arXiv subjects

Lothar Reichel

Publications and source records attributed to Lothar Reichel.

At least 19 recordsLinked to original sources

arXiv2026

Approximating matrix functions by block Krylov methods with randomized vectors

The need to evaluate expressions of the form $f(A)\mathbf{b}$, where $A$ is a square matrix, $f$ is a function, and $\mathbf{b}$ is a vector, arises in several areas of applied mathematics. When the matrix $A$ is very large, it is usually not attractive to evaluate $f(A)$. Instead, $f(A)\mathbf{b}$ often is approximated by computing an estimate in a Krylov subspace that depends on $A$ and $\mathbf{b}$, and only requires that $f$ be evaluated at a small matrix. This paper explores the application of several variants of randomized block Krylov methods to the approximation of $f(A)\mathbf{b}$. Computed examples suggest that block Krylov methods with an initial block vector that contains $\mathbf{b}$ as well as a few randomly generated vectors may require less computing time and reduce the number of Krylov steps than standard Krylov methods.

math.NA↗

arXiv2026

Iterated Tikhonov regularization of large linear problems

Many solution methods for linear discrete ill-posed problems with error-contaminated data (right-hand side) apply Tikhonov regularization to compute a meaningful approximate solution. This solution depends on a regularization parameter. It is well known that iterated Tikhonov regularization often determines an approximate solution of higher quality than (standard) Tikhonov regularization. We consider the situation when an estimate of the norm of the error in the data is known and would like to apply iterative Tikhonov regularization to determine an approximate solution that satisfies the discrepancy principle. This requires a suitable choice of a regularization parameter. The standard approach to determine this parameter is to compute solutions for several values of the regularization parameter and choose a computed approximate solution that satisfies the discrepancy principle. This paper discusses iterated Tikhonov regularization based on partial Golub-Kahan bidiagonalization and describes how the regularization parameter can be determined without computing several approximate solutions by using the connection between Golub-Kahan bidiagonalization and Gauss quadrature. This approach reduces the computational effort required to compute a desired solution.

math.NA↗

arXiv2026

A Unified Trace-Optimization Framework for Multidimensionality Reduction

This paper presents a comprehensive overview of several multidimensional reduction methods focusing on Multidimensional Principal Component Analysis (MPCA), Multilinear Orthogonal Neighborhood Preserving Projection (MONPP), Multidimensional Locally Linear Embedding (MLLE), and Multidimensional Laplacian Eigenmaps (MLE). These techniques are formulated within a unified framework based on trace optimization, where the dimensionality reduction problem is expressed as maximization or minimization problems. In addition to the linear MPCA and MONPP approaches, kernel-based extensions of these methods also are presented. The latter methods make it possible to capture nonlinear relations between high-dimensional data. A comparative analysis highlights the theoretical foundations, assumptions, and computational efficiency of each method, as well as their practical applicability. The study provides insights and guidelines for selecting an appropriate dimensionality reduction technique suited to the application at hand.

math.NA↗

arXiv2025

Sensitivity of Perron and Fiedler eigenpairs to structural perturbations of a network

One can estimate the change of the Perron and Fiedler values for a connected network when the weight of an edge is perturbed by analyzing relevant entries of the Perron and Fiedler vectors. This is helpful for identifying edges whose weight perturbation causes the largest change in the Perron and Fiedler values. It also is important to investigate the sensitivity of the Perron and Fiedler vectors to perturbations. Applications of the perturbation analysis include the identification of edges that are critical for the structural robustness of the network.

