arXiv subjects

Reinhard Nabben

Publications and source records attributed to Reinhard Nabben.

9 recordsLinked to original sources

arXiv2026

Hidden decay in the inverses of acyclic matrices

It is well-known that the absolute values of the entries of the inverse of a diagonally dominant tridiagonal matrix decay away from the diagonal. Moreover, lower and upper bounds for these absolute values are known for a long time. The graph of a tridiagonal matrix can be viewed as a line. Thus tridiagonal matrices are special acyclic matrices, i.e. matrices whose graph is a tree. In this paper we explore the structure of inverses of acyclic matrices. However, there is no strict decay from the diagonal in the inverses of diagonally dominant acyclic matrices. The decay is somehow hidden. Here we identify this decay which can be described nicely. Moreover we give bounds for the absolute value of the entries of the inverses of acyclic matrices. All of our results generalize results for tridiagonal matrices.

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arXiv2026

Inverse acyclic Z-matrices, matrices of tree structure and shifted generalized ultrametric matrices

Inverses of acyclic or treediagonal matrices are nicely characterized in the landmarked paper by Klein \cite{KLe82}. But also in \cite{Nab01} a characterization of these inverses is given which is related to the structure of inverses of tridiagonal matrices. Here we use this characterization to determine and construct matrices whose inverses are acyclic $Z$-matrices. Moreover, we establish relations between several classes of matrices, namely the classes of $Z$-matrices, inverse $Z$-matrices, ultrametric matrices, generalized ultrametric matrices, shifted ultrametric matrices, acyclic matrices, inverse acyclic matrices, matrices of tree structure, cyclopes and generalized cyclopes.

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arXiv2026

Optimal transfer operators for nonsymmetric two-grid methods

Algebraic Multigrid (AMG) methods have been proven to be effective solvers for large-scale linear algebraic systems $Ax = b$ with Hermitian positive definite (HPD) matrix $A$. For such problems the convergence in the $A$-norm is well understood, but for nonsymmetric indefinite systems fewer results exist. Recently, convergence results for more general $B$-norms induced by certain HPD matrices were established. There, orthogonal projections built by compatible transfer operators are used. Here, we present a theoretical framework for the convergence of nonsymmetric algebraic two-grid methods for arbitrary $B$-inner products and induced $B$-norms which naturally includes the HPD case and all recent results for the nonsymmetric case. For this purpose, we consider two different two-grid error operators with the first one being the natural generalization of the error operator in the HPD case. The second operator has been studied before and is simpler, but requires the additional assumption of normality in some inner product of the smoothing step $M^{-1}A$ to achieve convergence. We prove new convergence results, generalize some previous results and explain the differences and similarities of both operators together with the necessity of the normality. Moreover, we establish optimal compatible interpolation and restriction operators for both two-grid methods that minimize the error norm.

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arXiv2026

Norm-based convergence bounds for nonsymmetric algebraic V-cycle multigrid methods

Recently a new approach to analyze and create algebraic multigrid methods (AMG) for nonsymmetric and indefinite matrices was established. Convergence is measured in general norms induced by a certain HPD matrix $B$ and $B$-orthogonal projections built by compatible transfer operators are used. Here we continue our theoretical framework, started in Nabben and Rooch (2026), for nonsymmetric algebraic multigrid methods using any HPD matrix $B$ to induce a norm. Our framework not only includes all recent results but also provides many new results. We consider two, slightly different, multigrid operators. The first one is the natural generalization of the error operator in the HPD case. The second operator is simpler to apply and has been studied before. However, an additional condition for the smoother $M^{-1}A$ is needed, which is in our terminology the $B$-normality. We explain the differences and similarities of both operators in detail and show, why the extra condition is needed. We consider arbitrary interpolation and restriction operators that result in $B$-orthogonal coarse-grid corrections and give sharp estimates for the norm of the error propagation matrices for the two-grid methods. We also show, that the norms are decreasing if we increase the size of the coarse space. Moreover, we are able to extend the landmark $V$-cycle bound by McCormick to the nonsymmetric case.

