Solution Of Discrete Ill-posed Problems Research Articles

Approximate Schur-Block ILU Preconditioners for Regularized Solution of Discrete Ill-Posed Problems

High order iterative methods with a recurrence formula for approximate matrix inversion are proposed such that the matrix multiplications and additions in the calculation of matrix polynomials for the hyperpower methods of orders of convergence p=4k+3, where k≥1 is integer, are reduced through factorizations and nested loops in which the iterations are defined using a recurrence formula. Therefore, the computational cost is lowered from κ=4k+3 to κ=k+4 matrix multiplications per step. An algorithm is proposed to obtain regularized solution of ill-posed discrete problems with noisy data by constructing approximate Schur-Block Incomplete LU (Schur-BILU) preconditioner and by preconditioning the one step stationary iterative method. From the proposed methods of approximate matrix inversion, the methods of orders p=7,11,15,19 are applied for approximating the Schur complement matrices. This algorithm is applied to solve two problems of Fredholm integral equation of first kind. The first example is the harmonic continuation problem and the second example is Phillip’s problem. Furthermore, experimental study on some nonsymmetric linear systems of coefficient matrices with strong indefinite symmetric components from Harwell-Boeing collection is also given. Numerical analysis for the regularized solutions of the considered problems is given and numerical comparisons with methods from the literature are provided through tables and figures.

Numerical aspects of the nonstationary modified linearized Bregman algorithm

The solution of discrete ill-posed problems has been a subject of research for many years. Among the many methods described in the literature, the Bregman algorithm has attracted a great deal attention and been widely investigated. Recently, a nonstationary preconditioned version of this algorithm, referred to as the nonstationary modified linearized Bregman algorithm, was proposed. The aim of this paper is to discuss numerical aspects of this algorithm and to compare computed results with known theoretical properties. We also discuss the effect of several parameters required by the algorithm on the computed solution.

Schultz matrix iteration based method for stable solution of discrete ill-posed problems

In this paper, we propose an iterative method for solving discrete ill-posed problems based on matrix iterations generated by Schultz method and known to converge to the Moore–Penrose pseudoinverse of a matrix. Practically, letting the Schultz matrix iteration be Xk, we construct the vector xk=Xkb where b is a data vector. Hence, by construction, the iterates converge to the minimum 2-norm solution of a least squares problem with coefficient matrix A and data vector b. We derive theoretical properties of the sequence xk and show that it is quadratically convergent. In the case of corrupted data, we analyze the semi-convergence behavior of the iterates and conclude that the iteration must be truncated to control the propagation of the noise error. As a result, we derive an error estimate for the case where the truncation parameter is chosen by the discrepancy principle. In addition, combining a projected approach with the new method, we propose variants of the method that are well suited for large-scale problems. Several numerical results are presented to illustrate the effectiveness of the method on well known test problems.

Підвищення точності розв’язку дискретних некоректних задач методом випадкових проекцій

Підвищення точності розв’язку дискретних некоректних задач методом випадкових проекцій

Conjugate Gradient Method for the Solution of Inverse Problems: Application in Linear Seismic Tomography

We consider the conjugate gradient method for the normal equations in the solution of discrete ill-posed problems arising from seismic tomography. We use a linear approach of traveltime tomography that is characterized by an ill-conditioned linear system whose unknowns are the slownesses in each block of the computational domain. The algorithms considered in this work regularize the linear system by stopping the conjugate gradient method in an early iteration. They do not depend on the singular-value decomposition and represent an attractive and economic alternative for large-scale problems. We review two recently proposed stopping criteria and propose a modified stopping criterion that takes into account the oscillations in the approximate solution.

A randomized method for solving discrete ill-posed problems

An approach is proposed to the stable solution of discrete ill-posed problems on the basis of a combination of random projection of the initial ill-conditioned matrix with an ill-defined numerical rank and the pseudo-inversion of the resultant matrix. To select the dimension of the projection matrix, we propose to use criteria for the selection of a model and a regularization parameter. The results of experimental studies based on the well-known examples of discrete ill-posed problems are presented. Their solution errors are close to the Tikhonov regularization error, but a matrix dimension reduction owing to projection reduces the expenditures for computations, especially at high noise levels.

