2001) 261-271.
[52]Y.C.Hon and T.Wei,Solving Cauchy problems of elliptic equations by the method of fundamental solutions,In Boundary elements ⅩⅩⅦ,volume 39 of WIT Trans.Model.Simul,57-65.WIT Press,Southampton,2005.
[53]T.Hrycak and V.Isakov,Increased stability in the continuation of solutions to the Helmholtz equation,Inverse Problems,20(2004) 697-712.
[54]V.Isakov,Inverse problems for partial differential equations,Springer-Verlag,New York,1998.
[55]M.I.Ivanchov,The inverse problem of determining the heat source power for a parabolic equation under arbitrary boundary conditions,J.Math.Sci.,88(1998) 432-436.
[56]C.R.Johnson,Computational and numerical methods for bioelectric field problems,Crit.Rev.Biomed.Eng.,25(1997) 1-81.
[57]T.Johansson and D.Lesnic,Determining a spacewise dependent heat source,J.Comput.Appl.Math.,209(2007) 66-80.
[58]T.Johansson and D.Lesnic,A variational method for identifying a spacewisedependent heat source,IMA J.Appl.Math.,72(2007) 748-760.
[59]D.Q.Kern and A.D.Kraus,Extended surface heat transfer,McGraw-Hill,New York,1972.
[60]A.Kirsch,An introduction to the mathematical theory of inverse problems,Springer,Berlin,1999.
[61]M.V.Klibanov and F.Santosa,A computational quasi-reversibility method for Cauchy problems for Laplace's equation,SIAM J.Appl.Math.,51(1991),1653-1675.
[62]Y.Knosowski,E.Lieres and A.Schneider,Regularization of a non-characteristic Cauchy problem for a parabolic equation in multiple dimensions,Inverse Problems,15(1999) 731-743.
[63]E.D.Kolaczyk,Wavelet methods for the inversion of certain homogeneous linear operators in thepresence of noisy data,PhD Thesis,Stanford University,Stanford,CA,1994.
[64]A.V.Kononov,CD.Riyanti,S.W.de Leeuw,C.W.Oosterlee and C.Vuik,Numerical performance of a parallel solution method for a heterogeneous 2D Helmholtz equation,Comput.Visual Sci.,11(2008) 139-146.
[65]L.Landweber,An iteration formula for Fredholm integral equation of the first kind,Am.J.Math.,73(1951),615-624.
[66]R.Lattes and J.L.Lions,The method of quasi-reversibility:applications to partial differential equations,Elsevier,New York,1969.
[67]M.M.Lavrent'ev,V.G.Romanov and S.P.Shishatskii,Ill-Posed Problems of Mathematical Physics and Analysis,Translations of Mathematical Monographs vol 64,Providence,RI:American Mathematical Society,1986.
[68]G.S.Li,Data compatibility and conditional stability for an inverse source problem in the heat equation,Appl.Math.Comput.,173(2006) 566-581.
[69]G.S.Li and M.Yamamoto,Stability analysis for determining a source term in a 1-D advection-dispersion equation ,J.Inv.Ill-posed Probl.,14(2006) 147-155.
[70]李洪芳,几类逆边值问题的正则化方法及最优性分析,博士学位论文,兰州大学,2007.
[71]李振平,几类椭圆型方程Cauchy问题的磨光化求解方法,硕士学位论文,兰州大学,2008.
[72]刘继军,不适定问题的正则化方法和应用,科学出版社,2005.
[73]L.Ling,M.Yamamoto,Y.C.Hon and T.Takeuchi,Indentification of source locations in two-demensional heat equations.Inverse Problems,22(2006) 1289-1305.
[74]L.Matin,L.Elliott,P.J.Heggs,D.B.Ingham,D.Lesnic and X.Wen,An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation.Comput.Methods Appl.Mech.Eng.,192(2003) 709-722.
[75]L.Matin,L.Elliott,P.J.Heggs,D.B.Ingham,D.Lesnic and X.Wen,Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations.Comput.Mech.,31(2003) 367-377.
[76]L.Marin,L.Elliott,P.J.Heggs,D.B.Ingham,D.Lesnic and X.Wen,BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method,Eng.Anal.Bound.Elem.,28(2004) 1025-1034.
