不适定问题的正则化方法
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摘要
本文针对几类具体的反问题采用不同的正则化方法,分析了其正则解的收敛性及收敛率.同时也将Hilbert空间中的Landweber迭代法推广到了Banach空间,研究了其方法的可行性.
在第二章中,主要研究了一类半线性热传导偏微分方程的反系数问题,在弱来源条件成立的条件下,利用最小二乘法讨论了抛物型方程逼近系数与逼近解的收敛率.
而第三章主要研究了电化学实验中,控制电压下,物质传输过程模型,得出一个非线性抛物型方程.从反系数的观点,我们分析了反系数问题中输入-输出映射的单调性.此外,把所得的非线性抛物型方程推广到了一个更为一般的情形.考虑了推广后的非线性抛物型方程拟解的存在性.
第四章则利用简化Tikhonov方法讨论了一维截面抛物型方程和二维热传导方程逼近解的收敛率.对于一般的Tikhonov方法,在证明收敛率时都用到了来源性条件.而本章中,对于几类具体的抛物型方程,利用傅立叶变换,然后再采用简化的Tikhonov正则化方法,在无需来源条件下,仍可得到逼近解的收敛率,且其收敛阶为最优阶.实验证明了简化Tikhonov正则化方法的实用性.
最后一章,将Hilbert空间的Landweber迭代法推广到了Banach空间,即这一章中,我们给出了迭代法的收敛性.
The dissertation investigates the regularizd methods for a few kinds of specifical inverse problems. We obtain the convergence and convergence rates of regular solution. At the same time, we extend Hilbert space in which Landweber iteration method are involved to Banach space. The generalized Landweber iteration method in Banach space is available.
In chapter 2, we focus on an inverse coefficient problem of a semilinear parabolic equation. Under the weak source conditon:using least square method, we can get the convergence and convergence rate of approximate coefficent and approximate solution.
In chapter 3, the problem related to controlled potential experiments in electrochemistry is studied. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view an inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.
In chapter 4, we devote to simplified Tikhonov regularization for a sideways parabolic equation, and a two-dimensional backward (inverse) heat conduction problem. We concentrate on the convergence rates of the simplified Tikhonov approximation of solutions of the sideways parabolic equations at 0≤x<1, and the two-dimensional backward (inverse) heat conduction problem at 0≤t In the end, we extend Hilbert space which Landweber iteration method are involved to Banach space, i.e,The generalized Landweber iteration method in Banach space is available.
在第二章中,主要研究了一类半线性热传导偏微分方程的反系数问题,在弱来源条件成立的条件下,利用最小二乘法讨论了抛物型方程逼近系数与逼近解的收敛率.
而第三章主要研究了电化学实验中,控制电压下,物质传输过程模型,得出一个非线性抛物型方程.从反系数的观点,我们分析了反系数问题中输入-输出映射的单调性.此外,把所得的非线性抛物型方程推广到了一个更为一般的情形.考虑了推广后的非线性抛物型方程拟解的存在性.
第四章则利用简化Tikhonov方法讨论了一维截面抛物型方程和二维热传导方程逼近解的收敛率.对于一般的Tikhonov方法,在证明收敛率时都用到了来源性条件.而本章中,对于几类具体的抛物型方程,利用傅立叶变换,然后再采用简化的Tikhonov正则化方法,在无需来源条件下,仍可得到逼近解的收敛率,且其收敛阶为最优阶.实验证明了简化Tikhonov正则化方法的实用性.
最后一章,将Hilbert空间的Landweber迭代法推广到了Banach空间,即这一章中,我们给出了迭代法的收敛性.
The dissertation investigates the regularizd methods for a few kinds of specifical inverse problems. We obtain the convergence and convergence rates of regular solution. At the same time, we extend Hilbert space in which Landweber iteration method are involved to Banach space. The generalized Landweber iteration method in Banach space is available.
In chapter 2, we focus on an inverse coefficient problem of a semilinear parabolic equation. Under the weak source conditon:using least square method, we can get the convergence and convergence rate of approximate coefficent and approximate solution.
In chapter 3, the problem related to controlled potential experiments in electrochemistry is studied. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view an inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.
In chapter 4, we devote to simplified Tikhonov regularization for a sideways parabolic equation, and a two-dimensional backward (inverse) heat conduction problem. We concentrate on the convergence rates of the simplified Tikhonov approximation of solutions of the sideways parabolic equations at 0≤x<1, and the two-dimensional backward (inverse) heat conduction problem at 0≤t
引文
[1] Engl H. W., Louis A. K., Rundell W. Inverse Problems in Geophysics[M]. SIAM,Philadelphia, 1996.
