求解非线性不适定问题的几种双参数Newton型正则化方法
|
摘要
本文主要研究非线性反问题和不适定问题的求解。许多实际应用领域常归结为非线性反问题的求解，比如说参数识别问题，反散射问题，逆Sturm-Liouville问题以及非线性第一类Fredholm方程的求解问题等。目前，关于线性反问题和不适定问题的理论工作已经相对比较完善，在实际应用中也取得良好效果，而非线性反问题和不适定问题的理论和实践都还有许多需要完善的地方，而且非线性不适定问题的理论工作开展的少，相互借鉴的地方有限。非线性不适定问题研究的难点在于它的非线性性、不适定性、及无限维性。求解问题的关键是如何构造正则化算子，如何构造参数选取准则使方法成为收敛的正则化方法。
本文主要给出了几种带双参数的Newton-型正则化方法。我们首先给出了Newton-隐式迭代法，此时内层的正则化参数为内迭代步数，由Hanke准则来确定。接着，给出了一种带双正则化参数的Newton型方法，双正则化参数由修正Hanke准则确定，并证明了带双参数算法的收敛性和稳定性。上述的Newton-隐式迭代法可以看作这种带双正则化参数Newton型方法的一种特殊情况。数值例子显示了算法的有效性。但由Hanke准则确定正则化参数的方法还不能证明迭代解的收敛速率。
其次，把Tikhonov正则化方法应用到该线性化方程，然后利用Samanskii思想把一步Newton迭代与多步简化Newton迭代相结合，便得到了求解非线性不适定问题的带双参数渐近简化牛顿方法，此时两个参数比值由Bakushinskii准则来确定。在外迭代，我们首先采用了先验选取准则确定迭代次数，分析了近似解的收敛性，在适当的源条件等假设下，得到了近似解的按阶最优收敛速率。
再次，在没有先验的光滑性信息情况下，为了获得最优收敛速率，利用先验选取准则则变得不再实用，而利用不仅依赖于误差界δ而且依赖于扰动数据y~δ的后验停止准则是必要的。接着我们就引进了Kaltenbacher型后验停止准则和Lepskij型后验停止准则，在Kaltenbacher型后验终止准则中，只能给出ν∈(0，1/2]时的最优收敛速率，在此终止准则条件下，为了获得ν＞1/2时的收敛速率，必须对非线性算子的非线性性假设加强，而这个加强的条件在实际问题中几乎不能验证。为克服此困难，引入了Lepskij型后验停止准则，给出了ν∈(0，1/2]∪(?)时的最优速率证明。
最后，值得一提的是，由于非线性不适定问题求解的难点之一是内层正则化参数的选取，针对此问题，在最后一章我们提出了一种新的内层正则化参数的选取准则，它能更好地拟合线性化方程右端项的误差水平。这个新的准则结合在内层采用隐式迭代法，由此得到的新方法与Tikhonov方法和Bakushinskii方法进行了比较，结果显示了该方法的优越性。最后我们分别对内层的线性化方程采用隐式迭代法或Tikhonov方法，把这个新准则与Hanke准则和Bakushinskii准则进行了数值比较，这个准则的优越性再次突现出来。
In this thesis, we have discussed how to solve the nonlinear ill-posed problems. The problems in many applied domains often can be formulated as a nonlinear inverse problem, for example, parameter identification problem, inverse scatting problem, inverse Sturm-Liouville problem and the first nonlinear Fredholm equation, etc.. Presently, the theory of linear ill-posed problem has been relatively perfect and has favorable effect in the actual application. However, the theory and the practice of nonlinear ill-posed problem need to be perfected. There are finite aspects to use for reference because only few theories of nonlinear inverse problems have been developed. The difficulties in the research of the nonlinear inverse problems are the nonlinearity, ill-posedness and infinity. Now the key for solving the problems is how to construct the regularization operator and how to choose the parameter to make the method into a regularization method.
Several Newton-type regularization methods with double parameters are presented in the thesis. Firstly we presented Newton-implicit iterative method and the inner regularization parameter is inner iterative number which is determined by Hanke rule. At the same time, a Newton-type method with double regularization parameters is given. The double regularization parameters are determined by the modified Hanke's rule. And Newton-implicit iterative method can be regarded as a special case of this Newton-type method with double regularization parameters. The convergence and the stability of the two methods are proved. And the numerical examples show the effectiveness of the method with double parameters. However, the convergence rates of the regularization method with Hanke rule can not be proved now.
Secondly, after we use the Tikhonov regularization to solve the linearized equation and use the idea of Samanskii to combine the one-step Newton iterate with the more-step simplified Newton iterate, we can derive the asymptotic simplified Newton method with double parameters. The ratio of the two parameters is determined by Bakushinskii rule. In the outer iteration, we firstly adoptαpriori rule determining the outer iteration number and analyze the convergence of the approximation solution and the optimal convergence rates of the approximation under the proper source condition.
Thirdly, under the condition with noαpriori smoothness, in order to obtain the optimal convergence rates, it is not practical to use theαpriori rule. And it is necessary to use the a posteriori stopping rule which not only depends on the error boundδbut also on the perturbed data y~δ. In the following, we introduce the Kaltenbacher-typeαposteriori stopping rule and the Lepskij-type a posteriori stopping rule. In the Kaltenbacher-typeαposteriori stopping rule, we can only give the optimal convergence rate in v∈(0,1/2]. Under this stopping rule, in order to obtain the convergence rate in v > 1/2, the assumptions on the nonlinearity of the nonlinear operator has to be strengthened. However, the strengthened condition can not be verified in the actual problems. To conquer this difficulty, we introduce the Lepskij-typeαposteriori stopping rule and prove the optimal convergence rates in v∈(0, 1/2]∪N.
In the end, it is worthwhile to mention that we present a new inner stopping rule in the last chapter because one of the difficulties to resolve the nonlinear ill-posed problem is how to choose the inner regularization parameter. And the new rule can simulate the error level of the righthand in the linearized equation nicely. Combine the new stopping rule and using the implicit iterative method in the inner iteration, we get a new method. We compare the new method with Tikhonov method and Bakushinskii method and the numerical examples show the superiority of the new method. Finally, we use implicit iterative method or Tikhonov method for solving the linearized equation and compare the new rule with Hanke rule and Bakushinskii rule, the superiority of the new rule is taken on.
