Lothar Reichel

Tikhonov regularization and the L-curve for large discrete ill-posed problems

Discretization of linear inverse problems generally gives rise to very ill-conditioned linear systems of algebraic equations. Typically, the linear systems obtained have to be regularized to make the computation of a meaningful approximate solution possible. Tikhonov regularization is one of the most popular regularization methods. A regularization parameter specifies the amount of regularization and, in general, an appropriate value of this parameter is not known a priori. We review available iterative methods, and present new ones, for the determination of a suitable value of the regularization parameter by the L-curve criterion and the solution of regularized systems of algebraic equations.

Krylov-subspace methods for the Sylvester equation

Abstract We describe Galerkin and minimal residual algorithms for the solution of Sylvesters equation AX – XB = C . The algorithms use Krylov subspaces forwhich orthogonal bases are generated by the Arnoldi process. For certain choices of Krylov subspaces the computation of the solution splits into the solution of many independent subproblems. This makes the algorithms suitable for implementation on parallel computers.

Adaptively Preconditioned GMRES Algorithms

The restarted GMRES algorithm proposed by Saad and Schultz [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869] is one of the most popular iterative methods for the solution of large linear systems of equations Ax=b with a nonsymmetric and sparse matrix. This algorithm is particularly attractive when a good preconditioner is available. The present paper describes two new methods for determining preconditioners from spectral information gathered by the Arnoldi process during iterations by the restarted GMRES algorithm. These methods seek to determine an invariant subspace of the matrix A associated with eigenvalues close to the origin and to move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved.

Eigenvalues and Pseudo-eigenvalues of Toeplitz Matrices

The eigenvalues of a nonhermitian Toeplitz matrix A are usually highly sensitive to perturbations, having condition numbers that increase exponentially with the dimension N. An equivalent statement is that the resolvent ( ZZ - A)- ’ of a Toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest-exponentially large as a function of N, even when z is far from the spectrum. Because of these facts, the meaningfulness of the eigenvalues of nonhermitian Toeplitz matrices for any but the most theoretical purposes should be considered suspect. In many applications it is more meaningful to investigate the e-pseudo-eigenvalues: the complex numbers z with ll(zZ - A)-‘11 > &-l. This paper analyzes the pseudospectra of Toeplitz matrices, and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated Toeplitz and Laurent operators. Our results are reasonably complete in the triangular case, and preliminary in the cases of nontriangular Toeplitz matrices, block Toeplitz matrices, and Toeplitz-like matrices with smoothly varying coefficients. Computed examples of pseudospectra are presented throughout, and applications in numerical analysis are mentioned.

Application of ADI Iterative Methods to the Restoration of Noisy Images

The restoration of two-dimensional images in the presence of noise by Wieners minimum mean square error filter requires the solution of large linear systems of equations. When the noise is white and Gaussian, and under suitable assumptions on the image, these equations can be written as a Sylvesters equation \[ T_1^{-1}\hat{F}+\hat{F}T_2=C \] for the matrix

A Hybrid GMRES algorithm for nonsymmetric linear systems

\hat{F}

Estimation of the L-Curve via Lanczos Bidiagonalization

representing the restored image. The matrices

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

T_1

TIKHONOV REGULARIZATION OF LARGE LINEAR PROBLEMS

and

Newton interpolation at Leja points

T_2