Iterative Solution Methods for Large Linear Discrete Ill-Posed Problems

Geophysical Inverse Theory and Regularization Problems

This paper proposes a new iterative integration regularization method for robust solution of ill-posed inverse problems. The proposed method is motivated from the fact that inversion of a positive definite matrix can be expressed in an integral form. Then, the development of the proposed method is mainly twofold. Firstly, two ways—including the linear iteration and the exponential ($$2^j$$) iteration—are invoked to compute the integral, of which the exponential iteration is often preferred due to its fast convergence. Secondly, after stability analysis, the proposed method is shown able to filter out the undesired effect of relatively small singular values, while preserving the desired terms of relatively large singular values, i.e., the proposed method has the guaranteed regularization effect. Numerical examples on three typical ill-posed problems are conducted with detailed comparison to some usual direct and iterative regularization methods. Final results have highlighted the proposed method: (a) due to the iterative nature, the proposed method often turns out to be more efficient than the conventional direct regularization methods including the Tikhonov regularization and the truncated singular value decomposition (TSVD), (b) the proposed method converges much faster than the Landweber method and (c) the regularization effect is guaranteed in the proposed method, while may not be in the conjugate gradient method for least squares problem (CGLS).

Invertible smoothing preconditioners for linear discrete ill-posed problems

In multiple linear regressions, it is known that least-squares estimates of the parameters are likely to be too large in absolute value and possibly of wrong sign, if explanatory variables are correlated. To reduce the undesirable effects of collinearity, the ridge estimator has been proposed as an alternative method to the least squares estimator. The biggest debate with the ridge estimator is the selection of the regularization parameter. Several methods for the selection of the regularization parameter have been discussed. Although most of these methods are based on minimizing the mean square error of the ridge estimator, bootstrap resampling method is also proposed to provide an optimal regularization parameter which gives an estimate of mean square error of prediction. The purpose of this paper is to provide a comprehensive study of regularization parameter selection methods which consider bootstrap approach as well as estimation and prediction approaches. The paper concludes with application of these procedures to simulation studies and real data.

The purpose of this paper is to bring a series of rather surprising results which have appeared in the technical literature to the attention of a wider audience. We describe the results informally here; precise definitions and results will be given below. Our subject is the solution of an ill-posed problem on a digital computer. Such a problem can at best only be solved approximately. We say a problem is solvable if we can compute an "-approximation for any positive " and is unsolvable if we cannot compute an "-approximation, even for arbitrarily large ". It was shown in [17] that a linear problem is unsolvable iff it is ill-posed. Are there circumstances under which one can avoid this negative conclusion? It is common to associate an operator S with an ill-posed problem. If S is linear, then a problem is ill-posed iff S is unbounded. Note that unboundedness is a worst case concept; the supremum of S operating on elements in the unit ball of its domain is infinite. This suggests introducing the concept of a problem’s being ill-posed on the average if the expectation of S with respect to a measure is infinite and being well-posed on the average if this expectation is finite. These concepts were introduced in [17], where it was shown that if the measure is Gaussian and the linear operator S is measurable, then a linear problem is unsolvable on the average iff it is ill-posed on the average. A natural next step is to find a linear ill-posed problem which is also ill-posed on the average. Several attempts to do this for Gaussian measures were unsuccessful. This suggested the question: Is every linear problem well-posed on the average for any Gaussian measure? This question was answered in the affirmative in [5] and [15]. The conclusion in the title follows immediately from this and the preceding result. What if the measure is non-Gaussian? Now the situation is more complicated. In particular, there are problems that are even ill-posed on the average. In this paper, we round out the picture by presenting examples of such problems for non-Gaussian measures.

This paper is concerned with the solution of ill-conditioned Systems of Linear Equations (SLE's) via the solution of equivalent SLE's which are well-conditioned. A matrix is first constructed from that of the given ill-conditioned system. Then, an adequate right-hand side is computed to make up the instance of an equivalent system. Formulae and algorithms for computing an instance of this equivalent SLE and solving it will be given and illustrated. © 2011 The Korean Mathematical Society.

This paper is a matrix iterative method presented to compute the solutions of the matrix equation, AXB=C, with unknown matrix X∈𝒮, where 𝒮 is the constrained matrices set like symmetric, symmetric-R-symmetric and (R, S)-symmetric. By this iterative method, for any initial matrix X 0∈𝒮, a solution X* can be obtained within finite iteration steps if exact arithmetics were used, and the solution X* with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution , which is nearest to a given matrix ˜X in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in [G.-X. Huang, F. Yin, and K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C , J. Comput. Appl. Math. 212 (2008), pp. 231–244; Y. Lei and A.-P. Liao, A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. Math. Comput. 188 (2007), pp. 499–513; Z.-Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C , Appl. Math. Comput. 170 (2005), pp. 711–723; Y.-X. Peng, X.-Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C , Appl. Math. Comput. 160 (2005), pp. 763–777].

An extrapolated TSVD method for linear discrete ill-posed problems with Kronecker structure

In engineering practice, the problem arises of reconstracting the stress–strain state of a structural element using the data measured on the accessible part of the it's boundary. For solving this problem in the framework of the elastic model, we have the Cauchy problem for the Lame equation. The Cauchy problem for an elliptic Lame equation is an ill-posed problem. An instability is the main difficulty for such problems. There exist numerous methods for numerical solution of such problems but few methods are known for analytical one. In this paper we present a new analytical method for the Cauchy problem for Lame equation of the elasticity theory in a rectangle wherein a missing boundary condition is recovered from a full measurement of the stresses and displacements on an accessible boundary. As a result, for the displacement components, we obtain solutions in the form of a sum of series with three regularization parameters.

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

In this paper, first, we use the Rich type iterative method for numerical solution of nonlinear fuzzy Fredholm integral equation. The main structure of this method is based on the Richardson iterative method and trapezoidal quadrature formula for Numerical solution of nonlinear fuzzy integral equations. Convergence, error analysis and numerical stability of the proposed method are also discussed.

Cascadic multilevel methods for the solution of linear discrete ill-posed problems with noise-reducing restriction and prolongation operators recently have been developed for the restoration of blur- and noise-contaminated images. This is a particular ill-posed problem. The multilevel methods were found to determine accurate restorations with fairly little computational work. This paper describes noise-reducing multilevel methods for the solution of general linear discrete ill-posed problems.

In literature, it has been reported that the convergence of some preconditioned stationary iterative methods using certain type upper triangular matrices as preconditioners are faster than the basic iterative methods. In this paper, a new preconditioned iterative method for the numerical solution of linear systems has been introduced, and the convergence analysis of the proposed method and an existing one have been done. Some numerical examples have also been given, which show the effectiveness of both of the methods.

An inverse coefficient problem for a mathematical model of desorption dynamics is considered. The inverse problem is reduced to a nonlinear operator equation for an unknown coefficient. The operator equation is used to construct an iterative numerical method for solving the inverse problem. To prove the convergence of the iterative method, the contraction mappings principle is used. Examples of the application of the iterative method for the numerical solution of the inverse problem are given.

A Highly Accurate Technique for Interpolations Using Very High-Order Polynomials, and Its Applications to Some Ill-Posed Linear Problems