本文主要给出了几种带双参数的Newton-型正则化方法。我们首先给出了Newton-隐式迭代法，此时内层的正则化参数为内迭代步数，由Hanke准则来确定。接着，给出了一种带双正则化参数的Newton型方法，双正则化参数由修正Hanke准则确定，并证明了带双参数算法的收敛性和稳定性。上述的Newton-隐式迭代法可以看作这种带双正则化参数Newton型方法的一种特殊情况。数值例子显示了算法的有效性。但由Hanke准则确定正则化参数的方法还不能证明迭代解的收敛速率。
其次，把Tikhonov正则化方法应用到该线性化方程，然后利用Samanskii思想把一步Newton迭代与多步简化Newton迭代相结合，便得到了求解非线性不适定问题的带双参数渐近简化牛顿方法，此时两个参数比值由Bakushinskii准则来确定。在外迭代，我们首先采用了先验选取准则确定迭代次数，分析了近似解的收敛性，在适当的源条件等假设下，得到了近似解的按阶最优收敛速率。
再次，在没有先验的光滑性信息情况下，为了获得最优收敛速率，利用先验选取准则则变得不再实用，而利用不仅依赖于误差界δ而且依赖于扰动数据y~δ的后验停止准则是必要的。接着我们就引进了Kaltenbacher型后验停止准则和Lepskij型后验停止准则，在Kaltenbacher型后验终止准则中，只能给出ν∈(0，1/2]时的最优收敛速率，在此终止准则条件下，为了获得ν＞1/2时的收敛速率，必须对非线性算子的非线性性假设加强，而这个加强的条件在实际问题中几乎不能验证。为克服此困难，引入了Lepskij型后验停止准则，给出了ν∈(0，1/2]∪(?)时的最优速率证明。
最后，值得一提的是，由于非线性不适定问题求解的难点之一是内层正则化参数的选取，针对此问题，在最后一章我们提出了一种新的内层正则化参数的选取准则，它能更好地拟合线性化方程右端项的误差水平。这个新的准则结合在内层采用隐式迭代法，由此得到的新方法与Tikhonov方法和Bakushinskii方法进行了比较，结果显示了该方法的优越性。最后我们分别对内层的线性化方程采用隐式迭代法或Tikhonov方法，把这个新准则与Hanke准则和Bakushinskii准则进行了数值比较，这个准则的优越性再次突现出来。
In this thesis, we have discussed how to solve the nonlinear ill-posed problems. The problems in many applied domains often can be formulated as a nonlinear inverse problem, for example, parameter identification problem, inverse scatting problem, inverse Sturm-Liouville problem and the first nonlinear Fredholm equation, etc.. Presently, the theory of linear ill-posed problem has been relatively perfect and has favorable effect in the actual application. However, the theory and the practice of nonlinear ill-posed problem need to be perfected. There are finite aspects to use for reference because only few theories of nonlinear inverse problems have been developed. The difficulties in the research of the nonlinear inverse problems are the nonlinearity, ill-posedness and infinity. Now the key for solving the problems is how to construct the regularization operator and how to choose the parameter to make the method into a regularization method.
Several Newton-type regularization methods with double parameters are presented in the thesis. Firstly we presented Newton-implicit iterative method and the inner regularization parameter is inner iterative number which is determined by Hanke rule. At the same time, a Newton-type method with double regularization parameters is given. The double regularization parameters are determined by the modified Hanke's rule. And Newton-implicit iterative method can be regarded as a special case of this Newton-type method with double regularization parameters. The convergence and the stability of the two methods are proved. And the numerical examples show the effectiveness of the method with double parameters. However, the convergence rates of the regularization method with Hanke rule can not be proved now.
Secondly, after we use the Tikhonov regularization to solve the linearized equation and use the idea of Samanskii to combine the one-step Newton iterate with the more-step simplified Newton iterate, we can derive the asymptotic simplified Newton method with double parameters. The ratio of the two parameters is determined by Bakushinskii rule. In the outer iteration, we firstly adoptαpriori rule determining the outer iteration number and analyze the convergence of the approximation solution and the optimal convergence rates of the approximation under the proper source condition.
Thirdly, under the condition with noαpriori smoothness, in order to obtain the optimal convergence rates, it is not practical to use theαpriori rule. And it is necessary to use the a posteriori stopping rule which not only depends on the error boundδbut also on the perturbed data y~δ. In the following, we introduce the Kaltenbacher-typeαposteriori stopping rule and the Lepskij-type a posteriori stopping rule. In the Kaltenbacher-typeαposteriori stopping rule, we can only give the optimal convergence rate in v∈(0,1/2]. Under this stopping rule, in order to obtain the convergence rate in v > 1/2, the assumptions on the nonlinearity of the nonlinear operator has to be strengthened. However, the strengthened condition can not be verified in the actual problems. To conquer this difficulty, we introduce the Lepskij-typeαposteriori stopping rule and prove the optimal convergence rates in v∈(0, 1/2]∪N.
In the end, it is worthwhile to mention that we present a new inner stopping rule in the last chapter because one of the difficulties to resolve the nonlinear ill-posed problem is how to choose the inner regularization parameter. And the new rule can simulate the error level of the righthand in the linearized equation nicely. Combine the new stopping rule and using the implicit iterative method in the inner iteration, we get a new method. We compare the new method with Tikhonov method and Bakushinskii method and the numerical examples show the superiority of the new method. Finally, we use implicit iterative method or Tikhonov method for solving the linearized equation and compare the new rule with Hanke rule and Bakushinskii rule, the superiority of the new rule is taken on.
引文
[1] Bakushinskii,A.B., The problem of the convergence of the iteratively regularized Gauβ-Newton method. Comput.Math.Phys., 32(1992):1353-1359.
