Solution Of Ill-conditioned Linear Systems Research Articles

Accelerating the solution of linear systems appearing in two-phase reservoir simulation by the use of POD-based deflation methods

We explore and develop a Proper Orthogonal Decomposition (POD)-based deflation method for the solution of ill-conditioned linear systems, appearing in simulations of two-phase flow through highly heterogeneous porous media. We accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method achieving speed-ups of factors up to five. The up-front extra computational cost of the proposed method depends on the number of deflation vectors. The POD-based deflation method is tested for a particular problem and linear solver; nevertheless, it can be applied to various transient problems, and combined with multiple solvers, e.g., Krylov subspace and multigrid methods.

An extension of the reciprocity functional approach to the estimate of two-dimensional spatially dependent thermal contact conductances using regularization via TSVD: numerical and experimental results

A method based on the reciprocity functional is presented to estimate a two-dimensional spatially dependent thermal contact conductance between two materials without the use of intrusive measurements. In order to perform the estimation, two auxiliary problems are needed. The first one determines the temperature jump and the second one the heat flux, both at the inaccessible interface between the two materials. Besides the solution of the auxiliary problems, the application of this method requires the solution of ill-conditioned linear systems. This paper is an extension of our previously presented technique to two-dimensional interfaces. This paper also presents some new treatment for the numerical solution of the auxiliary problems. The TSVD method is used to solve the linear systems and a methodology to find the best truncation parameter is proposed. In addition to a test case with computationally simulated temperature measurements, an experimental case is also evaluated with good results.

On the iterative refinement of the solution of ill-conditioned linear system of equations

ABSTRACTRecently, Salkuyeh and Fahim [A new iterative refinement of the solution of ill-conditioned linear system of equations, Int. Comput. Math. 88(5) (2011), pp. 950–956] have proposed a two-step iterative refinement of the solution of an ill-conditioned linear system of equations. In this paper, we first present a generalized two-step iterative refinement procedure to solve ill-conditioned linear system of equations and study its convergence properties. Afterward, it is shown that the idea of an orthogonal projection technique together with a basic stationary iterative method can be utilized to construct a new efficient and neat hybrid algorithm for solving the mentioned problem. The convergence of the offered hybrid approach is also established. Numerical examples are examined to demonstrate the feasibility of proposed algorithms and their superiority to some of existing approaches for solving ill-conditioned linear system of equations.

A new iterative refinement of the solution of ill-conditioned linear system of equations

In this paper, a new iterative refinement of the solution of an ill-conditioned linear system of equations are given. The convergence properties of the method are studied. Some numerical experiments of the method are given and compared with that of two of the available methods.

A note on the pin-pointing solution of ill-conditioned linear system of equations

Volokh and Vilnay [Appl. Math. Lett. 13 (2000), pp. 119–124] proposed a method for computing an accurate solution of nearly singular linear systems of equations. The method is processed in two stages. Their method uses the truncated singular value decomposition of the initial coefficient matrix at the first stage and the Gaussian elimination procedure for well-conditioned reduced system of linear equations at the second stage. In this note, we propose a multistage approach based on the Volokh and Vilnay's method. Some numerical results are given to show the efficiency of the proposed method.

Error estimates for the regularization of least squares problems

The a posteriori estimate of the errors in the numerical solution of ill-conditioned linear systems with contaminated data is a complicated problem. Several estimates of the norm of the error have been recently introduced and analyzed, under the assumption that the matrix is square and nonsingular. In this paper we study the same problem in the case of a rectangular and, in general, rank-deficient matrix. As a result, a class of error estimates previously introduced by the authors (Brezinski et al., Numer Algorithms, in press, 2008) are extended to the least squares solution of consistent and inconsistent linear systems. Their application to various direct and iterative regularization methods are also discussed, and the numerical effectiveness of these error estimates is pointed out by the results of an extensive experimentation.

On the Numerical Solution of Ill-Conditioned Linear Systems with Applications to Ill-Posed Problems

We consider the solution of ill-conditioned linear systems using the singular value decomposition, and show how this can improve the accuracy of the computed solution for certain kinds of right-hand sides. Then we indicate how this technique is especially appropriate for some classical ill-posed problems of mathematical physics.