Examples of Saturated Convergence Rates for Tikhonov Regularization

Tikhonov regularization is one of the most popular methods for solving linear operator equations of the first kind Au = f with bounded operator, which are ill-posed in general (Fredholm's integral equation of the first kind is a typical example). For problems with inexact data (both the operator and the right-hand side) the rate of convergence of regularized solutions to the generalised solution u ₊ (i.e.the minimal-norm least-squares solution) can be estimated under the condition that this solution has the source form: u ₊ im(A*A). It is well known that for Tikhonov regularization the highest-possible worst-case convergence rates increase with only for some values of , in general not greater than one. This phenomenon is called the saturation of convergence rate. In this article the analysis of this property of the method with a criterion of a priori regularization parameter choice is presented and illustrated by examples constructed for equations with compact operators.This revised version was published online in October 2005 with corrections to the Cover Date.     

A Modified Tikhonov Regularization for Linear Operator Equations

We construct with the aid of regularizing filters a new class of improved regularization methods, called modified Tikhonov regularization (MTR), for solving ill-posed linear operator equations. Regularizing properties and asymptotic order of the regularized solutions are analyzed in the presence of noisy data and perturbation error in the operator. With some accurate estimates in the solution errors, optimal convergence order of the regularized solutions is obtained by a priori choice of the regularization parameter. Furthermore, numerical results are given for several ill-posed integral equations, which not only roughly coincide with the theoretical results but also show that MTR can be more accurate than ordinary Tikhonov regularization (OTR).     

Nonstationary Iterated Tikhonov Regularization

A convergence rate is established for nonstationary iterated Tikhonov regularization, applied to ill-posed problems involving closed, densely defined linear operators, under general conditions on the iteration parameters. It is also shown that an order-optimal accuracy is attained when a certain a posteriori stopping rule is used to determine the iteration number.     

Tikhonov Regularization of Large Linear Problems

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.     

求解病态问题的一种改进的Tikhonov正则化--(1)正则化方法的建立

对于带有右端扰动数据的第一类紧算子方程的病态问题 ,本文应用正则化子建立了一类新的正则化求解方法 ,称之为改进的Tikonov正则化 ;通过适当选取正则参数 ,证明了正则解具有最优的渐近收敛阶 .与通常的Tikhonov正则化相比 ,这种改进的正则化可使正则解取到足够高的最优渐近阶     

Economic Discretization Scheme for the Nonstationary Iterated Tikhonov Method

An adaptive discretization scheme of ill-posed problems is used for the nonstationary iterated Tikhonov regularization. It is shown that, for some classes of operator equations of the first kind, the proposed algorithm is more efficient compared with standard methods. Communicated by F. A. Potra This work was partially supported by DAAD under Grant A/01/06016     

On Regularization of a Source Identification Problem in a Parabolic PDE and its Finite Dimensional Analysis

We consider the inverse problem of identifying a general source term, which is a function of both time variable and the spatial variable, in a parabolic PDE from the knowledge of boundary measurements of the solution on some portion of the lateral boundary. We transform this inverse problem into a problem of solving a compact linear operator equation. For the regularization of the operator equation with noisy data, we employ the standard Tikhonov regularization, and its finite dimensional realization is done using a discretization procedure involving the space $L^2(0,\tau;L^2(Ω))$. For illustrating the specification of an a priori source condition, we have explicitly obtained the range space of the adjoint of the operator involved in the operator equation.     

A new zero-finder for Tikhonov regularization

Tikhonov regularization with the regularization parameter determined by the discrepancy principle requires the computation of a zero of a rational function. We describe a cubically convergent zero-finder for this purpose. AMS subject classification (2000)  65F22, 65H05, 65R32     

A Joint Tikhonov Regularization and Augmented Lagrange Approach for Ill-Posed State Constrained Control Problems with Sparse Controls

Abstract

We provide a modified augmented Lagrange method coupled with a Tikhonov regularization for solving ill-posed state constrained elliptic optimal control problems with sparse controls. We consider a linear quadratic optimal control problem without any additional L² regularization terms. The sparsity is guaranteed by an additional L¹ term. Here, the modification of the classical augmented Lagrange method guarantees us uniform boundedness of the multiplier that corresponds to the state constraints. We present a coupling between the regularization parameter introduced by the Tikhonov regularization and the penalty parameter from the augmented Lagrange method, which allows us to prove strong convergence of the controls and their corresponding states. Moreover, convergence results proving the weak convergence of the adjoint state and weak*-convergence of the multiplier are provided. Finally, we demonstrate our method in several numerical examples.     