[2] Bauer,F., Hohage,T., A Lepskij-Type Stopping Rule for Regularized Newton Methods. Inverse Problems, 21(2005):1975-1991.
[3] Baumeister,J., Stable Solutions of Inverse Problems. Vieweg-Verlog, Braunschweig, 1987.
[4] BechJ.V., Blackwell,B., C.St.Clair,Jr., Inverse Heat Conduction: Ill-Posed Problems, Wiley, New York, 1985.
[5] Binder,A., Hanke,M., Scherzer,O., On the Landweber iteration for nonlinear ill-posed problems. J.Inv.Ill-Posed Problems, 4(1996):381-389.
[6] Biaschke-Kaltenbacher,B. Some Newton type methods for the solution of nonlinear ill-posed problems. PhD thesis. Schriften der Johannes Kepler Universit(?)t Linz, Trauner, Linz, 1996.
[7] Blaschke-Kaltenbacher,B., Neubauer,A., Scherzer,O. On convergence rates for the iteratively regularized Gauβ-Newton method. IMA J.Numer.Anal., 17(1997):421-436.
[8] Burger,M., Osher,S., Convergence rates of convex variational regularization. Inverse Problems, 20(2004):1411-1421.
[9] Calvetti,D., Lewis,B., On the regularizing properties of the GMRES method. Numer.Math., 91(2002):605-625.
[10] CannonJ.R., Duchateau,P., An inverse problem for a nonlinear diffusion equation. SIAM J.Appl.Math., 39(1980):272-282.
[11] Chavent,G., Strategies for the regularization of nonlinear least squares problems. SIAM, Philade-phia, 1995:217-232.
[12] Chavent,G., Kunisch,K., On weakly nonlinear inverse problems. SIAM J.Appl.Math., 56(1996):542-572.
[13] Cheng.J., Hon,Y.C., Wei,T., Yamamoto,M., Numerical computation of a Cauchy Problem for Laplaces's equation. Z.Angew.Math.Mech., 81(2001):665-674.
[14] Colonius,F., Kunisch,K., Output least squares stability in elliptic systems. Appl.Math.Optim., 19(1989):33-63.
[15] Colonius,F., Kunisch,K., Stability for parameter estimation in two point boundary value problems. J.Reine Angewandte Math., 370(1986):l-29.
[16] Colton,D., Analytic Theory of Partial Differential Equations, Pitman, Boston, 1980.
[17] Engl,H.W., On the Choice of the Regularization Parameter for Iterated Tikhonov Regularization of Ill-Posed Problems. J.Approx.Theory.,49(1987):55-63.
[18] Engl,H.W., Regularization methods for the stable solutions of inverse problems. Surv.Math.Ind., 3(1993):71-143.
[19] Engl.H.W., Discrepancy principles for Tiknonov regularization of ill-posed problems leading to optimal convergence rates. J.Optim.Theor.Appl., 52(1987):209-215.
[20] Engl,H.W., Kunisch,K., Neubauer,A. Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Problems, 5(1989):523-540.
[21] Engl,H.W., Kunisch.K., Neubauer,A. Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems. SIAM J.Numer.Anal., 30(1993):1796-1883.
[22] Engl,H.W., Hanke,M., Neubauer,A., Regularization of inverse problems. Dordrecht:kluwer, 1996.
[23] Engl,H.W., Scherzer,O., Convergence rates results for iterative methods for nonlinear ill-posed problems. D.Colton, etc.(eds), Surveys on Solution Method for Inverse Problems, Spring, Vienna/New York, 2000:7-34.
[24] Gfrerer,H., An a-posteriori parameter choice for ordinary and iterated Tiknonov regularization of ill-posed problems leading to optimal convergence rates. Mathematics of Computation, 49(1987):507-522.
[25] Girard,D.A., A fast 'Monte-Carlo Cross-Validation' procedure for large least squares problems with noisy data. Numer.Math., 56(1989):l-23.
[26] Golub,G.H., Heath.M., Wahba,G., Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 21(1979):215-223.
[27] Groetsch,C.W., Inverse Problems in the Mathematical Sciences. Vieweg, Braunschweig, 1993.
[28] Groetsch,C.W., The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.
[29] Hadamard,J., Lectures on the Cauchy Problems in Linear Partial Differential Equations. Yale University Press, New Haven, 1923.
[30] Hanke,M., Accelerated Landweber iterations for the solution of ill-posed equations. Numer.Math., 60(1991):341-373.
[31] Hanke,M., A regularization Levenberg-Marquardt scheme, with applications to inverse groundwa-ter filtration problem. Inverse Problems, 13(1997):79-95.
[32] Hanke,M., Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems. 18(1997):971-993.
[33] Hanke,M., Conjugate gradient type method for ill-posed problems. Longman Scientific Technical, Harlow, Essex, 1995.
[34] Hanke,M., Hansen,P.C., Regularization methods for large-scale problems. Surv.Math.Ind., 3(1993):253-315.
[35] Hanke M, Neubauer A and Scherzer 0, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer.Math., 72(1995)21 -37.
[36] Hansen,P.C, Analysis of discrete ill-posed problems by means of the L-curve. SIAM Review, 34(1992):561-584.
[37] Hansen,P.C, Regularization GSVD and truncated GSVD. BIT, 29(1989):491 -504.
[38] Hansen,P.C, The truncated SVD as a method for regularization. BIT, 27(1987):534-553.
[39] He Guoqiang, An inverse problem for the burgers equation. Journal of Computational Mathematics, ll(1993):103-112.
[40] He Guoqiang, Liu Linxian, A kind of implicit iterative methods for ill-posed operator equation. Journal of Computational Mathematics, 17(1999):275-284.
[41] He Guoqiang, Wang Xinge and Liu Linxian, Implicit iterative methods with variable control parameters for ill-posed operator equations. Acta Mathematica Scientia, 20B(2000):485-494.
[42] He Guoqiang, Chen Y.M., Inverse Problem for the Burgers' Equation. Journal of Computational Mathematics, 11(1999):275-284.