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arXiv2025

Factorized sparse approximate inverse preconditioning for singular M-matrices

Here we consider the factorized sparse approximate inverse (FSAI) preconditioner. We apply the FSAI preconditioner to singular irreducible M-matrices. These matrices arise e.g. in discrete Markov chain modeling or as graph Laplacians. We show, that there are some restrictions on the nonzero pattern needed for a stable construction of the FSAI preconditioner in this case. With these restrictions FSAI is well-defined. Moreover, we proved that the FSAI preconditioner shares some important properties with the original system. The lower triangular matrix $L_G$ and the upper triangular matrix $U_G$, generated by FSAI, are non-singular and non-negative. The diagonal entries of $L_GAU_G$ are positive and $L_GAU_G$, the preconditioned matrix, is a singular M-matrix. Even more, we establish that a (1,2)-inverse is computed for the complete nonzero patter.

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arXiv2020

A two-level shifted Laplace preconditioner for Helmholtz problems: Field-of-values analysis and wavenumber-independent convergence

One of the main tools for solving linear systems arising from the discretization of the Helmholtz equation is the shifted Laplace preconditioner, which results from the discretization of a perturbed Helmholtz problem $-Δu - (k^2 + i \varepsilon )u = f$ where $0 \neq \varepsilon \in \mathbb{R}$ is an absorption parameter. In this work we revisit the idea of combining the shifted Laplace preconditioner with two-level deflation and apply it to Helmholtz problems discretized with linear finite elements. We use the convergence theory of GMRES based on the field of values to prove that GMRES applied to the two-level preconditioned system with a shift parameter $\varepsilon \sim k^2$ converges in a number of iterations independent of the wavenumber $k$,provided that the coarse mesh size $H$ satisfies a condition of the form $Hk^{2} \leq C$ for some constant $C$ depending on the domain but independent of the wavenumber $k$. This behaviour is sharply different to the standalone shifted Laplacian, for which wavenumber-independent GMRES convergence has been established only under the condition that $\varepsilon \sim k$ by [M.J. Gander, I.G. Graham and E.A. Spence, Numer. Math., 131 (2015), 567-614]. Finally, we present numerical evidence that wavenumber-independent convergence of GMRES also holds for pollution-free meshes, where the coarse mesh size satisfies $Hk^{3/2} \leq C $, and inexact coarse grid solves.

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arXiv2019

On Optimal Algebraic Multigrid Methods

In this note we present an alternative way to obtain optimal interpolation operators for two-grid methods applied to Hermitian positive definite linear systems. Falgout and Vassilevski in [SIAM J. Numer. Anal, 42 (2004), pp. 1669-1693] and Zikatanov [Numer. Linear Algebra Appl., 15 (2008), pp. 439-454] have characterized the $A$-norm of the error propagation operator of algebraic multigrid methods. These results have been recently used by Xu and Zikatanov [Acta Numer., 26 (2017), pp. 591-721] and Brannick, Cao et al. [SIAM J. Sci. Comp, 40 (2018), pp. 591-721] to determine optimal interpolation operators. Here we use a characterization not of the $A$-norm but of the spectrum of the error propagation operator of two-grid methods, which was proved by García Ramos, Nabben and Kehl and holds for arbitrary matrices. For Hermitian positive definite systems this result leads to optimal interpolation operators with respect to the $A$-norm in a short way, moreover, it also leads to optimal interpolation operators with respect to the spectral radius. For the symmetric two-grid method (with pre- and post-smoothing) the optimal interpolation operators are the same. But for a two-grid method with only post-smoothing the optimal interpolations (and hence the optimal algebraic multigrid methods) can be different. Moreover, using the characterization of the spectrum, we can show that the found optimal interpolation operators are also optimal with respect to the condition number of the multigrid preconditioned system.

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arXiv2017

Block diagonal dominance of matrices revisited: bounds for the norms of inverses and eigenvalue inclusion sets

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.

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arXiv2012

A framework for deflated and augmented Krylov subspace methods

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.

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