We investigate iterated Tikhonov methods coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. In the case of noisy data we propose a modification, the so called loping iterated Tikhonov-Kaczmarz method, where a sequence of relaxation parameters is introduced and a different stopping rule is used. Convergence analysis for this method is also provided.
In this article we propose a novel nonstationary iterated Tikhonov (NIT) type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: I) A 2D elliptic parameter identification problem (Inverse Potential Problem); II) An i…
We investigate Tikhonov regularization methods for nonlinear ill-posed problems in Banach spaces, where the penalty term is described by Bregman distances. We prove convergence and stability results. Moreover, using appropriate source conditions, we are able to derive rates of convergence in terms of Bregman distances. We also analyze an iterated Tikhonov method for nonlinear problems, where the penalization is given by an appropriate convex functional.
This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641--658]. A t-product Arnoldi (t-Arnoldi) process is defined and applied to reduce a large-scale Tikhonov regularization problem for third order tensors to a problem of small size. The data may be represented by a laterally oriented matrix or a third order tensor, and the regularization operator is a third order tensor. The dis…
Error Estimates for Arnoldo-Tikhonov Regularization for Ill-Posed Operator Equations
Institute for Industrial Mathematics, Kepler University（工业数学研究所，开普勒大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（约翰·拉德纳计算与应用数学研究所，奥地利科学院）；Department of Mathematical Sciences, Kent State University（凯特州立大学数学科学系）
We consider a modified Tikhonov-type functional for the solution of ill-posed nonlinear inverse problems. Motivated by applications in the field of production engineering, we allow small deviations in the solution, which are modeled through a tolerance measure in the regularization term of the functional. The existence, stability, and weak convergence of minimizers are proved for such a functional, as well as the convergence rates in the Bregman distance. We present an example for illustrating the effect of tolerances on the regularized solution and examine parameter choice rules for finding t…
For the Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where Hölder-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence of regularized solutions without source conditions are completely missing. A result of the present work is to …
Inverse initial data reconstruction for a memory convection-diffusion equation via Legendre spatial reduction and Tikhonov regularization
University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；International University - Vietnam National University（越南国立大学国际大学）；Texas A&M University（德克萨斯农工大学）
Many problems in Science and Engineering give rise to linear integral equations of the first kind with a smooth kernel. Discretization of the integral operator yields a matrix, whose singular values cluster at the origin. We describe the approximation of such matrices by adaptive cross approximation, which avoids forming the entire matrix. The choice of the number of steps of adaptive cross approximation is discussed. The discretized right-hand side represents data that commonly are contaminated by measurement error. Solution of the linear system of equations so obtained is not meaningful beca…
A global minimization algorithm for Tikhonov functionals with sparsity constraints
Jiaxing University（嘉兴学院）；Harbin Institute of Technology（哈尔滨工业大学）；Technische Universität Chemnitz（开姆尼茨工业大学）；Johannes Kepler University（约翰·开普勒大学）；Industrial Mathematics Institute（工业数学研究所）；Johann Radon Institute for Computational and Appied Mathematics（约翰·拉登计算与应用数学研究所）
Regularization matrices for discrete ill-posed problems in several space-dimensions
University School（大学学校）；Geomathematics Key Laboratory of Sichuan, College of Management Science, Chengdu University of Technology（四川地球数学重点实验室，成都理工大学管理学院）；Department of Mathematics, SAPIENZA Università di Roma（罗马智慧大学数学系）；Department of Mathematical Sciences, Kent State University（肯特州立大学数理科学系）
In this short note, we formulate the convergence rates of the well known Tikhonov regularization scheme for solving the nonlinear ill-posed problems in Banach spaces. For deriving the convergence rates, we employ the novel smoothness concept of conditional stability estimates in terms of weaker norms. Moreover, we show that our results are applicable on two ill-posed inverse problems.
These lecture notes for a graduate class present the regularization theory for linear and nonlinear ill-posed operator equations in Hilbert spaces. Covered are the general framework of regularization methods and their analysis via spectral filters as well as the concrete examples of Tikhonov regularization, Landweber iteration, regularization by discretization for linear inverse problems. In the nonlinear setting, Tikhonov regularization and iterative regularization (Landweber, Levenberg-Marquardt, and iteratively regularized Gauß-Newton methods) are discussed. The necessary background from fu…
This paper is concerned with solving ill-posed tensor linear equations. These kinds of equations may appear from finite difference discretization of high-dimensional convection-diffusion problems or when partial differential equations in many dimensions are discretized by collocation spectral methods. Here, we propose the Tensor Golub--Kahan bidiagonalization (TGKB) algorithm in conjunction with the well known Tikhonov regularization method to solve the mentioned problems. Theoretical results are presented to discuss on conditioning of the Stein tensor equation and to reveal that how the TGKB …
Consider a Urysohn integral equation $x - \mathcal{K} (x) = f$, where $f$ and the integral operator $\mathcal{K}$ with kernel of the type of Green's function are given. In the computation of approximate solutions of the given integral equation by Galerkin method, all the integrals are needed to be evaluated by some numerical integration formula. This gives rise to the discrete version of the Galerkin method. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1$ with respect to a uniform partition is chosen to be the approximating space. For the appropriate choice of a numerical…
