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1.
In this paper, we study the multi-parameter Tikhonov regularization method which adds multiple different penalties to exhibit multi-scale features of the solution. An optimal error bound of the regularization solution is obtained by a priori choice of multiple regularization parameters. Some theoretical results of the regularization solution about the dependence on regularization parameters are presented. Then, an a posteriori parameter choice, i.e., the damped Morozov discrepancy principle, is introduced to determine multiple regularization parameters. Five model functions, i.e., two hyperbolic model functions, a linear model function, an exponential model function and a logarithmic model function, are proposed to solve the damped Morozov discrepancy principle. Furthermore, four efficient model function algorithms are developed for finding reasonable multiple regularization parameters, and their convergence properties are also studied. Numerical results of several examples show that the damped discrepancy principle is competitive with the standard one, and the model function algorithms are efficient for choosing regularization parameters.  相似文献   

2.
We study multi-parameter regularization (multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regularization parameters from the viewpoint of augmented Tikhonov regularization, and derive a new parameter choice strategy called the balanced discrepancy principle. A priori and a posteriori error estimates are provided to theoretically justify the principles, and numerical algorithms for efficiently implementing the principles are also provided. Numerical results on deblurring are presented to illustrate the feasibility of the balanced discrepancy principle.  相似文献   

3.
In this paper we establish the error estimates for multi-penalty regularization under the general smoothness assumption in the context of learning theory. One of the motivation for this work is to study the convergence analysis of two-parameter regularization theoretically in the manifold learning setting. In this spirit, we obtain the error bounds for the manifold learning problem using more general framework of multi-penalty regularization. We propose a new parameter choice rule “the balanced-discrepancy principle” and analyze the convergence of the scheme with the help of estimated error bounds. We show that multi-penalty regularization with the proposed parameter choice exhibits the convergence rates similar to single-penalty regularization. Finally on a series of test samples we demonstrate the superiority of multi-parameter regularization over single-penalty regularization.  相似文献   

4.
The reconstruction of solutions in statistical inverse problems in Hilbert spaces requires regularization, which is often based on a parametrized family of proposal estimators. The choice of an appropriate parameter in this family is crucial. We propose a modification of the classical discrepancy principle as an adaptive parameter selection. This varying discrepancy principle evaluates the misfit in some weighted norm, and it also has an incorporated emergency stop. These ingredients allow the order optimal reconstruction when the solution owns nice spectral resolution. Theoretical analysis is accompanied with numerical simulations, which highlight the features of the proposed varying discrepancy principle.  相似文献   

5.
Image restoration is an inverse problem that has been widely studied in recent years. The total variation based model by Rudin-Osher-Fatemi (1992) is one of the most effective and well known due to its ability to preserve sharp features in restoration. This paper addresses an important and yet outstanding issue for this model in selection of an optimal regularization parameter, for the case of image deblurring. We propose to compute the optimal regularization parameter along with the restored image in the same variational setting, by considering a Karush Kuhn Tucker (KKT) system. Through establishing analytically the monotonicity result, we can compute this parameter by an iterative algorithm for the KKT system. Such an approach corresponds to solving an equation using discrepancy principle, rather than using discrepancy principle only as a stopping criterion. Numerical experiments show that the algorithm is efficient and effective for image deblurring problems and yet is competitive.  相似文献   

6.
In this paper, we propose a new method for image restoration problems, which are degraded by impulsive noise, with nonconvex data fitting term and nonconvex regularizer.The proposed method possesses the advantages of nonconvex data fitting and nonconvex regularizer simultaneously, namely, robustness for impulsive noise and efficiency for restoring neat edge images.Further, we propose an efficient algorithm to solve the “Nonconvex+Nonconvex” structure problem via using the alternating direction minimization, and prove that the algorithm is globally convergent when the regularization parameter is known. However, the regularization parameter is unavailable in general. Thereby, we combine the algorithm with the continuation technique and modified Morozov’s discrepancy principle to get an improved algorithm in which a suitable regularization parameter can be chosen automatically. The experiments reveal the superior performances of the proposed algorithm in comparison with some existing methods.  相似文献   