[43] He,G., Meng, Z., A Newton type iterative method for heat-conduction inverse problems. Applied Mathematics and Mechanics(Eng.Ed),(SCI)(EI), accepted.
[44] Hofmann,B., Gorenflo,R., On autoconvolution and regularization. Inverse Problems, 10(1994):353-373.
[45] Hofmann,B., Scherzer,O., Factors influencing the ill-posedness of nonlinear problems. Inverse Problems, 10(1994):1277-1297.
[46] Hohage,T., Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scatting problem. Inverse Problems, 13(1997):1279-1299.
[47] Hou,Zongyi, Jin,Qinian, Tikhonov regularization for nonlinear ill-posed problems. Nonlinear Anal-ysis:Theory, Methods and Applications, 28(1997): 1799-1809.
[48] Hou,Zongyi, Yang Hongqi, Regularization method for improving optimal convergence rate of the regularized solution of ill-posed problems. Acta Mathematica Scientia, 18(1998):177-185.
[49] Isakov,V., Inverse Problems for Partial Differential Equations, Springer, Berlin, New York, 1998.
[50] Ito,K., Kunisch.K., On the choice of the regularization parameter in nonlinear inverse problems. SIAM J.Optimization, 2(1992):376-404.
[51] Ito,K., Kunisch,K., On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems. J.Math.Anal.Appl., 188(1994): 1040-1066.
[52] Jin,Qinian, On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems. Mathematics of Computation, 69(2000):1603-1623.
[53] Jin,Qinian, Hou,Zongyi, Finite-dimensional approximations to the solutions of nonlinear ill-posed problems. Appl.Anal., 62(1996):253-261.
[54] Jin,Qinian, Hou,Zongyi, On the choice of the regularization parameter for ordinary and iterated Tikhonov regularization of nonlinear ill-posed problems. Inverse Problems, 13(1997):815-827.
[55] Kaltenbacher,B., Some Newton type methods for the regularization of nonlinear ill-posed problems. Inverse Problems, 13(1997):729-753.
[56] Kaltenbacher,B., A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems. Numerical Mathematics, 79(1998):501-528.
[57] Kaltenbacher,B., Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems. Inverse Problems, 16(2000): 1523-1539.
[58] Kaltenbacher,B., A projection regularized Newton method for nonlinear ill-posed problems with application to parameter identification problems with finite element discretization. SIAM J.Numer.Anal., 37(2000): 1885-1908.
[59] Kaltenbacher,B., Some Newton type methods for the solution of nonlinear ill-posed problems. PhD thesis, Schriften der Johannes Kepler University Linz, Tranner, 1996.
[60] Kaltenbacher,B., On Broyden 's Method for Nonlinear Ill-Posed Problems. Numer.Funct.Anal.Opt., 19(1998):807-833.
[61] Kaltenbacher,B., Neubauer,A., Convergence of projected iterative regularization methods for nonlinear problems with smooth solutions. Inverse Problems, 22(2006): 1105-1119.
[62] Kaltenbacher,B., Neubauer,A., Scherzer,O., On convergence rates for the iterative!)' regularized Gauss-Newton method. IMA J.Numer.Anal., 17(1997):421-436.
[63] Keller,J.B., Inverse problems. Am.Math.Mon., 83(1976): 107-118.
[64] Kilmer,M.E., O'Leary,D.P., Choosing regularization parameters in iterative methods for ill-posed problems. SIAM J.Matrix Anal.Appl., 22(2001):1204-1221.
[65] Kravaris,C, Seinfeld,J.H., Multidimensional Inverse Problems of Mathematical Physics, American Mathematical Society Translations Providence, Rhod Island, 1986.
[66] Kress,R., Rundell.W., A quasi-Newton method in inverse obstacle scattering. Inverse Problems, 10(1994):1145-1158.
[67] Lanczos,C., Linear Differential Operators, Van Nostrand, New York, 1961.
[68] Larrentiev,M.M., Romanov,V.G., Vasiliev, Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Mathematics Volume 167, Springer-Verlag, Berlin, 1970.
[69] Lechleiter,A., Reider,A, Newton regularizations for impedance tomography: a numerical study. Inverse Problems, 22(2006):1967-1987.
[70] Levenberg,K., A method for the solution of certain nonlinear problems in least squares. Qart.Appl.Math.,2(1944):164-166.
[71] Liu,X.Y., Chen,Y.M., Convergence of a generalized pulse-spectum technique(GPST) for inverse problems of 1-D diffusion equations in space-time domain. Math.Comput., 51(1988):477-489.
[72] Lukas,M.A., Asymptotic optimality of generalized cross-validation for choosing the regularization parameter. Numer.Math., 66(1993):41-66.
[73] Maass,P, Pereverzev,S., Ramlau,R., Solodky,S.G.,An adaptive discretization scheme for Tikhonov regularization with a posteriori parameter selection. Numer.Math., 87(2001):3485-3502.
[74] Marquardt,D.W., An algorithm for least-squares estimation of nonlinear inequalities. SIAM J.Appl.Math., 11(1963):431-441.
[75] Meng,Z.H., He,G.Q., A Newton-implicit iterative method for nonlinear inverse problems, submitted.
[76] Morozov,V.A., On the solution of functional equations by the method of regularization. Soviet Math.Dokl., 7(1966):414-417.
[77] Murio,D.A., The Mollification Method and the Numerical Solution of Ill-Posed Problems, John Wileg, NewYork, 1993.
[78] Natter,E, W(?)bbeling,F., Mathematical methods on image reconstruction. Philadephia SIAM, 2001.
[79] Nashed,M.Z., Chen,X., Convergence of Newton-like methods for singular operator equations using outer inverses. Numer.Math., 66(1993):235-257.
[80] Neubauer,A., Tikhonov regularization for nonlinear ill-posed problems: optimal convergence rates and finite dimensional approximation. Inverse Problems, 5(1989):541-557.
[81] Neubauer,A., Scherzer,O., Finite-dimensional approximation of Tikhonov regularized solution of nonlinear ill-posed problems. Numer.Funct.Anal.Optim., 11(1990):85-99.