[77]L.Marin and D.Lesnic,The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations,Computers and Structures,83(2005) 267-278.
[78]Y.Meyer,Wavelets and Operators,Cambridge University Press,Cambridge,1992.
[79]D.A.Murio,C.E.Mej(?)a,S.Zhan,Discrete mollification and automatic numerical differentiation,Computers Math.Appl.35(1998) 1-16.
[80]L.E.Payne,Bounds in the Cauchy problem for the Laplace' s equation,Arch.Rat.Mech.Anal.,5(1960) 35-45.
[81]A.Peterson,S.Ray and R.Mittra,Computational Methods for Electromagnet-ics.IEEE Press,New York,1998.
[82]A.I.Prilepko and V.V.Solov'ev,Solvability theorems and Rote's method for inverse problems for a parabolic equation,J.Diff.Equ.,23(1998) 1230-1237.
[83]钱志,数学物理反问题的正则化,博士学位论文,兰州大学,2008.
[84]Z.Qian,C.L.Fu and X.T.Xiong,Fourth-order modified method for the Cauchy problem for the Laplace equation,J.Comput.Appl.Math.,192(2006) 205-218.
[85]Z.Qian,C.L.Fu,X.L.Feng,A modified method for high order numerical derivatives,Appl.Math.Comput.,182(2006) 1191-1120.
[86]Z.Qian,C.L.Fu,Regularization strategies for a two-dimensional inverse heat conduction problem,Inverse Problems,23(2007) 1053-1068.
[87]H.H.Qin,T.Wei,Modified regularization method for the Cauchy problem of the Helmholtz equation,Applied Mathematical Modelling,33(5)(2009) 2334-2348.
[88]H.H.Qin,T.Wei,R.Shi,Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation,J.Comput.Appl.Math.,224(1)(2009) 39-53.
[89]H.H.Qin,D.W.Wen,Tikhonov type regularization method for the Cauchy problem of the modified Helmholtz equation,Appl.Math.Comput.,203(2)(2008) 617-628.
[90]C.Y.Qiu and C.L.Fu,Wavelets and regularization of the Cauchy Problem for the Laplace equation,J.Math.Anal.Appl.,338(2008) 1440-1447.
[91]R.Qu,A new approach to numerical differentiation and integration,Math.comput.,24(10)(1996) 55-68.
[92]A.G.Ramm and A.B.Smirnova,On stable numerical differentiation,Math.Comput.,70(2001) 1131-1153.(electronic)
[93]V.S.Ryaben'kii,S.V.Tsynov and S.V.Utyuzhnikov,Inverse source problem and active shielding for composite domains,Appl.Math.Lett.,20(2007) 511-515.
[94]T.Regi(?)ska,Sideways heat equation and wavelets,J.Comput.Appl.Math.,63(1995) 209-214.
[95]T.Regi(?)ska and L.Eld(?)n,Solving the sideways heat equation by a wavelet-Galerkin method,Inverse Problems,13(1997) 1093-1106.
[96]T.Regi(?)ska and L.Eld(?)n,Stability and convergence of a wavelet-Galerkin method for the sideways heat equation,J.Inv.Ill-posed Problems,8(2000)31-49.
[97]T.Regi(?)ska,Application of wavelet shrinkage to solving sideways heat equation,BIT,41(5)(2001) 1101-1110.
[98]T.Regi(?)ska and K.Regi(?)ski,A Cauchy problem for the Helmholtz equation:application to analysis of light propagation in solids,Inverse Problems,22(2006)975-989.
[99]T.Regi(?)ska and A.Wakulicz,Wavelet moment method for Cauchy problem for the Helmholtz equation,J.Comput.Appl.Math.,223(1)(2009) 218-229.
[100]V.S.Ryaben'kii,S.V.Tsynov and S.V.Utyuzhnikov,Inverse source problem and active shielding for composite domains,Appl.Math.Lett.,20(2007) 511-515.
[101]P.C.Sabatier,Inverse problems:an introduction,Inverse problems,1(1985)1-4.
[102]E.G.Savateev,On problems of determining the source function in a parabolic equation,J.Inv.Ill-posed problems,3(1995) 83-102.