Tikhonov–Phillips regularization with operator dependent seminorms

We focus on the solution of discrete ill-posed problems to recover the original information from blurred signals in the presence of Gaussian white noise more accurately. We derive seminorms for the Tikhonov–Phillips regularization based on the underlying blur operator H. In this way it is possible to improve the reconstruction using spectral information of H. Reconstructions on various 1D discrete ill-posed inverse problems demonstrate the effect of the presented approach.

The adaptive augmented GMRES method for solving ill-posed problems

The GMRES method is an iterative method that provides better solutions when dealing with large linear systems of equations with a non-symmetric coefficient matrix. The GMRES method generates a Krylov subspace for the solution, and the augmented GMRES method allows augmentation of the Krylov subspaces by a user supplied subspace which represents certain known features of the desired solution. The augmented GMRES method performs well with suitable augmentation, but performs poorly with unsuitable augmentation. The adaptive augmented GMRES method automatically selects a suitable subspace from a set of candidates supplied by the user. This study shows that this method maintains the performance level of augmented GMRES and lightens the burden it puts on its users. Numerical experiments compare robustness as well as the efficiency of various heuristic strategies. References M. L. Baart. The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned linear least-squares problems. IMA Journal of Numerical Analysis, 2:241--247, 1982. doi:10.1093/imanum/2.2.241 J. Baglama and L. Reichel. Augmented GMRES-type method. Numerical Linear Algebra with Applications, 14:337--350, 2007. doi:10.1002/nla.518 D. Calvetti, B. Lewis, and L. Reichel. GMRES-type methods for inconsistent systems. Linear Algebra and its Applications, 316:157--169, 2000. doi:10.1016/S0024-3795(00)00064-1 D. Calvetti, B. Lewis, and L. Reichel. On the choice of subspace for iterative methods for linear discrete ill-posed problems. Int. J. Appl. Math. Comput. Sci, 11:1069--1092, 2001. http://www.math.kent.edu/ reichel/publications/subspaceselect.pdf G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press, 2nd edition, 1989. P. C. Hansen. Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems. Numerical Algorithms, 6:1--35, 1994. doi:10.1007/BF02149761 P. C. Hansen. Rank Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia, 1998. Y. Saad. and M. H. Schultz. GMRES: A generalized minimal residual method for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856--869, 1986. http://www.stanford.edu/class/cme324/saad-schultz.pdf

Regularization Tools version 4.0 for Matlab 7.3

This communication describes version 4.0 of Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems. The new version allows for under-determined problems, and it is expanded with several new iterative methods, as well as new test problems and new parameter-choice methods.

Regularization Tools Version 3.0 for Matlab 5.2

This communication describes Version 3.0 of Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems.

REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems

The package REGULARIZATION TOOLS consists of 54 Matlab routines for analysis and solution of discrete ill-posed problems, i.e., systems of linear equations whose coefficient matrix has the properties that its condition number is very large, and its singular values decay gradually to zero. Such problems typically arise in connection with discretization of Fredholm integral equations of the first kind, and similar ill-posed problems. Some form of regularization is always required in order to compute a stabilized solution to discrete ill-posed problems. The purpose of REGULARIZATION TOOLS is to provide the user with easy-to-use routines, based on numerical robust and efficient algorithms, for doing experiments with regularization of discrete ill-posed problems. By means of this package, the user can experiment with different regularization strategies, compare them, and draw conclusions from these experiments that would otherwise require a major programming effert. For discrete ill-posed problems, which are indeed difficult to treat numerically, such an approach is certainly superior to a single black-box routine. This paper describes the underlying theory gives an overview of the package; a complete manual is also available.