[2] Hanke M. A regularization Levenberg-Marquart scheme, with application to inverse groundwater filtration problems[J]. Inverse Problems, 1997, 13: 79-95.
[3] Zidarov D. Inverse gravimetric problem in geoprospecting and geodesy[M]. Amsterdam: Elsevier, 1980.
[4] Engl H.W., Louis A. K., Rundell W.,eds. Inverse Problems in Medichal Imaging and Nondestructive Testing[M], Springer. New York: Wien, 1996.
[5] Colton D., Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. Berlin: Springer, 1992.
[6] Engl H.W., Rundell W., eds. Inverse Problems in Diffusion Processes[M], SIAM,Philadelphia, 1995.
[7] Hasanov A., Liu Z.H. An inverse coefficient problem for a nonlinear parabolic variational inequality[J]. Applied Mathematics Letters, 2008, 21(6): 563-570.
[8] Klibanov M.V., Timonov A. Carleman Estimates for Coeffient Inverse Problems and Numerical Applications[M], Inverse and Ill-Posed Problems Series, VSP,Netherlands, 2004.
[9] Kiigler P., Engl H.W. Identification of a temperature dependent heat conductivity by Tikhonov regularization[J], J. Inv. Ill-posed Problems, 2002, 10: 67-90.
[10]Kravaris C, Seinfeld J.H. Identification of parameters in distributed parameter systems by regularization[J]. SIAM.J.Control Optim., 1985, 23:217-241.
[11] Liu Z.H. Identification of parameters in semilinear parabolic equations, Acta Mathematica Scientia, 1999, 19(2):175-180.
[12] Liu Z.H. On the identification of coefficients of semilinear parabolic equations[J]. Acta. Math. Appl. Sinica., 1994, 10: 356-367.
[13]Richter G.R. Numerical identification of a spacially varying diffusion coefficient[J]. Math.Comput., 1981, 36:375-386.
[14]Scherzer A. A Modified Landweber Iteration for Solving Parameter Estimation Problems[J]. Applied Mathematics and Optimization, 1998, 38: 45-68.
[15]Aubin G.., Kornprobst P.. Mathematical Problems in Image Processing[M]. Berlin: Springer, 2001.
[16]Bertero M., Boccacci P. Introduction to Inverse Problems in Imaging[M]. London: Istitute of Physics Publishing, 1998.
[17]Isaacson D., Newell J. C. Electrical impedance tomography[J], SIAM Review, 1999,41:85-101.
[18]Natterer F. The Mathematics of Computerized Tomography[M]. Stuttgart: Teubner, 1986.
[19] Black F., Scholes M. The pricing of options and corporate liabilities[J]. J. of Political Economy, 1973, 81: 637-659.
[20] Bouchouev I., Isakov V. Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets[J]. Inverse Problems, 1999, 15:95-116.
[21]Dupire B. Pricing with a smile[J], RISK, 1994, 7: 18-20.
[22]Egger H., Engl H.W. Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates[J]. Inverse Problems, 2005, 21:1027-1045.
[23]Lishang J., Youshan T. Identifying the volatility of underlying assets from option prices[J]. Inverse Problems, 2001, 17: 137-155.
[24]Airapetyan R., Ramm A.G.. Dynamical systems and discrete methods for solving nonlinear ill-posed problems[J]. Appl.Math.Reviews, Ed. G. Anastassiou, World Sci. Publishers, 2000, 1(1): 491-536.
[25]Richter G.R. An inverse problem for the steady state diffusion equation[J]. SIAM J.Appl.Math., 1981,41:210-221.
[26]Keller J.B. Inverse Problems[J]. Amer.Math.Monthly, 1976, 83: 107-118.
[27]Qian Z, C.L. Fu. Regularization strategies for a two-dimensional inverse heat conduction problem[J]. Inverse probl., 2007, 23:1053-1068.
[28]Ramm A.G.. Stable solutions of some ill-posed problems[J]. Math. Meth. in the appl. Sci, 1981, 3: 336-363.
[29]Ramm A.G.. A numerical method for some nonlinear problems[J]. Math. Models and Meth. in Appl.Sci, 1999, 9(2): 325-335.
[30] Ramm A.G, Smirnova A. B. A numerical method for solving nonlinear ill-posed problems[J]. Nonlinear Funct. Anal, and Optimiz, 1999, 20(3): 317-332.
[31]Ramm A.G.., Smirnova A.B. Continuous regularized Gauss-Newton-type algorithm for nonlinear ill-posed equations with simultaneous updates of inverse derivative[J], Intern. Jour, of Pure and Appl Math, 2002, 2(1): 23-34.
