Solution of Ill-Posed Problems via Adaptive Grid Regularization: Convergence Analysis

A new regularization method, adaptive grid regularization, has been presented. Numerical results there show in a convincing way that this method is a powerful tool to identify discontinuities of solutions of ill-posed problems. It is the aim of this paper to give a convergence analysis for this new method.     

Numerical differentiation from a viewpoint of regularization theory

In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.

     

On the Monotone Error Rule for Parameter Choice in Iterative and Continuous Regularization Methods

We consider in Hilbert spaces linear ill-posed problems Ax = y with noisy data y satisfying y –y. Regularized approximations x _r to the minimum-norm solution x of Ax = y are constructed by continuous regularization methods or by iterative methods. For the choice of the regularization parameter r (the stopping index n in iterative methods) the following monotone error rule (ME rule) is used: we choose r = r _ME (n = n _ME) as the largest r-value with the guaranteed monotonical decrease of the error x _r – x for r [0, r _ME] (x _n – x <#60; x _n–1 – x for n = 1, 2, ..., n _ME). Main attention is paid to iterative methods of gradient type and to nonstationary implicit iteration methods. As shown, the ME rule leads for many methods to order optimal error bounds. Comparisons with other rules for the choice of the stopping index are made and numerical examples are given.This revised version was published online in October 2005 with corrections to the Cover Date.     

一种带噪声的周期函数数值微分方法

本文利用Thkhonov正则化方法讨论了带有噪声离散数据的周期函数的数值微分问题,证明了该方法存在唯一的三次周期样条函数解,并给出了其误差估计,而且从理论和数值例子说明了此方法的有效性．     

Choice of Regularization Parameter in Adaptive Filtering Problems

Computational Mathematics and Mathematical Physics - The paper proposes a new method for choosing a regularization parameter when solving an integral equation of convolution type in problems of...     

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.     

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.     

Numerical differentiation by radial basis functions approximation

Based on radial basis functions approximation, we develop in this paper a new com-putational algorithm for numerical differentiation. Under an a priori and an a posteriori choice rules for the regularization parameter, we also give a proof on the convergence error estimate in reconstructing the unknown partial derivatives from scattered noisy data in multi-dimension. Numerical examples verify that the proposed regularization strategy with the a posteriori choice rule is effective and stable to solve the numerical differential problem. *The work described in this paper was partially supported by a grant from CityU (Project No. 7001646) and partially supported by the National Natural Science Foundation of China (No. 10571079).     

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.     

用正则化方法求解声波散射反问题

研究了从声波散射场的远场模式的信息来再现散射物边界形状的反问题.首先构造表达散射物特征的指示函数,然后利用该函数之特性,建立求解该类反问题的基本方程,从而确定散射物的边界形状.在这个算法中,不需预先知道散射物的边界类型和形状等知识,从T ikhonov正则化方法进行的数值计算结果表明了该方法是有效的和实用的.     

Direct Implementation of Tikhonov Regularization for the First Kind Integral Equation

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.     

Regularization of Discrete Ill-Posed Problems

The paper concerns conditioning aspects of finite-dimensional problems arising when the Tikhonov regularization is applied to discrete ill-posed problems. A relation between the regularization parameter and the sensitivity of the regularized solution is investigated. The main conclusion is that the condition number can be decreased only to the square root of that for the nonregularized problem. The convergence of solutions of regularized discrete problems to the exact generalized solution is analyzed just in the case when the regularization corresponds to the minimal condition number. The convergence theorem is proved under the assumption of the suitable relation between the discretization level and the data error. As an example the method of truncated singular value decomposition with regularization is considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

Computation of Control for Linear Approximately Controllable System Using Tikhonov Regularization

Computation of control for a controlled partial differential equation is a di?cult task, especially when the control problem is ill posed. In this paper, we propose a method of computing the regularized control of a diffusion control system using Tikhonov regularization approach when the system is approximately controllable. The method proposed here for choosing regularization parameter guarantees the convergence of the proposed control.     

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

We study the choice of the regularization parameter for linear ill-posed problems in the presence of noise that is possibly unbounded but only finite in a weaker norm, and when the noise-level is unknown. For this task, we analyze several heuristic parameter choice rules, such as the quasi-optimality, heuristic discrepancy, and Hanke-Raus rules and adapt the latter two to the weakly bounded noise case. We prove convergence and convergence rates under certain noise conditions. Moreover, we analyze and provide conditions for the convergence of the parameter choice by the generalized cross-validation and predictive mean-square error rules.     

Hull-White模型中参数校准的正则化方法

基于Hull-White模型,研究由零息债券的市场价格进行参数校准的问题.构造函数将问题转化为正则化问题,并利用正则化方法得到解的存在性,稳定性和所满足的必要条件.最后利用必要条件进行数值计算,给出了数值模拟算例和实证分析,数值结果表明了方法中引入正则项的有效性,且改善了其参数的稳定性,具有实际意义.     

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     

双调和方程柯西问题的修正的Tikhonov正则化方法

本文研究了双调和方程柯西问题,这类是不适定的,即问题的解(如果存在)不连续依赖于测量数据.首先在精确解的先验假设下给出问题的条件稳定性结果.接着利用修正的Tikhonov正则化方法求解此不适定问题.在先验和后验正则化参数选取规则下,给出正则解和精确解之间的误差估计式.最后给出几个数值例子验证此正则化方法求解此类反问题的有效性.     

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.     

Regularization Using QR Factorization and the Estimation of the Optimal Parameter

In this paper we propose a direct regularization method using QR factorization for solving linear discrete ill-posed problems. The decomposition of the coefficient matrix requires less computational cost than the singular value decomposition which is usually used for Tikhonov regularization. This method requires a parameter which is similar to the regularization parameter of Tikhonov's method. In order to estimate the optimal parameter, we apply three well-known parameter choice methods for Tikhonov regularization.This revised version was published online in October 2005 with corrections to the Cover Date.