[82] Otega,J.M., Rheinboldt,W.C, Iterative Solution of Nonlinear Equations in Several Variables, New York: Acad. Press, 1970.
[83] Plato,R.,Vainikko,G.M., On the regularization of projection methods for solving ill-posed problems. Numer.Math., 57(1990):63-79.
[84] Ramlau,R., TIGRA-an iterative algorithm for regularizing nonlinear ill-posed problems. Inverse Problems, 19(2003):433-465.
[85] Ramlau,R., Morozov's discrepancy principle for Tikhonov regularization of nonlinear operators. Numer.Funct.Anal.Optim., 23(2002):147-172.
[86] Ramlau,R., A steepest descent algorithm for the global minimization of the Tikhonov functional. Inverse Problems, 18(2002):381-405.
[87] Ramlau,R., Teschke,G., Tikhonov replacement functionals for iteraively solving nonlinear operator equations. Inverse Problems, 21(2005):1571-1592.
[88] Richter,G.R., An inverse problem for the steady state diffusion equation. SIAM J.Appl.Math., 41(1981):210-221.
[89] Richter,G.R., Numerical identification of a spatially varying diffusion coefficient. Math.Comput., 36(1981):375-386.
[90] Samanskii,V., On a modification of the Newton method(in Russian). Ukrain Mat.Z., 19(1967): 133-138.
[91] Scherzer,O., A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems. Num.Func.Anal.Opt., 17(1996): 197-214.
[92] Scherzer,O., A parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems leading to optimal convergence rates. Appl.Math., 38(1993):479-487.
[93] Scherzer,O., The use of Morozov's discrepancy principle for Tikhonov regularization for solving nonlinear ill-posed problems. Computing, 51(1993):45-60.
[94] Scherzer,O., Convergence rate of iterated Tikhonov solution of nonlinear ill-posed problems. Numer.Math., 66(1993):259-279.
[95] Scherzer,O., A modified Landweber iteration for solving parameter estimation problems. Appl.Math.Optimiz.,38(1998):45-68.
[96] Scherzer,O., Engl,H.W., Kunisch,K., Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems. SIAM J.Numer.Anal. 30(1993):1796-l 838.
[97] Schock,E., Semi-iterative methods for the approximate solution of ill-posed equations. Numer. Math., 60(1991):341-373.
[98] Seidman,T.I., Vogel,C.R., Wellposedness and convergence of some regularization methods for nonlinear ill-posed problems. Inverse Problems, 5(1989):227-238.
[99] Solodky,S.G., On a quasi-optimal regularized projection method for solving operator equations of the first kind. 21(2005): 1473-1485.
[100] Tautenhahn,U., On the asymptotical regularization of nonlinear ill-posed problems. Inverse Problems, 10(I994):1405-14I8.
[101] Tautenhahn,U., Jin,Qinian, Tikhonov regularization and a posteriori rules for solving nonlinear ill-posed problems. Inverse Problems, 19(2003): 1-21.
[102] Tautenhahn,U., Hamarik,U., The use of monotonicity for choosing the regularization parameter in ill-posed problems. Inverse Problems, 15(1999):1487-1505.
[103] Tikhonov,A.N., Arsenin,U.Y., Solutions of Ill-Posed Problems. John Wiley and Sons, New York, 1977.
[104] Tikhonov,A.N., Goncharskii,A.V.(Eds.), Ill-Posed Problems in the Natural Science. MIR, Moscow, 1989.
[105] Tikhonov,A.N., Leonov,A.S., Yagola,A.G., Nonlinear Ill-Posed Problems, London: Chapman and Hall, 1998.
[106] Vainikko,G.M., The discrepancy principle for a class os regularization methods. USSR Comp.Math.Math.Phys.,22(1982):l-19.
[107] Vainikko,G.M., The critical level of discrepancy in regularization methods. USSR Comp.Math.Math.Phys., 23(1983):1-9.
[108] Vogel,C.R., Optimal choice of a truncation level for the truncated SVD solution of linear first kind integral equation when data are noisy. SIAM J.Numer.Anal., 23(1986): 109-117.
[109] Vogel,C.R., Non-convergence of the L-curve regularization parameter selection method. Inverse Problems, 12(1996):535-547.
[110] Wahba,G., Spline models for observational data. SIAM, Philiadephia, 1990.
[111] Wang Yanfei, Yuan Yaxiang, A trust region algorithm for distributed parameter identification problems. J.Comput.Math., 21(2003)759-772.
[112] Wang Yanfei, Yuan Yaxiang, On the regularity of trust region algorithms for nonlinear ill-posed inverse problems. Proceedings of the Third Asian Mathematical Conference 2002, (World Scientific, Singapore, 2002)(2002)562-580.
[113] Wang Yanfei, Yuan Yaxiang, A trust region method for solving distributed parameter identification problems. Report, AMSS, Chinese Academy of Sciences.
[114] Wang Yanfei, Yuan Yaxiang, Hongchao Zhang, A trust region-CG algorithm for deblurring problem in atmospheric image reconstruction. Scicence in China, Ser. A, 45(2002):731-740.
[115] Zhang,N., He,G., A-smooth regularization for ill-posed equations with perturbed operators and noisy data. J.Shanghai Univ.(Eng. Ed.), 7(2003):35-40.
[116]贺国强，A-光滑正则化算子，高校应用数学学报，7(1992)：568-578。
[117]黄光远，刘小军，数学物理反问题，山东科技出版社，济南，1993。
[118]江泽坚，吴智泉，实变函数论，高等教育出版社，北京，1994。
[119]李世雄，刘家琦，小波变换和反演数学基础，地质出版社，北京，1994。
[120]刘高连，叶轮机械气体动力学基础，机械工业出版社，1980。
[121]肖庭延，宋金来，计量经济模型参数的正则化估计，数量经济技术研究，10(1998)：41-46。
[122]肖庭延，于慎根，王彦飞，反问题的数值解法，科学出版社，北京，2003。
[123]袁亚湘，孙文瑜，最优化理论与方法，科学出版社，北京，1997。
[2] Bauer,F., Hohage,T., A Lepskij-Type Stopping Rule for Regularized Newton Methods. Inverse Problems, 21(2005):1975-1991.