[32]Gorenflo R, Vesella S. Abel Integral Equations: Analysis and Applications. Lecture Notes in Math. 1461 [M]. Berlin: Springer, 1991.
[33]Kammerer W.J., Nashed M.Z. Iterative methods for best approximate solutions of linear integral equations of the first and second kinds[J]. J. Math. Anal. Appl., 1972,40:547-573.
[34] Kress R. Linear Integral Equations[M]. Berlin: Springer, 1989.
[35]Isakov V. Inverse Problems in Partial Differential Equations[M]. Berlin: Springer, 1998.
[36]Isakov V. Inverse Problems for Partial Differential Equations[M]. Springer, 1998.
[37]Alfio Q., Riccardo S., Fansto S. Numerical Mathematics[M]. Springer, 2000.
[38]Bakushinsky A.B. Iterative methods of gradient type for nonregular operator equations[J]. Comput.Math.Math.Phys. 1998,38: 1884-1887.
[39]Bakushinskii A.B., Goncharskii A.V. Iterative methods for the solution of incorrect problems[M]. Moscow: Nauka, 1989.
[40] Bakushinskii A.B., Goncharskii A.V. Ill-Posed Problems: Theory and Applications[M]. Dordrecht: Kluwer, 1994.
[41 ] Bakushinsky A.B., Kokurin M.Yu. Iterative Methods for Approximate Soution of Inverse Problems[M]. Springer, 2004.
[42]Brakhage H. On ill-posed problems and the method of conjugate gradients, in: H.W. Engl and C.W. Groetsch, eds., Inverse and Ill-posed Problems[J], Academic Press, Boston, London, New York,, 1987, 165-175.
[43] Charles W. Groetsch. Inverse Problems in the Mathematical Sciences [M]. 1993, Viewge.
[44]Curtis R.Vogel. Computational Methods for Inverse Problems[M]. SIAM, Philadelphia, 2002.
[45] Deuflhard P., Engl H. W., Scherzer O. A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions[J]. Inverse Problems, 1998,14: 1081-1106.
[46]Egger H. Accelerated Newton-Landweber iterations for regularizing nonlinear inverse problems, SFB-Report 2005-3 [M], Linz, January 2005.
[47]Eicke B., Louis A. K., Plato R. The instability of some gradient methods for ill-posed problems[J]. Numer. Math., 1990, 58: 129-134.
[48]Groetsch C.W. Inverse Problems in the Mathematical Sciences[M]. Vieweg, Braunschweig, 1993.
[49]Hofmann B. Mathematik inverser Probleme[M]. Teubner, Stuttgart, Leipzig, 1999.
[50] Hohage T. Logarithmic convergence rates of the iteratively regularized Gauss Newton method for an inverse potential and an inverse scattering problem[J].Inverse Problems, 1997, 13: 1279-1299.
[51] Hohage T. Regularization of exponentially ill-posed problems[J]. Numer. Funct.Anal. Optim., 2000,21: 439-464.
[52] Kaltenbacher B., Neubauer A., Ramm A.G. Convergence rates of the continuous regularized Gauss-Newton method[J], Jour. Inv. Ill-Posed Probl., 2002, 10(3):261-280.
[53]Khan T., Smirnova A. lD inverse problem in diffusion based optical tomography using iteratively regularized Gauss-Newton algorithm[J]. Applied Mathematics Computation, 2005, 161: 149-170.
[54] Smirnova A., Rosemary A.R. and Khan T. Convergence and application of a modified iteratively regularized Gauss-Newton algorithm[J]. Inverse Problems,2007,23: 1547-1563.
[55] 肖庭延,于慎根,王彦飞.反问题的数值解法[M].科学出版社,2003.
[56] Cannon J.R., Hill CD. Existence, uniqueness, stability and monotone dependence in a Stefan problem for the heat equation[J], J.Math.Mech., 1967, 17:1-19.
[57] Engl H.W., Hanke M., Neubauer A. Regularization of inverse problems[M].Dordrecht: Kluwer, 1996.
[58] Hanke M., Neubauer A., Scherzer O. A convergence analysis of the Landweber iteration for nonlinear ill-posed problems[J]. Numer. Math., 1995, 72: 21-37.
[59] Kaltenbacher B. Some Newton-type methods for the regularization of nonlinear ill-posed problems[J]. Inverse Problems, 1997, 13: 729-753.
[60] Egger H. Preconditioning iterative regularization methods in Hilbert scales[M]. Dissertion, Johannes Kepler Universit(?)Linz.
