Silvia Noschese

Tridiagonal Toeplitz matrices: properties and novel applications‡

SUMMARY The eigenvalues and eigenvectors of tridiagonal Toeplitz matrices are known in closed form. This property is in the first part of the paper used to investigate the sensitivity of the spectrum. Explicit expressions for the structured distance to the closest normal matrix, the departure from normality, and the ϵ-pseudospectrum are derived. The second part of the paper discusses applications of the theory to inverse eigenvalue problems, the construction of Chebyshev polynomial-based Krylov subspace bases, and Tikhonov regularization. Copyright

Differentiation of Multivariable Composite Functions and Bell Polynomials

We generalize the Bell polynomials in order to derive an operational tool for the differentiation of composite functions in several variables. In particular we show a formula that relates the Bell polynomials for multivariable composite functions to the classical ones. Some applications are suggested.

Inverse problems for regularization matrices

Discrete ill-posed problems are difficult to solve, because their solution is very sensitive to errors in the data and to round-off errors introduced during the solution process. Tikhonov regularization replaces the given discrete ill-posed problem by a nearby penalized least-squares problem whose solution is less sensitive to perturbations. The penalization term is defined by a regularization matrix, whose choice may affect the quality of the computed solution significantly. We describe several inverse matrix problems whose solution yields regularization matrices adapted to the desired solution. Numerical examples illustrate the performance of the regularization matrices determined.

Computing the Structured Pseudospectrum of a Toeplitz Matrix and Its Extreme Points

The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low-rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [N. Guglielmi and M. Overton, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166--1192] for the unstructured case, but their extension to structured pseudospectra and analysis presents several difficulties. Natural generalizations of the algorithms, allowing us to draw significant sections of the structured pseudospectra in proximity of extremal points, are also discussed. Since no algorithms are available in the literature to draw such structured pseudospectra, the approach we present seems promising to extend existing software tools (Eigtool, Seigtool) to structured pseudospectra representation for Toeplitz matrices. We discuss local convergence properties of the algorithms and show some applications to a few illustrative...

Rescaling the GSVD with application to ill-posed problems

The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by XT, is the same in both products. Available software for computing the GSVD scales the diagonal matrices and XT so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of XT. The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution.

Regularization matrices determined by matrix nearness problems

Abstract This paper is concerned with the solution of large-scale linear discrete ill-posed problems with error-contaminated data. Tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. This matrix should be chosen to promote the recovery of known important features of the desired solution, such as smoothness and monotonicity. We describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem. The constructed regularization matrix is the closest matrix in the Frobenius norm with a prescribed null space to a given matrix. Numerical examples illustrate the performance of the regularization matrices so obtained.

Some matrix nearness problems suggested by Tikhonov regularization

Abstract The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size are Tikhonov regularization and truncated singular value decomposition (TSVD). By considering matrix nearness problems related to Tikhonov regularization, several novel regularization methods are derived. These methods share properties with both Tikhonov regularization and TSVD, and can give approximate solutions of higher quality than either one of these methods.

Generalized circulant Strang-type preconditioners†

SUMMARY Strangs proposal to use a circulant preconditioner for linear systems of equations with a Hermitian positive definite Toeplitz matrix has given rise to considerable research on circulant preconditioners. This paper presents an {eiφ}-circulant Strang-type preconditioner. Copyright

The structured distance to normality of Toeplitz matrices with application to preconditioning

A formula for the distance of a Toeplitz matrix to the subspace of {eiϕ}-circulant matrices is presented, and applications of {eiϕ}-circulant matrices to preconditioning of linear systems of equations with a Toeplitz matrix are discussed. Copyright

A modified truncated singular value decomposition method for discrete ill‐posed problems

SUMMARY Truncated singular value decomposition is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank-k approximant, which we denote by Ak. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L-curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as Ak. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by Ak.Copyright