[3] Baumeister,J., Stable Solutions of Inverse Problems. Vieweg-Verlog, Braunschweig, 1987.
[4] BechJ.V., Blackwell,B., C.St.Clair,Jr., Inverse Heat Conduction: Ill-Posed Problems, Wiley, New York, 1985.
[5] Binder,A., Hanke,M., Scherzer,O., On the Landweber iteration for nonlinear ill-posed problems. J.Inv.Ill-Posed Problems, 4(1996):381-389.
[6] Biaschke-Kaltenbacher,B. Some Newton type methods for the solution of nonlinear ill-posed problems. PhD thesis. Schriften der Johannes Kepler Universit(?)t Linz, Trauner, Linz, 1996.
[7] Blaschke-Kaltenbacher,B., Neubauer,A., Scherzer,O. On convergence rates for the iteratively regularized Gauβ-Newton method. IMA J.Numer.Anal., 17(1997):421-436.
[8] Burger,M., Osher,S., Convergence rates of convex variational regularization. Inverse Problems, 20(2004):1411-1421.
[9] Calvetti,D., Lewis,B., On the regularizing properties of the GMRES method. Numer.Math., 91(2002):605-625.
[10] CannonJ.R., Duchateau,P., An inverse problem for a nonlinear diffusion equation. SIAM J.Appl.Math., 39(1980):272-282.
[11] Chavent,G., Strategies for the regularization of nonlinear least squares problems. SIAM, Philade-phia, 1995:217-232.
[12] Chavent,G., Kunisch,K., On weakly nonlinear inverse problems. SIAM J.Appl.Math., 56(1996):542-572.
[13] Cheng.J., Hon,Y.C., Wei,T., Yamamoto,M., Numerical computation of a Cauchy Problem for Laplaces's equation. Z.Angew.Math.Mech., 81(2001):665-674.
[14] Colonius,F., Kunisch,K., Output least squares stability in elliptic systems. Appl.Math.Optim., 19(1989):33-63.
[15] Colonius,F., Kunisch,K., Stability for parameter estimation in two point boundary value problems. J.Reine Angewandte Math., 370(1986):l-29.
[16] Colton,D., Analytic Theory of Partial Differential Equations, Pitman, Boston, 1980.
[17] Engl,H.W., On the Choice of the Regularization Parameter for Iterated Tikhonov Regularization of Ill-Posed Problems. J.Approx.Theory.,49(1987):55-63.
[18] Engl,H.W., Regularization methods for the stable solutions of inverse problems. Surv.Math.Ind., 3(1993):71-143.
[19] Engl.H.W., Discrepancy principles for Tiknonov regularization of ill-posed problems leading to optimal convergence rates. J.Optim.Theor.Appl., 52(1987):209-215.
[20] Engl,H.W., Kunisch,K., Neubauer,A. Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Problems, 5(1989):523-540.
[21] Engl,H.W., Kunisch.K., Neubauer,A. Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems. SIAM J.Numer.Anal., 30(1993):1796-1883.
[22] Engl,H.W., Hanke,M., Neubauer,A., Regularization of inverse problems. Dordrecht:kluwer, 1996.
[23] Engl,H.W., Scherzer,O., Convergence rates results for iterative methods for nonlinear ill-posed problems. D.Colton, etc.(eds), Surveys on Solution Method for Inverse Problems, Spring, Vienna/New York, 2000:7-34.
[24] Gfrerer,H., An a-posteriori parameter choice for ordinary and iterated Tiknonov regularization of ill-posed problems leading to optimal convergence rates. Mathematics of Computation, 49(1987):507-522.
[25] Girard,D.A., A fast 'Monte-Carlo Cross-Validation' procedure for large least squares problems with noisy data. Numer.Math., 56(1989):l-23.
[26] Golub,G.H., Heath.M., Wahba,G., Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 21(1979):215-223.
[27] Groetsch,C.W., Inverse Problems in the Mathematical Sciences. Vieweg, Braunschweig, 1993.
[28] Groetsch,C.W., The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.
[29] Hadamard,J., Lectures on the Cauchy Problems in Linear Partial Differential Equations. Yale University Press, New Haven, 1923.
[30] Hanke,M., Accelerated Landweber iterations for the solution of ill-posed equations. Numer.Math., 60(1991):341-373.
[31] Hanke,M., A regularization Levenberg-Marquardt scheme, with applications to inverse groundwa-ter filtration problem. Inverse Problems, 13(1997):79-95.
[32] Hanke,M., Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems. 18(1997):971-993.
[33] Hanke,M., Conjugate gradient type method for ill-posed problems. Longman Scientific Technical, Harlow, Essex, 1995.
[34] Hanke,M., Hansen,P.C., Regularization methods for large-scale problems. Surv.Math.Ind., 3(1993):253-315.
[35] Hanke M, Neubauer A and Scherzer 0, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer.Math., 72(1995)21 -37.
[36] Hansen,P.C, Analysis of discrete ill-posed problems by means of the L-curve. SIAM Review, 34(1992):561-584.
[37] Hansen,P.C, Regularization GSVD and truncated GSVD. BIT, 29(1989):491 -504.
[38] Hansen,P.C, The truncated SVD as a method for regularization. BIT, 27(1987):534-553.
[39] He Guoqiang, An inverse problem for the burgers equation. Journal of Computational Mathematics, ll(1993):103-112.
[40] He Guoqiang, Liu Linxian, A kind of implicit iterative methods for ill-posed operator equation. Journal of Computational Mathematics, 17(1999):275-284.
[41] He Guoqiang, Wang Xinge and Liu Linxian, Implicit iterative methods with variable control parameters for ill-posed operator equations. Acta Mathematica Scientia, 20B(2000):485-494.
[42] He Guoqiang, Chen Y.M., Inverse Problem for the Burgers' Equation. Journal of Computational Mathematics, 11(1999):275-284.