[61] Bakushinskii A.W. The problem of the convergence of the iteratively regularized Gauss-Newton method[J]. Comput.Math.Math.Phys.1992,32:1353-1359.
[62] Blaschke B., Neubauer A., Scherzer O.. On convergence rates for the iteratively regularized Gauss-Newton method[J]. IMA Journal of Numerical Analysis. 1997,17:421-436.
[63] Jacob B. Dynamics of fluids in porous media. American Elsevier[J], New York.1972.214-215.
[64] Dimir A., Hasanov A. Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach[J]. J. Math. Anal. Appl.,2007.
[65]Engl H.W., Zou J. A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction[J]. Inverse Problems. 2000, 16: 1907-1923.
[66]Hasanov A. Simultaneous determination of source terms in a linear parabolic problem from the final over-determination:Weak solution approach[J]. J. Math.Anal. Appl., 2007, 766-779.
[67]Hasanov A., Liu Z .H. An inverse coefficient problem for a nonlinear parabolic variational inequality[J]. Appl. Math. Lett. 2007.
[68] Bard A. J., Faulkner L.R. Electrochemical Methods: Fundamental and Application[J], Wiley, 1980.
[69]Cohn S., Pfabe K., Redepenning J. A similarity solution to a problem in nonlinear ion transport with a nonlocal condition[J]. Math. Mod. Meth. Appl. Sci., 1999, 9:445-461.
[70]Hasanov A., Hasanoglu S. Analytical formulaes and comparative analysis for linear models in chronoamperometry under conditions of diffusiion and migration[J]. J.Math.Chem. 2008, 44(1): 133-141.
[71]Saveant J.M. Electron hopping between localized sites. Effect of ion pairing on diffusion and migration. General rate laws and steady-state response[J]. J. Phys. Chem., 1988, 92:4526-4532.
[72] Pfabe K. A problem in nonlinear ion transport. PhD Thesis[M], Univesity of Nebraska, Lincoln, 1995.
[73]Hasanov A., Hasanoglu S. Comparative analysis of linear and nonlinear models for ion transport problem in chronoamperometry [J]. J. Math. Chem. 2008, 44(3):731-742.
[74]Hasanoglu S., Erdem A. Relationship between current response and time in ion transport problem including diffusion and convection: 2. Numerical approach[J], J.Math.Chem.2008, 43(4): 1458-1469.
[75]Ladyzenskaja O.A., Solonnikov V.A., Uralceva N.N. Linear and quasi-linear equations of parabolic type[J]. Transl. Math. Mono., AMS. Providence RI, 1968,23:417-571.
[76]Carasso A. Determing surface temperatures from interior observations [J]. SIAM J.Appl.Math., 1982, 47: 558-574.
[77]Hao D.N., Reinhardt H.J. On a sideways parabolic equation[J]. Inverse problems, 1997, 13: 297-309.
[78]Alifanov O.M. Inverse Heat Transfer Problems[J]. New York : Springer, 1994.
[79] Beck J., Blackwell B., Clair C. S. Inverse Heat Conductions[M], Wiley, Sussex,1985.
[80] Eld (?)n L. Numerical solution of the sideways heat equation by difference approximation in time[J]. Inverse Problems, 1995, 11:913-923.
[81]Eld(?)n L., Berntsson F., Regi(?)ska T. Wavelet and Fourier methods for solving the sideways heat equation[J]. SIAM J.SCI.Comput., 2000, 21: 2187-2205.
[82] Fu C.L. Simplified Tikhonov and Fourier regularization methods on general sideways parabolic equation[J]. J. Comput. Appl. Math., 2004, 167: 449-463.
[83]Hao D.N., Reinhardt H.J., Schneider A. Numerical solution to a sideways parabolic equation[J]. Inter. J. Numer. Methods in Engr., 2001, 50: 1253-1267.
[84] Tu J.S., Beck J.V. Solution of inverse heat conduction problems using a combined function specification and regularization method and a direct finite element solver, Proc.2nd Ann. Inverse Problems in Engineering Seminar ed J.V.Beck[J] (East Lansing, MI:College of Engineering, Michigan State University). 2008, 1-25.
[85]Engl H.W. Regularization of Inverse Problem[M]. Bostom: Kluwer Academic Publishers, 2000.
[86] Liu Z.H., Li J., Li Z.W. Regularization method with two parameters for nonlinear ill-posed problems[J]. Science in China Series A: Mathematics, 2008, 51: 70-78.