[103]A.Schneider,Wavelet-based mollication methods for some ill-posed problems,Dissertation Universit(a|¨)t Siegen,1996.
[104]T.Schrhter and U.Tautenhahn,On the 'optimal' regularization methods for solving linear ill-posed problems.Z.Anal.Anw.,13(1994) 697-710.
[105]V.V.Solov'ev,Solvability of the inverse problem of finding a source,using overdetermination on the upper base for a parabolic equation,Diff.Equ.,25(1990) 1114-1119.
[106]孙炯,王忠,线性算子的谱分析,科学出版社,北京,2005.
[107]U.Tautenhahn,Optimal stable solution of Cauchy problems for elliptic equations,Z.Anal.Anw.15(1996) 961-984.
[108]U.Tautenhahn,Optimal stable approximations for the sideways heat equation,J.Inv.Ill-posed Problems,5(1997) 287-307.
[109]U.Tautenhahn,Optimality for ill-posed problems under general source conditions,Numer.Funct.Anal.Optimiz.,19(1998) 377-398.
[110]U.Tautenhahn and R.Gorenflo,On Optimal regularization methods for fractional differentiation,J.Anal.Appl.,18(1999) 449-467.
[111]A.N.Tikhonov,V.Y.Arsenin,Solutions of ill-posed problems,Winston and Sons,Washington,1977.
[112]D.D.Trong,N.T.Long and P.N.D.Alain,Nonhomogeneous heat equation:Identification and regularization for the inhomogeneous term,J.Math.Anal.Appl.,312(2005) 93-104.
[113]G.Vainikko,On the optimality of methods for ill-posed problems,Z.Anal.Anw.,64(1987) 351-362.
[114]C.Vani and A.Avudainayagam,Regularized solution of the Cauchy problem for the Laplace equation using Meyer Wavelets,Math.Comput.Model.,36(2002)1151-1159.
[115]Y.B Wang,X.Z.Jia and J.Cheng,A numerical differentiation method and its application to reconstruction of discontinuity,Inverse Problems,18(2002)1461-1476.
[116]T.Wei,Y.C.Hon and L.Ling,Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators,Eng.Anal.Bound.Elem.,31(2007) 373-385.
[117]T.Wei,H.H.Qin and R.Shi,Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation,Inverse Problems,24(3)(2008)035003(18pp).
[118]A.S.Wood,G.E.Tupholme,M.I.H.Bhatti and P.J.Heggs,Steady-state heat transfer through extended plane surfaces,Int.Commun.Heat Mass Transfer 22(1995) 99-109.
[119]熊向团,抛物型方程反问题的正则化理论和算法,博士学位论文,兰州大学,2007.
[120]X.T.Xiong,C.L.Fu and Z.Qian,Two numerical methods for solving a backward heat conduction problem,Appl.Math.Comput.,179(2006) 370-377.
[121]X.T.Xiong and C.L.Fu,Central difference regularization method for the Cauchy problem of the Laplace's equation,Appl.Math.Comput.,181(2006)675-684.
[122]X.T.Xiong and C.L.Fu,Determining surface temperature and heat flux by a wavelet dual least squares method,J.Comput.Appl.Math.,201(2007) 198-207.
[123]X.T.Xiong and C.L.Fu,Error estimates on a backward heat equation by a wavelet dual least squares method,Int.J.Wavelets,Multiresolut.Inf.Process.,5(2007) 389-397.
[124]X.T.Xiong and C.L.Fu,Two approximate methods of a Cauchy problem for the Helmholtz equation,Comput.Appl.Math.,26(2007) 285-307.
[125]M.Yamamoto,Condtional stability in determation of force terms of heat equations in a rectangle,Math.Comput.Modelling,18(1993) 79-88.
[126]L.Yah,C.L.Fu,and F.L.Yang,The method of fundamental solutions for the inverse heat source problem,Eng.Anal.Boundary Elem.,32(3)(2008) 216-222.
[127]杨帆,几类未知源识别问题与正则化方法与算法,硕士学位论文,兰州大学,2007.
[128]Z.Yi and D.A.Murio,Source Term identification in 1-D IHCP,Comput.Math.Appl.,47(2004) 1921-1933.