[43] He,G., Meng, Z., A Newton type iterative method for heat-conduction inverse problems. Applied Mathematics and Mechanics(Eng.Ed),(SCI)(EI), accepted.
[44] Hofmann,B., Gorenflo,R., On autoconvolution and regularization. Inverse Problems, 10(1994):353-373.
[45] Hofmann,B., Scherzer,O., Factors influencing the ill-posedness of nonlinear problems. Inverse Problems, 10(1994):1277-1297.
[46] Hohage,T., Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scatting problem. Inverse Problems, 13(1997):1279-1299.
[47] Hou,Zongyi, Jin,Qinian, Tikhonov regularization for nonlinear ill-posed problems. Nonlinear Anal-ysis:Theory, Methods and Applications, 28(1997): 1799-1809.
[48] Hou,Zongyi, Yang Hongqi, Regularization method for improving optimal convergence rate of the regularized solution of ill-posed problems. Acta Mathematica Scientia, 18(1998):177-185.
[49] Isakov,V., Inverse Problems for Partial Differential Equations, Springer, Berlin, New York, 1998.
[50] Ito,K., Kunisch.K., On the choice of the regularization parameter in nonlinear inverse problems. SIAM J.Optimization, 2(1992):376-404.
[51] Ito,K., Kunisch,K., On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems. J.Math.Anal.Appl., 188(1994): 1040-1066.
[52] Jin,Qinian, On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems. Mathematics of Computation, 69(2000):1603-1623.
[53] Jin,Qinian, Hou,Zongyi, Finite-dimensional approximations to the solutions of nonlinear ill-posed problems. Appl.Anal., 62(1996):253-261.
[54] Jin,Qinian, Hou,Zongyi, On the choice of the regularization parameter for ordinary and iterated Tikhonov regularization of nonlinear ill-posed problems. Inverse Problems, 13(1997):815-827.
[55] Kaltenbacher,B., Some Newton type methods for the regularization of nonlinear ill-posed problems. Inverse Problems, 13(1997):729-753.
[56] Kaltenbacher,B., A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems. Numerical Mathematics, 79(1998):501-528.
[57] Kaltenbacher,B., Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems. Inverse Problems, 16(2000): 1523-1539.
[58] Kaltenbacher,B., A projection regularized Newton method for nonlinear ill-posed problems with application to parameter identification problems with finite element discretization. SIAM J.Numer.Anal., 37(2000): 1885-1908.
[59] Kaltenbacher,B., Some Newton type methods for the solution of nonlinear ill-posed problems. PhD thesis, Schriften der Johannes Kepler University Linz, Tranner, 1996.
[60] Kaltenbacher,B., On Broyden 's Method for Nonlinear Ill-Posed Problems. Numer.Funct.Anal.Opt., 19(1998):807-833.
[61] Kaltenbacher,B., Neubauer,A., Convergence of projected iterative regularization methods for nonlinear problems with smooth solutions. Inverse Problems, 22(2006): 1105-1119.
[62] Kaltenbacher,B., Neubauer,A., Scherzer,O., On convergence rates for the iterative!)' regularized Gauss-Newton method. IMA J.Numer.Anal., 17(1997):421-436.
[63] Keller,J.B., Inverse problems. Am.Math.Mon., 83(1976): 107-118.
[64] Kilmer,M.E., O'Leary,D.P., Choosing regularization parameters in iterative methods for ill-posed problems. SIAM J.Matrix Anal.Appl., 22(2001):1204-1221.
[65] Kravaris,C, Seinfeld,J.H., Multidimensional Inverse Problems of Mathematical Physics, American Mathematical Society Translations Providence, Rhod Island, 1986.
[66] Kress,R., Rundell.W., A quasi-Newton method in inverse obstacle scattering. Inverse Problems, 10(1994):1145-1158.
[67] Lanczos,C., Linear Differential Operators, Van Nostrand, New York, 1961.
[68] Larrentiev,M.M., Romanov,V.G., Vasiliev, Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Mathematics Volume 167, Springer-Verlag, Berlin, 1970.
[69] Lechleiter,A., Reider,A, Newton regularizations for impedance tomography: a numerical study. Inverse Problems, 22(2006):1967-1987.
[70] Levenberg,K., A method for the solution of certain nonlinear problems in least squares. Qart.Appl.Math.,2(1944):164-166.
[71] Liu,X.Y., Chen,Y.M., Convergence of a generalized pulse-spectum technique(GPST) for inverse problems of 1-D diffusion equations in space-time domain. Math.Comput., 51(1988):477-489.
[72] Lukas,M.A., Asymptotic optimality of generalized cross-validation for choosing the regularization parameter. Numer.Math., 66(1993):41-66.
[73] Maass,P, Pereverzev,S., Ramlau,R., Solodky,S.G.,An adaptive discretization scheme for Tikhonov regularization with a posteriori parameter selection. Numer.Math., 87(2001):3485-3502.
[74] Marquardt,D.W., An algorithm for least-squares estimation of nonlinear inequalities. SIAM J.Appl.Math., 11(1963):431-441.
[75] Meng,Z.H., He,G.Q., A Newton-implicit iterative method for nonlinear inverse problems, submitted.
[76] Morozov,V.A., On the solution of functional equations by the method of regularization. Soviet Math.Dokl., 7(1966):414-417.
[77] Murio,D.A., The Mollification Method and the Numerical Solution of Ill-Posed Problems, John Wileg, NewYork, 1993.
[78] Natter,E, W(?)bbeling,F., Mathematical methods on image reconstruction. Philadephia SIAM, 2001.
[79] Nashed,M.Z., Chen,X., Convergence of Newton-like methods for singular operator equations using outer inverses. Numer.Math., 66(1993):235-257.
[80] Neubauer,A., Tikhonov regularization for nonlinear ill-posed problems: optimal convergence rates and finite dimensional approximation. Inverse Problems, 5(1989):541-557.
[81] Neubauer,A., Scherzer,O., Finite-dimensional approximation of Tikhonov regularized solution of nonlinear ill-posed problems. Numer.Funct.Anal.Optim., 11(1990):85-99.