[87] Liu Z.H. Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities[J], Inverse Problems. 2005, 21: 13-20,
[88] Tikhonov A.N., Arsenin V.Ya. Solutions of Ill-posed Problems[M]. New York: Wiley, 1977.
[89]Reginska T., Eld(?)n L. Stability and convergence of wavelet-Galerkin method for the sideways heat equation[J]. Journal of Inverse and Ill-posed Problems., 2000, 8: 31-49.
[90]Guo L., Murio D.A. A mollified space-marching finite-difference algorithm for the two-dimensional inverse heat conduction problem with slab symmetry [J]. Inverse Problems, 1991, 7: 247-259.
[91]Nair M.T., Tautenhahn U. Lavrentiev regularization for linear ill-posed problems under general source conditions[J]. Journal of Analysis and Application, 2004, 23:167-185.
[92]Tadi M. An iterative method for the solution of ill-posed parabolic systems[J]. Appl. Math. Comput., 2008, 21: 843-851.
[93]Jourhmane M., Mera N.S. An iterative algorithm for the backward heat conduction problem based on variable relaxtion factors[J]. Inverse Probl.Eng. 2002,10:293-308.
[94] Morozov V.A. On the solution of functional equations by the method of regularization[J]. Soviet Math. Dokl., 1966, 7: 414-417.
[95]Mera N.S. The method of fundamental solutions for the backward heat conduction problem[J]. Inverse probl.Sci.Eng., 2005, 13: 79-98.
[96] Mera N.S., Elliott L, Ingham D.B., Lesnic D. An iterative boundary element method for solving the one dimensional backward heat conduction problem[J]. Int.J.Heat Mass Transfer., 2001, 44:1937-1946.
[97] Yildiz B., Yetis H., Sever A. A stability estimate on the regularized solution of the backward heat equatio[J]. Appl.Math.Comput, 2003, 135: 561-567.
[98] Liu L., Murio D.A. Numerical experiments in 2-D IHCP on bounded domains:part I.the 'interior' cube problem[J]. Comput. Math. Appl., 1996, 31: 15-32.
[99]Alber Ya.L, Ryazantseva I.P. Nonlinear ill-posed problems of monotone type[J]. Springer, Dordrecht, 2006.
[100] Adams R.A., Sobolev Spaces[M]. London : Academic Press, 1975.
[101] Zeidler E. Nonlinear Functional Analysis and its Applications II/A-B[M]. Springer, New York, 1990.
[2] Hanke M. A regularization Levenberg-Marquart scheme, with application to inverse groundwater filtration problems[J]. Inverse Problems, 1997, 13: 79-95.
[3] Zidarov D. Inverse gravimetric problem in geoprospecting and geodesy[M]. Amsterdam: Elsevier, 1980.
[4] Engl H.W., Louis A. K., Rundell W.,eds. Inverse Problems in Medichal Imaging and Nondestructive Testing[M], Springer. New York: Wien, 1996.
[5] Colton D., Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. Berlin: Springer, 1992.
[6] Engl H.W., Rundell W., eds. Inverse Problems in Diffusion Processes[M], SIAM,Philadelphia, 1995.
[7] Hasanov A., Liu Z.H. An inverse coefficient problem for a nonlinear parabolic variational inequality[J]. Applied Mathematics Letters, 2008, 21(6): 563-570.
[8] Klibanov M.V., Timonov A. Carleman Estimates for Coeffient Inverse Problems and Numerical Applications[M], Inverse and Ill-Posed Problems Series, VSP,Netherlands, 2004.
[9] Kiigler P., Engl H.W. Identification of a temperature dependent heat conductivity by Tikhonov regularization[J], J. Inv. Ill-posed Problems, 2002, 10: 67-90.
[10]Kravaris C, Seinfeld J.H. Identification of parameters in distributed parameter systems by regularization[J]. SIAM.J.Control Optim., 1985, 23:217-241.
[11] Liu Z.H. Identification of parameters in semilinear parabolic equations, Acta Mathematica Scientia, 1999, 19(2):175-180.
[12] Liu Z.H. On the identification of coefficients of semilinear parabolic equations[J]. Acta. Math. Appl. Sinica., 1994, 10: 356-367.
[13]Richter G.R. Numerical identification of a spacially varying diffusion coefficient[J]. Math.Comput., 1981, 36:375-386.
[14]Scherzer A. A Modified Landweber Iteration for Solving Parameter Estimation Problems[J]. Applied Mathematics and Optimization, 1998, 38: 45-68.
[15]Aubin G.., Kornprobst P.. Mathematical Problems in Image Processing[M]. Berlin: Springer, 2001.
[16]Bertero M., Boccacci P. Introduction to Inverse Problems in Imaging[M]. London: Istitute of Physics Publishing, 1998.
