The Optimal Approximations for Solving Linear Ill-Posed Problems

New projection discrete schemes for ill-posed problems are constructed. We show that for equations with self-adjoint operators the use of self-adjoint projection schemes is not optimal in the sense of the amount of discrete information.     

Stochastic Methods for Ill-Posed Problems

This paper is devoted to the numerical analysis of ill-posed problems of evolution equations in Banach spaces using certain classes of stochastic one-step methods. The linear stability properties of these methods are studied. Regularisation is given by the choice of the regularisation parameter as = , where _n is the stepsize and provides the convergence on smooth initial data. The case of the approximation of well-posed problems is also considered.     

Algorithms and Methods for Solving Scheduling Problems and Other Extremum Problems on Large-Scale Graphs

We consider a large-scale directed graph G = (V, E) whose edges are endowed with a family of characteristics. A subset of vertices of the graph, V′ ⊂ V, is selected and some additional conditions are imposed on these vertices. An algorithm for reducing the optimization problem on the graph G to an optimization problem on the graph G′ = (V′, E′) of a lower dimension is developed. The main steps of the solution and some methods for constructing an approximate solution to the problem on the transformed graph G′ are presented.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 235–251, 2003.     

多元函数条件极值的四种求解方法

多元函数条件极值是高等数学的重要内容之一,本文从等式约束、区域约束、拉格朗日乘子法和单位向量约束下二次型最值问题四个角度切入,力图全面介绍高等数学中有关多元函数极值的问题.     

Well-Posed and Ill-Posed Optimal Control Problems

In this paper, an optimal control problem with variable parameters and variable initial data is considered for some systems of ordinary differential equations. On the basis of variational methods, some sufficient conditions, under which the optimal processes depend continuously on the initial data and parameters of the system, are proved.     

求解非线性不适定问题的隐式迭代法

将处理线性不适定算子方程的隐式迭代法推广到非线性不适定问题,证明了迭代解误差序列的单调性,并进一步利用迭代误差的单调性得出求解非线性不适定问题隐式迭代法对精确方程和扰动方程的收敛性.     

A Regularized Iterative Scheme for Solving Nonlinear Ill-Posed Problems

In this article, we consider a regularized iterative scheme for solving nonlinear ill-posed problems. The convergence analysis and error estimates are derived by choosing the regularization parameter according to both a priori and a posteriori methods. The iterative scheme is stopped using an a posteriori stopping rule, and we prove that the scheme converges to the solution of the well-known Lavrentiev scheme. The salient features of the proposed scheme are: (i) convergence and error estimate analysis require only weaker assumptions compared to standard assumptions followed in literature, and (ii) consideration of an adaptive a posteriori stopping rule and a parameter choice strategy that gives the same convergence rate as that of an a priori method without using the smallness assumption, the source condition. The above features are very useful from theory and application points of view. We also supply the numerical results to illustrate that the method is adaptable. Further, we compare the numerical result of the proposed method with the standard approach to demonstrate that our scheme is stable and achieves good computational output.     

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.     

General Regularizing Functionals for Solving Ill-Posed Problems in Lebesgue Spaces

We study sufficient conditions for general integral functionals in Lebesgue spaces to possess regularizing properties required for solving nonlinear ill-posed problems. We select special classes of such functionals: uniformly convex and quasiuniformly convex (in the extended sense). We give a series of examples and, in particular, a functional that can be used in a generalized version of the maximum entropy method in Lebesgue spaces.     

REGULARIZATION METHODS FOR NONLINEAR ILL-POSED PROBLEMS WITH ACCRETIVE OPERATORS

This article is devoted to the regularization of nonlinear ill-posed problems with accretive operators in Banach spaces. The data involved are assumed to be known approximately. The authors concentrate their discussion on the convergence rates of regular solutions.     

Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

Recently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

     

Automatic Optimal Regularization for Ill-Posed Problems with Stochastical Noise without Imposing Smoothness Assumptions

We consider the compact operator A : 𝒳 → 𝒴 for the separable Hilbert spaces 𝒳 and 𝒴. The problem Ax = y is called ill-posed when the singular values s_k , k = 1, 2, … of the operator A tend to zero. Classically one assumes that y is biased with “deterministic noise”; we will also consider “stochastic noise” where the noise element is a weak Gaussian random variable. There classical stopping rules (e.g. Morozov) do not work. We will show that both for the “deterministic noise” case as well for the “stochastical noise” case we can regularize in an (asymptotically almost) optimal way without knowledge of the smoothness of the solution using Lepskij's method. Furthermore the method also works for estimated error levels and error behavior. So we can assure regularization which is just dependent on measurements obtainable in reality, e.g. satellite problems. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

应用正则化子建立求解不适定问题的正则化方法的探讨

根据紧算子的奇异系统理论,提出一种新的正则化子进而建立了一类新的求解不适定问题的正则化方法。分别通过正则参数的先验选取和后验确定方法,证明了正则解的收敛性并得到了其最优的渐近收敛阶;验证了应用Newton迭代法计算最佳参数的可行性。最后建立了当算子与右端均有扰动时相应的正则化求解策略。文中所述方法完善了一般优化正则化策略的构造理论。     

On a Family of Gradient-Type Projection Methods for Nonlinear Ill-Posed Problems

Abstract

We propose and analyze a family of successive projection methods whose step direction is the same as the Landweber method for solving nonlinear ill-posed problems that satisfy the Tangential Cone Condition (TCC). This family encompasses the Landweber method, the minimal error method, and the steepest descent method; thus, providing an unified framework for the analysis of these methods. Moreover, we define new methods in this family, which are convergent for the constant of the TCC in a range twice as large as the one required for the Landweber and other gradient type methods. The TCC is widely used in the analysis of iterative methods for solving nonlinear ill-posed problems. The key idea in this work is to use the TCC in order to construct special convex sets possessing a separation property, and to successively project onto these sets. Numerical experiments are presented for a nonlinear two-dimensional elliptic parameter identification problem, validating the efficiency of our method.     

On Lanczos Based Methods for the Regularization of Discrete Ill-Posed Problems

We study hybrid methods for the solution of linear ill-posed problems. Hybrid methods are based on he Lanczos process, which yields a sequence of small bidiagonal systems approximating the original ill-posed problem. In a second step, some additional regularization, typically the truncated SVD, is used to stabilize the iteration. We investigate two different hybrid methods and interpret these schemes as well-known projection methods, namely least-squares projection and the dual least-squares method. Numerical results are provided to illustrate the potential of these methods. This gives interesting insight in to the behavior of hybrid methods in practice.This revised version was published online in October 2005 with corrections to the Cover Date.     

Dynamical System Method for Solving Ill-Posed Operator Equations

Two dynamical system methods are studied for solving linear ill-posed problems with both operator and right-hand nonexact. The methods solve a Cauchy problem for a linear operator equation which possesses a global solution. The limit of the global solution at infinity solves the original linear equation. Moreover, we also present a convergent iterative process for solving the Cauchy problem.     

解不适定问题的快速多层方法

Many industrial and engineering applications require numerically solving ill-posed problems. Regularization methods are employed to find approximate solutions of these problems. The choice of regularization parameters by numerical algorithms is one of the most important issues for the success of regularization methods. When we use some discrepancy principles to determine the regularization parameter,     

A Combination of Variational and Penalty Methods for Solving a Class of Fractional Optimal Control Problems

This paper develops an approximate method, based on the combination of epsilon penalty and variational methods, for solving a class of multidimensional fractional optimal control problems. The fractional derivative is in the Caputo sense. In the presented method, utilizing the epsilon method, the given optimal control problem transforms into an unconstrained optimization problem; then, the equivalent variational equality is derived for the given unconstrained problem. The variational equality is approximately solved by applying a spectral method.     

Mixed Methods for Optimal Control Problems

In this paper, we investigate a posteriori error estimates of amixed finite elementmethod for elliptic optimal control problems with an integral constraint. The gradient for ourmethod belongs to the square integrable space instead of the classical H(div; Ω) space. The state and co-state are approximated by the P ₀ ² -P₁ (velocity–pressure) pair and the control variable is approximated by piecewise constant functions. Using duality argument method and energy method, we derive the residual a posteriori error estimates for all variables.     

A New Gradient Method for Ill-Posed Problems

In this paper, we present a new gradient method for linear and nonlinear ill-posed problems F(x) = y. Combined with the discrepancy principle as stopping rule it is a regularization method that yields convergence to an exact solution if the operator F satisfies a tangential cone condition. If the exact solution satisfies smoothness conditions, then even convergence rates can be proven. Numerical results show that the new method in most cases needs less iteration steps than Landweber iteration, the steepest descent or minimal error method.