math.NA↗

arXiv2025

Network connectivity analysis via shortest paths

Complex systems of interacting components often can be modeled by a simple graph $\mathcal{G}$ that consists of a set of $n$ nodes and a set of $m$ edges. Such a graph can be represented by an adjacency matrix $A\in\R^{n\times n}$, whose $(ij)$th entry is one if there is an edge pointing from node $i$ to node $j$, and is zero otherwise. The matrix $A$ and its positive integer powers reveal important properties of the graph and allow the construction of the path length matrix $L$ for the graph. The $(ij)$th entry of $L$ is the length of the shortest path from node $i$ to node $j$; if there is no path between these nodes, then the value of the entry is set to $\infty$. We are interested in how well information flows via shortest paths of the graph. This can be studied with the aid of the path length matrix. The path length matrix allows the definition of several measures of communication in the network defined by the graph such as the global $K$-efficiency, which considers shortest paths that are made up of at most $K$ edges for some $K<n$, as well as the number of such shortest paths. Novel notions of connectivity introduced in this paper help us understand the importance of specific edges for the flow of information through the graph. This is of interest when seeking to simplify a network by removing selected edges or trying to assess the sensitivity of the flow of information to changes due to exterior causes such as a traffic stoppage on a road network.

physics.soc-ph↗

arXiv2025

Low-Rank Regularized Convex-Non-Convex Problems for Image Segmentation or Completion

This work proposes a novel convex-non-convex formulation of the image segmentation and the image completion problems. The proposed approach is based on the minimization of a functional involving two distinct regularization terms: one promotes low-rank structure in the solution, while the other one enforces smoothness. To solve the resulting optimization problem, we employ the alternating direction method of multipliers (ADMM). A detailed convergence analysis of the algorithm is provided, and the performance of the methods is demonstrated through a series of numerical experiments.

math.NA↗

arXiv2025

The iterated Golub-Kahan-Tikhonov method

The Golub-Kahan-Tikhonov method is a popular solution technique for large linear discrete ill-posed problems. This method first applies partial Golub-Kahan bidiagonalization to reduce the size of the given problem and then uses Tikhonov regularization to compute a meaningful approximate solution of the reduced problem. It is well known that iterated variants of this method often yield approximate solutions of higher quality than the standard non-iterated method. Moreover, it produces more accurate computed solutions than the Arnoldi method when the matrix that defines the linear discrete ill-posed problem is far from symmetric. This paper starts with an ill-posed operator equation in infinite-dimensional Hilbert space, discretizes the equation, and then applies the iterated Golub-Kahan-Tikhonov method to the solution of the latter problem. An error analysis that addresses all discretization and approximation errors is provided. Additionally, a new approach for choosing the regularization parameter is described. This solution scheme produces more accurate approximate solutions than the standard (non-iterated) Golub-Kahan-Tikhonov method and the iterated Arnoldi-Tikhonov method.

math.NA↗

arXiv2024

Communication in Multiplex Transportation Networks

Complex networks are made up of vertices and edges. The edges, which may be directed or undirected, are equipped with positive weights. Modeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices in distinct layers represent different kinds of objects. Multiplex networks are special vertex-aligned multilayer networks, in which vertices in distinct layers are identified with each other and inter-layer edges connect each vertex with its copy in other layers and have a fixed weight $\gamma>0$ associated with the ease of communication between layers. This paper discusses two different approaches to analyze communication in a multiplex. One approach focuses on the multiplex global efficiency by using the multiplex path length matrix, the other approach considers the multiplex total communicability. The sensitivity of both the multiplex global efficiency and the multiplex total communicability to structural perturbations in the network is investigated to help to identify intra-layer edges that should be strengthened to enhance communicability.

math.NA↗

arXiv2024

Edge Importance in Complex Networks

Complex networks are made up of vertices and edges. The latter connect the vertices. There are several ways to measure the importance of the vertices, e.g., by counting the number of edges that start or end at each vertex, or by using the subgraph centrality of the vertices. It is more difficult to assess the importance of the edges. One approach is to consider the line graph associated with the given network and determine the importance of the vertices of the line graph, but this is fairly complicated except for small networks. This paper compares two approaches to estimate the importance of edges of medium-sized to large networks. One approach computes partial derivatives of the total communicability of the weights of the edges, where a partial derivative of large magnitude indicates that the corresponding edge may be important. Our second approach computes the Perron sensitivity of the edges. A high sensitivity signals that the edge may be important. The performance of these methods and some computational aspects are discussed. Applications of interest include to determine whether a network can be replaced by a network with fewer edges with about the same communicability.