An efficient discretization scheme for solving ill-posed problems

In this paper, we consider a finite-dimensional approximation scheme combined with Tikhonov regularization for solving ill-posed problems. Error estimates are obtained by an a priori parameter choice strategy and the results show that the amount of discrete information required for solving the problem is far less than the traditional finite-dimensional approach.     

Convergence analysis of a regularized approximation for solving Fredholm integral equations of the first kind

In this paper, we suggest a convergence analysis for solving Fredholm integral equations of the first kind using Tikhonov regularization under supremum norm. We also provide an a priori parameter choice strategy for choosing the regularization parameter and obtain an error estimate.     

Identifying an unknown source in the Poisson equation by the method of Tikhonov regularization in Hilbert scales

In this paper, we consider the problem for identifying the unknown source in the Poisson equation. The Tikhonov regularization method in Hilbert scales is extended to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. The user does not need to estimate the smoothness parameter and the a priori bound of the exact solution when the a posteriori choice rule is used. Numerical examples show that the proposed method is effective and stable.     

A numerical technique for solving IHCPs using Tikhonov regularization method

This study is intended to provide a numerical algorithm for solving a one-dimensional inverse heat conduction problem. The given heat conduction equation, the boundary conditions, and the initial condition are presented in a dimensionless form. The numerical approach is developed based on the use of the solution to the auxiliary problem as a basis function. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution.     

Convergence Rates of the Hilbert Uniqueness Method via Tikhonov Regularization

In this paper, we identify the Hilbert uniqueness method for a boundary control problem with the calculation of the pseudo inverse. Because of its ill-posedness, we approximate it by a regularized Hilbert uniqueness method, which we prove to be identical with Tikhonov regularization. By this equivalence, we can find sufficient conditions for convergence and convergence rates, which require approximation rates in Müntz spaces. We show that these conditions are fulfilled by an a priori bound in Sobolev norms on the exact solution.     

Iterated Tikhonov regularization with a general penalty term

Tikhonov regularization is one of the most popular approaches to solving linear discrete ill‐posed problems. The choice of the regularization matrix may significantly affect the quality of the computed solution. When the regularization matrix is the identity, iterated Tikhonov regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov regularization. This paper provides an analysis of iterated Tikhonov regularization with a regularization matrix different from the identity. Computed examples illustrate the performance of this method.     

A modified Tikhonov regularization method for an axisymmetric backward heat equation

This paper deals with the inverse time problem for an axisymmetric heat equation. The problem is ill-posed. A modified Tikhonov regularization method is applied to formulate regularized solution which is stably convergent to the exact one. estimate between the approximate solution and exact technical inequality and improving a priori smoothness Meanwhile, a logarithmic-HSlder type error solution is obtained by introducing a rather assumption.     

反向热传导问题的一种新的正则化方法

讨论一个高维反向热传导问题,这是一个经典的严重不适定问题.关于这一问题我们给出一种新的正则化方法-改进的Tikhonov正则化方法,以恢复解对数据的连续依赖性.通过构造一个重要的不等式和提高先验光滑条件,获得正则解在0相似文献   

Regularization of Ill-Posed Problems by Envelope Guided Conjugate Gradients

Abstract

We propose a new way to iteratively solve large scale ill-posed problems by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain, iteratively, approximations to the Tikhonov L-curve and its corner. Monitoring the change of the approximate L-curves allows us to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired solution can be reconstructed with a low number of iterations. We apply the technique to an idealized image reconstruction problem in positron emission tomography.     

基于混沌粒子群算法的Tikhonov正则化参数选取

Tikhonov正则化方法是求解不适定问题最为有效的方法之一,而正则化参数的最优选取是其关键.本文将混沌粒子群优化算法与Tikhonov正则化方法相结合,基于Morozov偏差原理设计粒子群的适应度函数,利用混沌粒子群优化算法的优点,为正则化参数的选取提供了一条有效的途径.数值实验结果表明,本文方法能有效地处理不适定问题,是一种实用有效的方法.