7.
Summary In this paper we consider a class of regularization methods for a discretized version of an operator equation (which includes the case that the problem is ill-posed) with approximately given right-hand side. We propose an a priori- as well as an a posteriori parameter choice method which is similar to the discrepancy principle of Ivanov-Morozov. From results on fractional powers of selfadjoint operators we obtain convergence rates, which are (in many cases) the same for both parameter choices.  相似文献   

8.
The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a sequential variant of the discrepancy principle is analysed. In many cases, such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here, we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.  相似文献   

9.
We consider Tikhonov regularization of linear ill-posed problems with noisy data. The choice of the regularization parameter by classical rules, such as discrepancy principle, needs exact noise level information: these rules fail in the case of an underestimated noise level and give large error of the regularized solution in the case of very moderate overestimation of the noise level. We propose a general family of parameter choice rules, which includes many known rules and guarantees convergence of approximations. Quasi-optimality is proved for a sub-family of rules. Many rules from this family work well also in the case of many times under- or overestimated noise level. In the case of exact or overestimated noise level we propose to take the regularization parameter as the minimum of parameters from the post-estimated monotone error rule and a certain new rule from the proposed family. The advantages of the new rules are demonstrated in extensive numerical experiments.  相似文献   

10.
ABSTRACT

In this paper, we establish an initial theory regarding the second-order asymptotical regularization (SOAR) method for the stable approximate solution of ill-posed linear operator equations in Hilbert spaces, which are models for linear inverse problems with applications in the natural sciences, imaging and engineering. We show the regularizing properties of the new method, as well as the corresponding convergence rates. We prove that, under the appropriate source conditions and by using Morozov's conventional discrepancy principle, SOAR exhibits the same power-type convergence rate as the classical version of asymptotical regularization (Showalter's method). Moreover, we propose a new total energy discrepancy principle for choosing the terminating time of the dynamical solution from SOAR, which corresponds to the unique root of a monotonically non-increasing function and allows us to also show an order optimal convergence rate for SOAR. A damped symplectic iterative regularizing algorithm is developed for the realization of SOAR. Several numerical examples are given to show the accuracy and the acceleration effect of the proposed method. A comparison with other state-of-the-art methods are provided as well.  相似文献   

11.
In this paper we propose a criterion based on risk minimization to stop the Landweber algorithm for estimating the solution of a linear system with noisy data. Under the hypothesis of white Gaussian noise, we provide an unbiased estimator of the risk and we use it for defining a variant of the classical discrepancy principle. Moreover, we prove that the proposed variant satisfies the regularization property in expectation. Finally, we perform some numerical simulations when the signal formation model is given by a convolution or a Radon transform, to show that the proposed method is numerically reliable and furnishes slightly better solutions than classical estimators based on the predictive risk, namely the Unbiased Predictive Risk Estimator and the Generalized Cross Validation.  相似文献   

12.
A common way to handle the Tikhonov regularization method for the first kind Fredholm integral equations, is first to discretize and then to work with the final linear system. This unavoidably inflicts discretization errors which may lead to disastrous results, especially when a quadrature rule is used. We propose to regularize directly the integral equation resulting in a continuous Tikhonov problem. The Tikhonov problem is reduced to a simple least squares problem by applying the Golub-Kahan bidiagonalization (GKB) directly to the integral operator. The regularization parameter and the iteration index are determined by the discrepancy principle approach. Moreover, we study the discrete version of the proposed method resulted from numerical evaluating the needed integrals. Focusing on the nodal values of the solution results in a weighted version of GKB-Tikhonov method for linear systems arisen from the Nyström discretization. Finally, we use numerical experiments on a few test problems to illustrate the performance of our algorithms.  相似文献   

