Convergence results of a modified regularized gradient method for nonlinear ill-posed problems

A modified iteratively regularized gradient method and its continuous version are proposed for nonlinear ill-posed problems, in which the Tikhonov regularization term is generated by a linear operator. The linear operator may have some physical meaning. And by employing the linear operator, scaling the problem can be avoided in the case that the nonlinear operator in the problem has larger gradient. Adopting a posteriori and a priori stopping rule respectively, we establish the convergence results by using a modified approximate source condition. The numerical results show that the linear operator effects the performance greatly.     

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.     

An augmented mixed finite element method with Lagrange multipliers: A priori and a posteriori error analyses

In this paper, we provide a priori and a posteriori error analyses of an augmented mixed finite element method with Lagrange multipliers applied to elliptic equations in divergence form with mixed boundary conditions. The augmented scheme is obtained by including the Galerkin least-squares terms arising from the constitutive and equilibrium equations. We use the classical Babuška–Brezzi theory to show that the resulting dual-mixed variational formulation and its Galerkin scheme defined with Raviart–Thomas spaces are well posed, and also to derive the corresponding a priori error estimates and rates of convergence. Then, we develop a reliable and efficient residual-based a posteriori error estimate and a reliable and quasi-efficient Ritz projection-based one, as well. Finally, several numerical results illustrating the performance of the augmented scheme and the associated adaptive algorithms are reported.     

Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems

This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under a Hölder-type sourcewise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt, Lobatto, and Radau methods.     

THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

In this article, we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation. This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem. We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule. Some numerical results in one-dimensional case and two-dimensional case show that our method is effcient and stable.     

Descentwise inexact proximal algorithms for smooth optimization

The proximal method is a standard regularization approach in optimization. Practical implementations of this algorithm require (i)?an algorithm to compute the proximal point, (ii)?a rule to stop this algorithm, (iii)?an update formula for the proximal parameter. In this work we focus on?(ii), when smoothness is present??so that Newton-like methods can be used for?(i): we aim at giving adequate stopping rules to reach overall efficiency of the method. Roughly speaking, usual rules consist in stopping inner iterations when the current iterate is close to the proximal point. By contrast, we use the standard paradigm of numerical optimization: the basis for our stopping test is a ??sufficient?? decrease of the objective function, namely a fraction of the ideal decrease. We establish convergence of the algorithm thus obtained and we illustrate it on some ill-conditioned problems. The experiments show that combining the proposed inexact proximal scheme with a standard smooth optimization algorithm improves the numerical behaviour of the latter for those ill-conditioned problems.     

Error correction iterative method for the stationary incompressible MHD flow

A new iterative finite element method for solving the stationary incompressible magnetohydrodynamics (MHD) equations is derived in this paper. The method consists of two steps at each iteration step, we need first to solve the MHD equations by the Oseen-type iterative scheme, and then an error correction strategy is applied to control the error arising from the linearization of the nonlinear MHD equations. The new method not only maintains the advantage of the standard Oseen-type scheme but also possesses a rapid rate of convergence. It is proved that the convergence rate of the proposed method is increased greatly under the uniqueness condition. The uniform stability and convergence of the new scheme are analyzed. Ample numerical experiments are performed to validate the accuracy and the efficiency of the new numerical scheme.     

Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.     

A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION

In this paper,we consider the Cauchy problem for the Laplace equation,which is severely ill-posed in the sense that the solution does not depend continuously on the data.A modified Tikhonov regularization method is proposed to solve this problem.An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained.Moreover,an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained.Numerical examples illustrate the validity and effectiveness of this method.     

A Modified Landweber Iteration for Solving Parameter Estimation Problems

In this paper a convergence analysis for a modified Landweber iteration for the solution of nonlinear ill-posed problems is presented. A priori and a posteriori stopping criteria for terminating the iteration are compared. Some numerical results for the solution of a parameter estimation problem are presented. Accepted 11 September 1996     

求解热源识别问题的修正吉洪诺夫正则化方法

该文研究了一个热源识别问题,通过引入修正吉洪诺夫方法来处理问题的不适定性,在一种先验和一种后验参数选取准则下,分别获得了问题的误差估计.数值例子进一步验证了方法的有效性和稳定性.     

求解椭圆方程柯西问题的修正吉洪诺夫正则化方法

研究了一类变系数椭圆方程的柯西问题,这类问题出现在很多实际问题领域.由于问题的不适定性,不可能通过经典的数值方法来求解上述问题,必须引入正则化手段.采用了一种修正吉洪诺夫正则化方法来求解上述问题.在一种先验和一种后验参数选取准则下,分别获得了问题的误差估计.数值例子进一步显示方法是稳定有效的.     

On the Convergence of Chebyshev’s Method for Multiple Polynomial Zeros

In this paper we investigate the local convergence of Chebyshev’s iterative method for the computation of a multiple polynomial zero. We establish two convergence theorems for polynomials over an arbitrary normed field. A priori and a posteriori error estimates are also provided. All of the results are new even in the case of simple zero.     

Tikhonov regularization method for a backward problem for the time-fractional diffusion equation

This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.     

Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem

Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates.     

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.     

A priori and a posteriori error analyses in the study of viscoelastic problems

In this work, the numerical approximation of a viscoelastic problem is studied. A fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize time derivatives. Then, two numerical analyses are presented. First, a priori estimates are proved from which the linear convergence of the algorithm is derived under suitable regularity conditions. Secondly, an a posteriori error analysis is provided extending some preliminary results obtained in the study of the heat equation. Upper and lower error bounds are obtained.     

IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

In this paper, we identify a space-dependent source for a fractional diffusion equation. This problem is ill-posed, i.e., the solution （if it exists） does not depend continuously on the data. The generalized Tikhonov regularization method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained, Numerical examples are presented to illustrate the validity and effectiveness of this method.     

An iterative method for backward time‐fractional diffusion problem

The aim of this work is to solve the backward problem for a time‐fractional diffusion equation with variable coefficients in a general bounded domain. The problem is ill‐posed in L ² norm sense. An iteration scheme is proposed to obtain a regularized solution. Two kinds of convergence rates are obtained using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one‐dimensional and two‐dimensional cases are provided to show the effectiveness of the proposed methods. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2029–2041, 2014     

Iteratively regularized Gauss–Newton method for operator equations with normally solvable derivative at the solution

We study the iteratively regularized Gauss–Newton method in a Hilbert space for solving irregular nonlinear equations with smooth operators having normally solvable derivatives at the solution. We consider both a priori and a posteriori stopping criteria for the iterations and establish accuracy estimates for resulting approximations. In the case where the a priori stopping rule is used, the accuracy of approximations arises to be proportional to the error level in input data. The latter result generalizes well-known estimates of this kind obtained for linear equations with normally solvable operators.