Lothar Reichel
Kent State University
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Featured researches published by Lothar Reichel.
Journal of Computational and Applied Mathematics | 2000
Daniela Calvetti; Serena Morigi; Lothar Reichel; Fiorella Sgallari
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.
Linear Algebra and its Applications | 1992
D.Y. Hu; Lothar Reichel
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.
SIAM Journal on Scientific Computing | 1998
James Baglama; Daniela Calvetti; Gene H. Golub; Lothar Reichel
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.
Linear Algebra and its Applications | 1992
Lothar Reichel; Lloyd N. Trefethen
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.
SIAM Journal on Matrix Analysis and Applications | 1996
Daniela Calvetti; Lothar Reichel
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
SIAM Journal on Matrix Analysis and Applications | 1992
Noël M. Nachtigal; Lothar Reichel; Lloyd N. Trefethen
\hat{F}
Bit Numerical Mathematics | 1999
Daniela Calvetti; Gene H. Golub; Lothar Reichel
representing the restored image. The matrices
SIAM Journal on Scientific Computing | 2005
James Baglama; Lothar Reichel
T_1
Bit Numerical Mathematics | 2003
Daniela Calvetti; Lothar Reichel
and
Bit Numerical Mathematics | 1990
Lothar Reichel
T_2