13.
Adaptive Parameter Selection for Total Variation Image Deconvolution   总被引:1,自引:0,他引:1  
In this paper, we propose a discrepancy rule-based method to automatically choose the regularization parameters for total variation image restoration problems. The regularization parameters are adjusted dynamically in each iteration. Numerical results are shown to illustrate the performance of the proposed method.  相似文献   

14.
In this paper, we are interested in the solution of nonlinear inverse problems of the form F(x)=y. We propose an implicit Landweber method, which is similar to the third-order midpoint Newton method in form, and consider the convergence behavior of the implicit Landweber method. Using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. We conclude with numerical examples confirming the theoretical results, including comparisons with the classical Landweber iteration and presented modified Landweber methods.  相似文献   

15.
Inexact Newton regularization methods have been proposed by Hanke and Rieder for solving nonlinear ill-posed inverse problems. Every such a method consists of two components: an outer Newton iteration and an inner scheme providing increments by regularizing local linearized equations. The method is terminated by a discrepancy principle. In this paper we consider the inexact Newton regularization methods with the inner scheme defined by Landweber iteration, the implicit iteration, the asymptotic regularization and Tikhonov regularization. Under certain conditions we obtain the order optimal convergence rate result which improves the suboptimal one of Rieder. We in fact obtain a more general order optimality result by considering these inexact Newton methods in Hilbert scales.  相似文献   

16.
Tikhonov Regularization of Large Linear Problems   总被引:1,自引:0,他引:1  
Many numerical methods for the solution of linear ill-posed problems apply Tikhonov regularization. This paper presents a new numerical method, based on Lanczos bidiagonalization and Gauss quadrature, for Tikhonov regularization of large-scale problems. An estimate of the norm of the error in the data is assumed to be available. This allows the value of the regularization parameter to be determined by the discrepancy principle.  相似文献   

17.
An operator regularization method is considered for ill-posed vector optimization of weakly lower semicontinuous essentially convex functionals on reflexive Banach spaces. The regularization parameter is chosen by a modified generalized discrepancy principle. A condition for the estimation of the convergence rate of regularized solutions is derived. This article was submitted by the author in English.  相似文献   

18.
In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite “strip”. This problem is well known to be severely ill-posed. The optimal error bound for the problem with only nonhomogeneous Neumann data is deduced, which is independent of the selected regularization methods. A framework of a modified Tikhonov regularization in conjunction with the Morozov’s discrepancy principle is proposed, it may be useful to the other linear ill-posed problems and helpful for the other regularization methods. Some sharp error estimates between the exact solutions and their regularization approximation are given. Numerical tests are also provided to show that the modified Tikhonov method works well.  相似文献   

19.
Summary. The ``L--curve' is a plot (in ordinary or doubly--logarithmic scale) of the norm of (Tikhonov--) regularized solutions of an ill--posed problem versus the norm of the residuals. We show that the popular criterion of choosing the parameter corresponding to the point with maximal curvature of the L--curve does not yield a convergent regularization strategy to solve the ill--posed problem. Nevertheless, the L--curve can be used to compute the regularization parameters produced by Morozov's discrepancy principle and by an order--optimal variant of the discrepancy principle proposed by Engl and Gfrerer in an alternate way. Received June 29, 1993 / Revised version received February 2, 1994  相似文献   

20.
《Journal of Complexity》2006,22(3):371-381
We discuss adaptive strategies for choosing regularization parameters in Tikhonov–Phillips regularization of discretized linear operator equations. Two rules turn out to be based entirely on data from the underlying regularization scheme. Among them, only the discrepancy principle allows us to search for the optimal regularization parameter from the easiest problem. This potential advantage cannot be achieved by the standard projection scheme. We present a modified scheme, in which the discretization level varies with the successive regularization parameters, which has the advantage, mentioned before.  相似文献   

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