【作者】 孟泽红；

【导师】 贺国强；

【作者基本信息】 上海大学 ， 计算数学， 2007， 博士

【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。更多还原

【Abstract】 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.更多还原