Converse and saturation results on Tikhonov regularization together witha posteriori parameter choice

Some converse and saturation results on Tikhonov regularization of nonlinear ill-posed problems are proved and the aposteriori parameter choice yielding optimal rates of convergence is discussed Project supported by the National Natural Science Foundation of China (Grant No. 9801018).     

On the quasi-optimal rules for the choice of the regularization parameter in case of a noisy operator

A usual way to characterize the quality of different a posteriori parameter choices is to prove their order-optimality on the different sets of solutions. In paper by Raus and H?marik (J Inverse Ill-Posed Probl 15(4):419–439, 2007) we introduced the property of the quasi-optimality to characterize the quality of the rule of the a posteriori choice of the regularization parameter for concrete problem Au = f in case of exact operator and discussed the quasi-optimality of different well-known rules for the a posteriori parameter choice as the discrepancy principle, the modification of the discrepancy principle, balancing principle and monotone error rule. In this paper we generalize the concept of the quasi-optimality for the case of a noisy operator and concretize results for the mentioned parameter choice rules.     

The balancing principle in solving semi-discrete inverse problems in Sobolev scales by Tikhonov method

In this article we discuss a regularization of semi-discrete ill-posed problem appearing as a result of application of a collocation method to Fredholm integral equation of the first kind. In this context we analyse Tikhonov regularization in Sobolev scales and prove error bounds under general source conditions. Moreover, we study an a posteriori regularization parameter choice by means of the balancing principle.     

A regularization method for the cauchy problem of the modified Helmholtz equation

In the present paper, an iteration regularization method for solving the Cauchy problem of the modified Helmholtz equation is proposed. The a priori and a posteriori rule for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method. Copyright © 2014 John Wiley & Sons, Ltd.     

Tikhonov regularization method for a system of equilibrium problems in Banach spaces

The purpose of this paper is to investigate the Tikhonov regularization method for solving a system of ill-posed equilibrium problems in Banach spaces with a posteriori regularization-parameter choice. An application to convex minimization problems with coupled constraints is also given.     

Recovering the initial distribution for space-fractional diffusion equation by a logarithmic regularization method

In this paper, the backward problem for space-fractional diffusion equation is investigated. We proposed a so-called logarithmic regularization method to solve it. Based on the conditional stability and an a posteriori regularization parameter choice rule, the convergence rate estimates are given under a-priori bound assumption for the exact solution.     

Multi-parameter Tikhonov regularization and model function approach to the damped Morozov principle for choosing regularization parameters

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.     

A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems

In this paper, we conduct a goal-oriented a posteriori analysis for the error in a quantity of interest computed from a cell-centered finite volume scheme for a semilinear elliptic problem. The a posteriori error analysis is based on variational analysis, residual errors and the adjoint problem. To carry out the analysis, we use an equivalence between the cell-centered finite volume scheme and a mixed finite element method with special choice of quadrature.     

On the <Emphasis Type="Italic">a posteriori</Emphasis> choice of a regularization parameter in the solution of severely ill-posed problems

We consider the problem of approximate solution of severely ill-posed problems with perturbed right-hand sides. The approximation properties of a finite-dimensional version of the Tikhonov regularization in the combination with the a posteriori choice of a regularization parameter by means of the balancing principle are analyzed. It is shown that this approach provides an optimal order of accuracy. The efficiency of the theoretical results is checked by comparison with the earlier known methods.     

Approximation with polynomial kernels and SVM classifiers

This paper presents an error analysis for classification algorithms generated by regularization schemes with polynomial kernels. Explicit convergence rates are provided for support vector machine (SVM) soft margin classifiers. The misclassification error can be estimated by the sum of sample error and regularization error. The main difficulty for studying algorithms with polynomial kernels is the regularization error which involves deeply the degrees of the kernel polynomials. Here we overcome this difficulty by bounding the reproducing kernel Hilbert space norm of Durrmeyer operators, and estimating the rate of approximation by Durrmeyer operators in a weighted L¹ space (the weight is a probability distribution). Our study shows that the regularization parameter should decrease exponentially fast with the sample size, which is a special feature of polynomial kernels. Dedicated to Charlie Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 68T05, 62J02. Ding-Xuan Zhou: The first author is supported partially by the Research Grants Council of Hong Kong (Project No. CityU 103704).     