[17]Isaacson D., Newell J. C. Electrical impedance tomography[J], SIAM Review, 1999,41:85-101.
[18]Natterer F. The Mathematics of Computerized Tomography[M]. Stuttgart: Teubner, 1986.
[19] Black F., Scholes M. The pricing of options and corporate liabilities[J]. J. of Political Economy, 1973, 81: 637-659.
[20] Bouchouev I., Isakov V. Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets[J]. Inverse Problems, 1999, 15:95-116.
[21]Dupire B. Pricing with a smile[J], RISK, 1994, 7: 18-20.
[22]Egger H., Engl H.W. Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates[J]. Inverse Problems, 2005, 21:1027-1045.
[23]Lishang J., Youshan T. Identifying the volatility of underlying assets from option prices[J]. Inverse Problems, 2001, 17: 137-155.
[24]Airapetyan R., Ramm A.G.. Dynamical systems and discrete methods for solving nonlinear ill-posed problems[J]. Appl.Math.Reviews, Ed. G. Anastassiou, World Sci. Publishers, 2000, 1(1): 491-536.
[25]Richter G.R. An inverse problem for the steady state diffusion equation[J]. SIAM J.Appl.Math., 1981,41:210-221.
[26]Keller J.B. Inverse Problems[J]. Amer.Math.Monthly, 1976, 83: 107-118.
[27]Qian Z, C.L. Fu. Regularization strategies for a two-dimensional inverse heat conduction problem[J]. Inverse probl., 2007, 23:1053-1068.
[28]Ramm A.G.. Stable solutions of some ill-posed problems[J]. Math. Meth. in the appl. Sci, 1981, 3: 336-363.
[29]Ramm A.G.. A numerical method for some nonlinear problems[J]. Math. Models and Meth. in Appl.Sci, 1999, 9(2): 325-335.
[30] Ramm A.G, Smirnova A. B. A numerical method for solving nonlinear ill-posed problems[J]. Nonlinear Funct. Anal, and Optimiz, 1999, 20(3): 317-332.
[31]Ramm A.G.., Smirnova A.B. Continuous regularized Gauss-Newton-type algorithm for nonlinear ill-posed equations with simultaneous updates of inverse derivative[J], Intern. Jour, of Pure and Appl Math, 2002, 2(1): 23-34.
[32]Gorenflo R, Vesella S. Abel Integral Equations: Analysis and Applications. Lecture Notes in Math. 1461 [M]. Berlin: Springer, 1991.
[33]Kammerer W.J., Nashed M.Z. Iterative methods for best approximate solutions of linear integral equations of the first and second kinds[J]. J. Math. Anal. Appl., 1972,40:547-573.
[34] Kress R. Linear Integral Equations[M]. Berlin: Springer, 1989.
[35]Isakov V. Inverse Problems in Partial Differential Equations[M]. Berlin: Springer, 1998.
[36]Isakov V. Inverse Problems for Partial Differential Equations[M]. Springer, 1998.
[37]Alfio Q., Riccardo S., Fansto S. Numerical Mathematics[M]. Springer, 2000.
[38]Bakushinsky A.B. Iterative methods of gradient type for nonregular operator equations[J]. Comput.Math.Math.Phys. 1998,38: 1884-1887.
[39]Bakushinskii A.B., Goncharskii A.V. Iterative methods for the solution of incorrect problems[M]. Moscow: Nauka, 1989.
[40] Bakushinskii A.B., Goncharskii A.V. Ill-Posed Problems: Theory and Applications[M]. Dordrecht: Kluwer, 1994.
[41 ] Bakushinsky A.B., Kokurin M.Yu. Iterative Methods for Approximate Soution of Inverse Problems[M]. Springer, 2004.
[42]Brakhage H. On ill-posed problems and the method of conjugate gradients, in: H.W. Engl and C.W. Groetsch, eds., Inverse and Ill-posed Problems[J], Academic Press, Boston, London, New York,, 1987, 165-175.
[43] Charles W. Groetsch. Inverse Problems in the Mathematical Sciences [M]. 1993, Viewge.
[44]Curtis R.Vogel. Computational Methods for Inverse Problems[M]. SIAM, Philadelphia, 2002.
[45] Deuflhard P., Engl H. W., Scherzer O. A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions[J]. Inverse Problems, 1998,14: 1081-1106.
[46]Egger H. Accelerated Newton-Landweber iterations for regularizing nonlinear inverse problems, SFB-Report 2005-3 [M], Linz, January 2005.
