Jörg Liesen

Robert Luce, Peter Hildebrandt, Uwe Kuhlmann et al.

The key challenge of time-resolved Raman spectroscopy is the identification of the constituent species and the analysis of the kinetics of the underlying reaction network. In this work we present an integral approach that allows for determining both the component spectra and the rate constants simultaneously from a series of vibrational spectra. It is based on an algorithm for non-negative matrix factorization which is applied to the experimental data set following a few pre-processing steps. As a prerequisite for physically unambiguous solutions, each component spectrum must include one vibrational band that does not significantly interfere with vibrational bands of other species. The approach is applied to synthetic "experimental" spectra derived from model systems comprising a set of species with component spectra differing with respect to their degree of spectral interferences and signal-to-noise ratios. In each case, the species involved are connected via monomolecular reaction pathways. The potential and limitations of the approach for recovering the respective rate constants and component spectra are discussed.

André Gaul, Martin H. Gutknecht, Jörg Liesen et al.

We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) "removes" certain parts from the operator making it singular, while augmentation adds a subspace to the Krylov subspace (often the one that is generated by the singular operator); in contrast, preconditioning changes the spectrum of the operator without making it singular. Deflation and augmentation have been used in a variety of methods and settings. Typically, deflation is combined with augmentation to compensate for the singularity of the operator, but both techniques can be applied separately. We introduce a framework of Krylov subspace methods that satisfy a Galerkin condition. It includes the families of orthogonal residual (OR) and minimal residual (MR) methods. We show that in this framework augmentation can be achieved either explicitly or, equivalently, implicitly by projecting the residuals appropriately and correcting the approximate solutions in a final step. We study conditions for a breakdown of the deflated methods, and we show several possibilities to avoid such breakdowns for the deflated MINRES method. Numerical experiments illustrate properties of different variants of deflated MINRES analyzed in this paper.

Jörg Liesen, Olivier Sète, Mohamed M. S. Nasser

We present a numerical method for computing the logarithmic capacity of compact subsets of $\mathbb{C}$, which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have any special symmetry. The method relies on the conformal map onto lemniscatic domains and, computationally, on the solution of a boundary integral equation with the Neumann kernel. Our numerical examples indicate that the method is fast and accurate. We apply it to give an estimate of the logarithmic capacity of the Cantor middle third set and generalizations of it.

Carlos Echeverría, Jörg Liesen, Reinhard Nabben

We generalize the bounds on the inverses of diagonally dominant matrices obtained in [16] from scalar to block tridiagonal matrices. Our derivations are based on a generalization of the classical condition of block diagonal dominance of matrices given by Feingold and Varga in [11]. Based on this generalization, which was recently presented in [3], we also derive a variant of the Gershgorin Circle Theorem for general block matrices which can provide tighter spectral inclusion regions than those obtained by Feingold and Varga.

Vance Faber, Jörg Liesen, Petr Tichý

Given a matrix $A$ and iteration step $k$, we study a best possible attainable upper bound on the GMRES residual norm that does not depend on the initial vector $b$. This quantity is called the worst-case GMRES approximation. We show that the worst case behavior of GMRES for the matrices $A$ and $A^T$ is the same, and we analyze properties of initial vectors for which the worst-case residual norm is attained. In particular, we show that such vectors satisfy a certain "cross equality", and we characterize them as right singular vectors of the corresponding GMRES residual matrix. We show that the worst-case GMRES polynomial may not be uniquely determined, and we consider the relation between the worst-case and the ideal GMRES approximations, giving new examples in which the inequality between the two quantities is sharp at all iteration steps $k\geq 3$. Finally, we give a complete characterization of how the values of the approximation problems in the context of worst-case and ideal GMRES for a real matrix change, when one considers complex (rather than real) polynomials and initial vectors in these problems.

Olivier Sète, Jörg Liesen

The Faber--Walsh polynomials are a direct generalization of the (classical) Faber polynomials from simply connected sets to sets with several simply connected components. In this paper we derive new properties of the Faber--Walsh polynomials, where we focus on results of interest in numerical linear algebra, and on the relation between the Faber--Walsh polynomials and the classical Faber and Chebyshev polynomials. Moreover, we present examples of Faber--Walsh polynomials for two real intervals as well as some non-real sets consisting of several simply connected components.

Jörg Liesen, Robert Luce

We derive an algorithm of optimal complexity which determines whether a given matrix is a Cauchy matrix, and which exactly recovers the Cauchy points defining a Cauchy matrix from the matrix entries. Moreover, we study how to approximate a given matrix by a Cauchy matrix with a particular focus on the recovery of Cauchy points from noisy data. We derive an approximation algorithm of optimal complexity for this task, and prove approximation bounds. Numerical examples illustrate our theoretical results.