Stable numerical differentiation for the second order derivatives

Consider a numerical differential problem, which aims to compute the second order derivative of a function stably from its given noisy data. For this ill-posed problem, we introduce the Lavrent′ev regularization scheme by reformulating this differentiation problem as an integral equation of the first kind. The advantage of this proposed scheme is that we can give the regularizing solution by an explicit integral expression, therefore it is easy to be implemented. The a-priori and a-posterior choice strategies for the regularization parameter are considered, with convergence analysis and error estimate of the regularizing solution for noisy data based on the integral operator decomposition. The validity of the proposed scheme is shown by several numerical examples.     

An iteration regularization for a time-fractional inverse diffusion problem

In the present paper we consider a time-fractional inverse diffusion problem, where data is given at x = 1 and the solution is required in the interval 0 < x < 1. This problem is typically ill-posed: the solution (if it exists) does not depend continuously on the data. We give a new iteration regularization method to deal with this problem, and error estimates are obtained for a priori and a posteriori parameter choice rules, respectively. Furthermore, numerical implement shows the proposed method works effectively.     

A Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient: regularization and error estimates

In this paper, we consider a Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient We show that this problem is ill‐posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under a priori and a posteriori parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd.     

Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates

We propose a class ofa posteriori parameter choice strategies for Tikhonov regularization (including variants of Morozov's and Arcangeli's methods) that lead to optimal convergence rates toward the minimal-norm, least-squares solution of an ill-posed linear operator equation in the presence of noisy data.     

A Posteriori Error Estimators of Gradient Recovery Type for Elliptic Obstacle Problems

In this paper, we present a posteriori error estimates of gradient recovery type for elliptic obstacle problems. The a posteriori error estimates provide both lower and upper error bounds. It is shown to be equivalent to the discretization error in an energy type norm for general meshes. Furthermore, when the solution is smooth and the mesh is uniform, it is shown to be asymptotically exact. Some numerical results which demonstrate the theoretical results are also reported in this paper.     

Use of dynamic trees in a network simplex algorithm for the maximum flow problem

Goldfarb and Hao (1990) have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on ann-vertex,m-arc network in at mostnm pivots and O(n ² m) time. In this paper we describe how to extend the dynamic tree data structure of Sleator and Tarjan (1983, 1985) to reduce the running time of this algorithm to O(nm logn). This bound is less than a logarithmic factor larger than those of the fastest known algorithms for the problem. Our extension of dynamic trees is interesting in its own right and may well have additional applications.Research partially supported by a Presidential Young Investigator Award from the National Science Foundation, Grant No. CCR-8858097, an IBM Faculty Development Award, and AT&T Bell Laboratories.Research partially supported by the Office of Naval Research, Contract No. N00014-87-K-0467.Research partially supported by the National Science Foundation, Grant No. DCR-8605961, and the Office of Naval Research, Contract No. N00014-87-K-0467.     

The a posteriori Fourier method for solving the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data

In the present paper, the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data in an infinite “strip” domain is considered. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A conditional stability result is given. A new a posteriori Fourier method for solving this problem is proposed. The corresponding error estimate between the exact solution and its regularization approximate solution is also proved. Numerical examples show the effectiveness of the method and the comparison of numerical effect between the a posteriori and the a priori Fourier method are also taken into account.     

Multi-parameter regularization techniques for ill-conditioned linear systems

Summary.  When a system of linear equations is ill-conditioned, regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties: the ill-conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen. In this paper, we consider the case of several regularizations terms added simultaneously, thus overcoming the problem of the best choice of the regularization matrix. The error of this procedure is analyzed and numerical results prove its efficiency. Received January 15, 2002 / Revised version received July 31, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65F05 – 65F22     

Adaptive operator splitting finite element method for Allen–Cahn equation

In this paper, a new numerical method is proposed and analyzed for the Allen–Cahn (AC) equation. We divide the AC equation into linear section and nonlinear section based on the idea of operator splitting. For the linear part, it is discretized by using the Crank–Nicolson scheme and solved by finite element method. The nonlinear part is solved accurately. In addition, a posteriori error estimator of AC equation is constructed in adaptive computation based on superconvergent cluster recovery. According to the proposed a posteriori error estimator, we design an adaptive algorithm for the AC equation. Numerical examples are also presented to illustrate the effectiveness of our adaptive procedure.     

On the first homology of automorphism groups of manifolds with geometric structures

Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category. This research was partially supported by Grant-in-Aid for Scientific Research (No. 16540058), Japan Society for the Promotion of Science. This research was partially supported by Grant-in-Aid for Scientific Research (No. 14540093), Japan Society for the Promotion of Science.