Constantin Popa

Least-squares solution of overdetermined inconsistent linear systems using kaczmarz's relaxation

For numerical computation of the minimal Euclidean norm (least-squares) solution of overdetermined linear systems, usually direct solvers are used (like QR decomposition, see [4]). The iterative methods for such kind of problems need special assumptions about the system (consistency, full rank of the system matrix, some parameters they use or they give not the minimal length solution, [2,3,5,8,10,13]). In the present paper we purpose two iterative algorithms which generate sequences convergent to the minimal Euclidean length solution in the general case (inconsistent system and rank deficient matrix). The algorithms use only some combinations and properties of the well-known Kaczmarz iterative method ([13]) and need no special assumptions about the system.

Characterization of the solutions set of inconsistent least-squares problems by an extended Kaczmarz algorithm

We give a new characterization of the solutions set of the general (inconsistent) linear least-squares problem using the set of limit-points of an extended version of the classical Kaczmarz’s projections method. We also obtain a “ step error reduction formula” which, in some cases, can give us apriori information about the convergence properties of the algorithm. Some numerical experiments with our algorithm and comparisons between it and others existent in the literature, are made in the last section of the paper.

A method for improving orthogonality of rows and columns of matrices

In [5] Kovarik described a method for approximate orthogonalization of a finite set of linearly independent vectors from an arbitrary Hubert space. In this paper we generalize this method to the case when the vectors are rows (not necessary linearly independent) of an arbitrary rectangular real matrix. In this case we prove that, after the application of Kovariks algorithm, both rows and columns of the matrix are transformed in vectors which are “quasi-orthogonal”, in a sense that is clearly described. Some numerical experiments, on a matrix obtained from the discretization of a first kind integral equation are presented in the last section of the paper.

A hybrid Kaczmarz-Conjugate Gradient algorithm for image reconstruction

The present paper is a theoretical contribution to the field of iterative methods for solving inconsistent linear least squares problems arising in image reconstruction from projections in computerized tomography. It consists on a hybrid algorithm which includes in each iteration a CG-like step for modifying the right-hand side and a Kaczmarz-like step for producing the approximate solution. We prove convergence of the hybrid algorithm for general inconsistent and rank-deficient least-squares problems. Although the new algorithm has potential for more applied experiments and comparisons, we restrict them in this paper to a regularized image reconstruction problem involving a 2D medical data set.

A fast Kaczmarz-Kovarik algorithm for consistent least-squares problems

In some previous papers the author extended two algorithms proposed by Z. Kovarik for approximate orthogonalization of a finite set of linearly independent vectors from a Hilbert space, to the case when the vectors are rows (not necessary linearly independent) of an arbitrary rectangular matrix. In this paper we describe combinations between these two methods and the classical Kaczmarz’s iteration. We prove that, in the case of a consistent least-squares problem, the new algorithms so obtained converge to any of its solutions (depending on the initial approximation). The numerical experiments described in the last section of the paper on a problem obtained after the discretization of a first kind integral equation ilustrate the fast convergence of the new algorithms.

A general extending and constraining procedure for linear iterative methods

Algebraic reconstruction techniques (ARTs), on both their successive and simultaneous formulations, have been developed since the early 1970s as efficient ‘row-action methods’ for solving the image-reconstruction problem in computerized tomography. In this respect, two important development directions were concerned with, first, their extension to the inconsistent case of the reconstruction problem and, second, their combination with constraining strategies, imposed by the particularities of the reconstructed image. In the first part of this paper, we introduce extending and constraining procedures for a general iterative method of an ART type and we propose a set of sufficient assumptions that ensure the convergence of the corresponding algorithms. As an application of this approach, we prove that Cimminos simultaneous reflection method satisfies this set of assumptions, and we derive extended and constrained versions for it. Numerical experiments with all these versions are presented on a head phantom widely used in the image reconstruction literature. We also consider hard thresholding constraining used in sparse approximation problems and apply it successfully to a 3D particle image-reconstruction problem.

Extension of an approximate orthogonalization algorithm to arbitrary rectangular matrices

Abstract Z. Kovarik described in [SIAM J. Numer. Anal. 7 (3) (1970) 386] a method for approximate orthogonalization of a finite set of linearly independent vectors from an arbitrary (real or complex) Hilbert space. In this paper, we generalize Kovariks method in the case when the vectors are rows (not necessary linearly independent) of an arbitrary rectangular real matrix. In this case we prove that, both rows and columns of the matrix are transformed in vectors which are “quasi-orthogonal”, in a sense that is clearly described. Numerical experiments are presented in the last section of the paper.

Preconditioning conjugate gradient method for nonsymmetric systems

It is well known that the preconditioned conjugate gradient algorithms (PCG) work very well (for both symmetric and nonsymmetric problems) if the preconditioned is “good enough”. But, in many cases, “good enough” means that for solving (during the application of (PCG)) the systems in which the preconditioning matrix appears too much computational work is necessary. In this paper we present, for systems arising from finite differences or finite element discretizations, both a method of preconditioning and some possibilities to approximate the exact solutions of the systems in which the preconditioner appears. We present a theoretical study of the influence of these approxiamtions on the convergence behaviour of the (PCG) algorithm. In the last section we present some test on the one-dimensional steady state heat convective transfer problem.

Mesh independence of the condition number of discrete galerkin systems by preconditioning

In the paper [1] the author proves that preconditioning the discrete Galerkin system by the Choleski factors of the Gramian we obtain a mesh independent condition number. In the present paper we generalize this method by using the Choleski factors of a term from a special splitting of the Gramian. Thus, we obtain the method from [1] like a particular case. We also observe that such a kind of splitting can be obtained by an incomplete Choleski decomposition of the Gramian.

Regularized solution of LCP problems with application to rigid body dynamics

For Linear Complementarity Problems (LCP) with a positive semidefinite matrix M, iterative solvers can be derived by a process of regularization. In [3] the initial LCP is replaced by a sequence of positive definite ones, with the matrices M + αI. Here we analyse a generalization of this method where the identity I is replaced by a positive definite diagonal matrix D. We prove that the sequence of approximations so defined converges to the minimal D-norm solution of the initial LCP. This extension opens the possibility for interesting applications in the field of rigid multibody dynamics.