We are specially interested in the case that problem (1) is ill-posed; that is, the solutions of (1) do not depend continuously on the data. Now the regularization techniques are required. The traditional method is Ti...We are specially interested in the case that problem (1) is ill-posed; that is, the solutions of (1) do not depend continuously on the data. Now the regularization techniques are required. The traditional method is Tikhonov regularization. In recent years, the concept of entropy was introduced into the study of ill-posed problems and developed the maximum entropy method. It is found that the maximum entropy method has its展开更多
We propose a finite dimensional method to compute the solution of nonlinear ill-posed problems approximately and show that under certain conditions, the convergence can be guaranteed. Moreover, we obtain the rate of c...We propose a finite dimensional method to compute the solution of nonlinear ill-posed problems approximately and show that under certain conditions, the convergence can be guaranteed. Moreover, we obtain the rate of convergence of our method provided that the true solution satisfies suitable smoothness condition. Finally, we present two examples from the parameter estimation problems of differential equations and illustrate the applicability of our method.展开更多
In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is...In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is used that yields convergence of the Newton iteration to a solution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Some appropriate assumptions on the closedness and smoothness of the starting value and the solution are shown to lead to optimal convergence rates.展开更多
Some converse and saturation results on Tikhonov regularization of nonlinear ill-posed problems are proved and the a posteriori parameter choice yielding optimal rates of convergence is discussed.
基金Project supported by the National Natural Science Foundation of China.
文摘We are specially interested in the case that problem (1) is ill-posed; that is, the solutions of (1) do not depend continuously on the data. Now the regularization techniques are required. The traditional method is Tikhonov regularization. In recent years, the concept of entropy was introduced into the study of ill-posed problems and developed the maximum entropy method. It is found that the maximum entropy method has its
文摘We propose a finite dimensional method to compute the solution of nonlinear ill-posed problems approximately and show that under certain conditions, the convergence can be guaranteed. Moreover, we obtain the rate of convergence of our method provided that the true solution satisfies suitable smoothness condition. Finally, we present two examples from the parameter estimation problems of differential equations and illustrate the applicability of our method.
文摘In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is used that yields convergence of the Newton iteration to a solution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Some appropriate assumptions on the closedness and smoothness of the starting value and the solution are shown to lead to optimal convergence rates.
基金Project supported by the National Natural Science Foundation of China (Grant No. 9801018).
文摘Some converse and saturation results on Tikhonov regularization of nonlinear ill-posed problems are proved and the a posteriori parameter choice yielding optimal rates of convergence is discussed.