[47]Eicke B., Louis A. K., Plato R. The instability of some gradient methods for ill-posed problems[J]. Numer. Math., 1990, 58: 129-134.
[48]Groetsch C.W. Inverse Problems in the Mathematical Sciences[M]. Vieweg, Braunschweig, 1993.
[49]Hofmann B. Mathematik inverser Probleme[M]. Teubner, Stuttgart, Leipzig, 1999.
[50] Hohage T. Logarithmic convergence rates of the iteratively regularized Gauss Newton method for an inverse potential and an inverse scattering problem[J].Inverse Problems, 1997, 13: 1279-1299.
[51] Hohage T. Regularization of exponentially ill-posed problems[J]. Numer. Funct.Anal. Optim., 2000,21: 439-464.
[52] Kaltenbacher B., Neubauer A., Ramm A.G. Convergence rates of the continuous regularized Gauss-Newton method[J], Jour. Inv. Ill-Posed Probl., 2002, 10(3):261-280.
[53]Khan T., Smirnova A. lD inverse problem in diffusion based optical tomography using iteratively regularized Gauss-Newton algorithm[J]. Applied Mathematics Computation, 2005, 161: 149-170.
[54] Smirnova A., Rosemary A.R. and Khan T. Convergence and application of a modified iteratively regularized Gauss-Newton algorithm[J]. Inverse Problems,2007,23: 1547-1563.
[55] 肖庭延,于慎根,王彦飞.反问题的数值解法[M].科学出版社,2003.
[56] Cannon J.R., Hill CD. Existence, uniqueness, stability and monotone dependence in a Stefan problem for the heat equation[J], J.Math.Mech., 1967, 17:1-19.
[57] Engl H.W., Hanke M., Neubauer A. Regularization of inverse problems[M].Dordrecht: Kluwer, 1996.
[58] Hanke M., Neubauer A., Scherzer O. A convergence analysis of the Landweber iteration for nonlinear ill-posed problems[J]. Numer. Math., 1995, 72: 21-37.
[59] Kaltenbacher B. Some Newton-type methods for the regularization of nonlinear ill-posed problems[J]. Inverse Problems, 1997, 13: 729-753.
[60] Egger H. Preconditioning iterative regularization methods in Hilbert scales[M]. Dissertion, Johannes Kepler Universit(?)Linz.
[61] Bakushinskii A.W. The problem of the convergence of the iteratively regularized Gauss-Newton method[J]. Comput.Math.Math.Phys.1992,32:1353-1359.
[62] Blaschke B., Neubauer A., Scherzer O.. On convergence rates for the iteratively regularized Gauss-Newton method[J]. IMA Journal of Numerical Analysis. 1997,17:421-436.
[63] Jacob B. Dynamics of fluids in porous media. American Elsevier[J], New York.1972.214-215.
[64] Dimir A., Hasanov A. Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach[J]. J. Math. Anal. Appl.,2007.
[65]Engl H.W., Zou J. A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction[J]. Inverse Problems. 2000, 16: 1907-1923.
[66]Hasanov A. Simultaneous determination of source terms in a linear parabolic problem from the final over-determination:Weak solution approach[J]. J. Math.Anal. Appl., 2007, 766-779.
[67]Hasanov A., Liu Z .H. An inverse coefficient problem for a nonlinear parabolic variational inequality[J]. Appl. Math. Lett. 2007.
[68] Bard A. J., Faulkner L.R. Electrochemical Methods: Fundamental and Application[J], Wiley, 1980.
[69]Cohn S., Pfabe K., Redepenning J. A similarity solution to a problem in nonlinear ion transport with a nonlocal condition[J]. Math. Mod. Meth. Appl. Sci., 1999, 9:445-461.
[70]Hasanov A., Hasanoglu S. Analytical formulaes and comparative analysis for linear models in chronoamperometry under conditions of diffusiion and migration[J]. J.Math.Chem. 2008, 44(1): 133-141.
[71]Saveant J.M. Electron hopping between localized sites. Effect of ion pairing on diffusion and migration. General rate laws and steady-state response[J]. J. Phys. Chem., 1988, 92:4526-4532.
[72] Pfabe K. A problem in nonlinear ion transport. PhD Thesis[M], Univesity of Nebraska, Lincoln, 1995.
[73]Hasanov A., Hasanoglu S. Comparative analysis of linear and nonlinear models for ion transport problem in chronoamperometry [J]. J. Math. Chem. 2008, 44(3):731-742.
[74]Hasanoglu S., Erdem A. Relationship between current response and time in ion transport problem including diffusion and convection: 2. Numerical approach[J], J.Math.Chem.2008, 43(4): 1458-1469.