[82] Otega,J.M., Rheinboldt,W.C, Iterative Solution of Nonlinear Equations in Several Variables, New York: Acad. Press, 1970.
[83] Plato,R.,Vainikko,G.M., On the regularization of projection methods for solving ill-posed problems. Numer.Math., 57(1990):63-79.
[84] Ramlau,R., TIGRA-an iterative algorithm for regularizing nonlinear ill-posed problems. Inverse Problems, 19(2003):433-465.
[85] Ramlau,R., Morozov's discrepancy principle for Tikhonov regularization of nonlinear operators. Numer.Funct.Anal.Optim., 23(2002):147-172.
[86] Ramlau,R., A steepest descent algorithm for the global minimization of the Tikhonov functional. Inverse Problems, 18(2002):381-405.
[87] Ramlau,R., Teschke,G., Tikhonov replacement functionals for iteraively solving nonlinear operator equations. Inverse Problems, 21(2005):1571-1592.
[88] Richter,G.R., An inverse problem for the steady state diffusion equation. SIAM J.Appl.Math., 41(1981):210-221.
[89] Richter,G.R., Numerical identification of a spatially varying diffusion coefficient. Math.Comput., 36(1981):375-386.
[90] Samanskii,V., On a modification of the Newton method(in Russian). Ukrain Mat.Z., 19(1967): 133-138.
[91] Scherzer,O., A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems. Num.Func.Anal.Opt., 17(1996): 197-214.
[92] Scherzer,O., A parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems leading to optimal convergence rates. Appl.Math., 38(1993):479-487.
[93] Scherzer,O., The use of Morozov's discrepancy principle for Tikhonov regularization for solving nonlinear ill-posed problems. Computing, 51(1993):45-60.
[94] Scherzer,O., Convergence rate of iterated Tikhonov solution of nonlinear ill-posed problems. Numer.Math., 66(1993):259-279.
[95] Scherzer,O., A modified Landweber iteration for solving parameter estimation problems. Appl.Math.Optimiz.,38(1998):45-68.
[96] Scherzer,O., Engl,H.W., Kunisch,K., Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems. SIAM J.Numer.Anal. 30(1993):1796-l 838.
[97] Schock,E., Semi-iterative methods for the approximate solution of ill-posed equations. Numer. Math., 60(1991):341-373.
[98] Seidman,T.I., Vogel,C.R., Wellposedness and convergence of some regularization methods for nonlinear ill-posed problems. Inverse Problems, 5(1989):227-238.
[99] Solodky,S.G., On a quasi-optimal regularized projection method for solving operator equations of the first kind. 21(2005): 1473-1485.
[100] Tautenhahn,U., On the asymptotical regularization of nonlinear ill-posed problems. Inverse Problems, 10(I994):1405-14I8.
[101] Tautenhahn,U., Jin,Qinian, Tikhonov regularization and a posteriori rules for solving nonlinear ill-posed problems. Inverse Problems, 19(2003): 1-21.
[102] Tautenhahn,U., Hamarik,U., The use of monotonicity for choosing the regularization parameter in ill-posed problems. Inverse Problems, 15(1999):1487-1505.
[103] Tikhonov,A.N., Arsenin,U.Y., Solutions of Ill-Posed Problems. John Wiley and Sons, New York, 1977.
[104] Tikhonov,A.N., Goncharskii,A.V.(Eds.), Ill-Posed Problems in the Natural Science. MIR, Moscow, 1989.
[105] Tikhonov,A.N., Leonov,A.S., Yagola,A.G., Nonlinear Ill-Posed Problems, London: Chapman and Hall, 1998.
[106] Vainikko,G.M., The discrepancy principle for a class os regularization methods. USSR Comp.Math.Math.Phys.,22(1982):l-19.
[107] Vainikko,G.M., The critical level of discrepancy in regularization methods. USSR Comp.Math.Math.Phys., 23(1983):1-9.
[108] Vogel,C.R., Optimal choice of a truncation level for the truncated SVD solution of linear first kind integral equation when data are noisy. SIAM J.Numer.Anal., 23(1986): 109-117.
[109] Vogel,C.R., Non-convergence of the L-curve regularization parameter selection method. Inverse Problems, 12(1996):535-547.
[110] Wahba,G., Spline models for observational data. SIAM, Philiadephia, 1990.
[111] Wang Yanfei, Yuan Yaxiang, A trust region algorithm for distributed parameter identification problems. J.Comput.Math., 21(2003)759-772.
[112] Wang Yanfei, Yuan Yaxiang, On the regularity of trust region algorithms for nonlinear ill-posed inverse problems. Proceedings of the Third Asian Mathematical Conference 2002, (World Scientific, Singapore, 2002)(2002)562-580.
[113] Wang Yanfei, Yuan Yaxiang, A trust region method for solving distributed parameter identification problems. Report, AMSS, Chinese Academy of Sciences.
[114] Wang Yanfei, Yuan Yaxiang, Hongchao Zhang, A trust region-CG algorithm for deblurring problem in atmospheric image reconstruction. Scicence in China, Ser. A, 45(2002):731-740.
[115] Zhang,N., He,G., A-smooth regularization for ill-posed equations with perturbed operators and noisy data. J.Shanghai Univ.(Eng. Ed.), 7(2003):35-40.
[116]贺国强，A-光滑正则化算子，高校应用数学学报，7(1992)：568-578。
[117]黄光远，刘小军，数学物理反问题，山东科技出版社，济南，1993。
[118]江泽坚，吴智泉，实变函数论，高等教育出版社，北京，1994。
[119]李世雄，刘家琦，小波变换和反演数学基础，地质出版社，北京，1994。
[120]刘高连，叶轮机械气体动力学基础，机械工业出版社，1980。
[121]肖庭延，宋金来，计量经济模型参数的正则化估计，数量经济技术研究，10(1998)：41-46。
[122]肖庭延，于慎根，王彦飞，反问题的数值解法，科学出版社，北京，2003。
[123]袁亚湘，孙文瑜，最优化理论与方法，科学出版社，北京，1997。