[75]Ladyzenskaja O.A., Solonnikov V.A., Uralceva N.N. Linear and quasi-linear equations of parabolic type[J]. Transl. Math. Mono., AMS. Providence RI, 1968,23:417-571.
[76]Carasso A. Determing surface temperatures from interior observations [J]. SIAM J.Appl.Math., 1982, 47: 558-574.
[77]Hao D.N., Reinhardt H.J. On a sideways parabolic equation[J]. Inverse problems, 1997, 13: 297-309.
[78]Alifanov O.M. Inverse Heat Transfer Problems[J]. New York : Springer, 1994.
[79] Beck J., Blackwell B., Clair C. S. Inverse Heat Conductions[M], Wiley, Sussex,1985.
[80] Eld (?)n L. Numerical solution of the sideways heat equation by difference approximation in time[J]. Inverse Problems, 1995, 11:913-923.
[81]Eld(?)n L., Berntsson F., Regi(?)ska T. Wavelet and Fourier methods for solving the sideways heat equation[J]. SIAM J.SCI.Comput., 2000, 21: 2187-2205.
[82] Fu C.L. Simplified Tikhonov and Fourier regularization methods on general sideways parabolic equation[J]. J. Comput. Appl. Math., 2004, 167: 449-463.
[83]Hao D.N., Reinhardt H.J., Schneider A. Numerical solution to a sideways parabolic equation[J]. Inter. J. Numer. Methods in Engr., 2001, 50: 1253-1267.
[84] Tu J.S., Beck J.V. Solution of inverse heat conduction problems using a combined function specification and regularization method and a direct finite element solver, Proc.2nd Ann. Inverse Problems in Engineering Seminar ed J.V.Beck[J] (East Lansing, MI:College of Engineering, Michigan State University). 2008, 1-25.
[85]Engl H.W. Regularization of Inverse Problem[M]. Bostom: Kluwer Academic Publishers, 2000.
[86] Liu Z.H., Li J., Li Z.W. Regularization method with two parameters for nonlinear ill-posed problems[J]. Science in China Series A: Mathematics, 2008, 51: 70-78.
[87] Liu Z.H. Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities[J], Inverse Problems. 2005, 21: 13-20,
[88] Tikhonov A.N., Arsenin V.Ya. Solutions of Ill-posed Problems[M]. New York: Wiley, 1977.
[89]Reginska T., Eld(?)n L. Stability and convergence of wavelet-Galerkin method for the sideways heat equation[J]. Journal of Inverse and Ill-posed Problems., 2000, 8: 31-49.
[90]Guo L., Murio D.A. A mollified space-marching finite-difference algorithm for the two-dimensional inverse heat conduction problem with slab symmetry [J]. Inverse Problems, 1991, 7: 247-259.
[91]Nair M.T., Tautenhahn U. Lavrentiev regularization for linear ill-posed problems under general source conditions[J]. Journal of Analysis and Application, 2004, 23:167-185.
[92]Tadi M. An iterative method for the solution of ill-posed parabolic systems[J]. Appl. Math. Comput., 2008, 21: 843-851.
[93]Jourhmane M., Mera N.S. An iterative algorithm for the backward heat conduction problem based on variable relaxtion factors[J]. Inverse Probl.Eng. 2002,10:293-308.
[94] Morozov V.A. On the solution of functional equations by the method of regularization[J]. Soviet Math. Dokl., 1966, 7: 414-417.
[95]Mera N.S. The method of fundamental solutions for the backward heat conduction problem[J]. Inverse probl.Sci.Eng., 2005, 13: 79-98.
[96] Mera N.S., Elliott L, Ingham D.B., Lesnic D. An iterative boundary element method for solving the one dimensional backward heat conduction problem[J]. Int.J.Heat Mass Transfer., 2001, 44:1937-1946.
[97] Yildiz B., Yetis H., Sever A. A stability estimate on the regularized solution of the backward heat equatio[J]. Appl.Math.Comput, 2003, 135: 561-567.
[98] Liu L., Murio D.A. Numerical experiments in 2-D IHCP on bounded domains:part I.the 'interior' cube problem[J]. Comput. Math. Appl., 1996, 31: 15-32.
[99]Alber Ya.L, Ryazantseva I.P. Nonlinear ill-posed problems of monotone type[J]. Springer, Dordrecht, 2006.
[100] Adams R.A., Sobolev Spaces[M]. London : Academic Press, 1975.
[101] Zeidler E. Nonlinear Functional Analysis and its Applications II/A-B[M]. Springer, New York, 1990.