physics.soc-ph↗

arXiv2023

Averaged Nyström interpolants for the solution of Fredholm integral equations of the second kind

Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. This paper discusses Nyström methods based on Gauss quadrature rules for the solution of such integral equations. It is important to be able to estimate the error in the computed solution, because this allows the choice of an appropriate number of nodes in the Gauss quadrature rule used. This paper explores the application of averaged and weighted averaged Gauss quadrature rules for this purpose, and introduces new stability properties for them.

math.NA↗

arXiv2023

Convergence analysis and parameter estimation for the iterated Arnoldi-Tikhonov method

The Arnoldi-Tikhonov method is a well-established regularization technique for solving large-scale ill-posed linear inverse problems. This method leverages the Arnoldi decomposition to reduce computational complexity by projecting the discretized problem into a lower-dimensional Krylov subspace, in which it is solved. This paper explores the iterated Arnoldi-Tikhonov method, conducting a comprehensive analysis that addresses all approximation errors. Additionally, it introduces a novel strategy for choosing the regularization parameter, leading to more accurate approximate solutions compared to the standard Arnoldi-Tikhonov method. Moreover, the proposed method demonstrates robustness with respect to the regularization parameter, as confirmed by the numerical results.

math.NA↗

arXiv2023

Enhancing multiplex global efficiency

Modeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices are connected by both inter-layer and intra-layer edges. In this paper, we investigate multiplex networks, in which vertices in different layers are identified with each other, and the only inter-layer edges are those that connect a vertex with its copy in other layers. Let the third-order adjacency tensor $\mathcal{A}\in\R^{N\times N\times L}$ and the parameter $γ\geq 0$, which is associated with the ease of communication between layers, represent a multiplex network with $N$ vertices and $L$ layers. To measure the ease of communication in a multiplex network, we focus on the average inverse geodesic length, which we refer to as the multiplex global efficiency $e_\mathcal{A}(γ)$ by means of the multiplex path length matrix $P\in\R^{N\times N}$. This paper generalizes the approach proposed in \cite{NR23} for single-layer networks. We describe an algorithm based on min-plus matrix multiplication to construct $P$, as well as variants $P^K$ that only take into account multiplex paths made up of at most $K$ intra-layer edges. These matrices are applied to detect redundant edges and to determine non-decreasing lower bounds $e_\mathcal{A}^K(γ)$ for $e_\mathcal{A}(γ)$, for $K=1,2,\dots,N-2$. Finally, the sensitivity of $e_\mathcal{A}^K(γ)$ to changes of the entries of the adjacency tensor $\mathcal{A}$ is investigated to determine edges that should be strengthened to enhance the multiplex global efficiency the most.

math.NA↗

arXiv2023

A tensor formalism for multilayer network centrality measures using the Einstein product

Complex systems that consist of different kinds of entities that interact in different ways can be modeled by multilayer networks. This paper uses the tensor formalism with the Einstein tensor product to model this type of networks. Several centrality measures, that are well known for single-layer networks, are extended to multilayer networks using tensors and their properties are investigated. In particular, subgraph centrality based on the exponential and resolvent of a tensor are considered. Krylov subspace methods are introduced for computing approximations of different measures for large multilayer networks.

math.NA↗

arXiv2023

Spectral computation with third-order tensors using the t-product

The tensor t-product, introduced by Kilmer and Martin [26], is a powerful tool for the analysis of and computation with third-order tensors. This paper introduces eigentubes and eigenslices of third-order tensors under the t-product. The eigentubes and eigenslices are analogues of eigenvalues and eigenvectors for matrices. Properties of eigentubes and eigenslices are investigated and numerical methods for their computation are described. The methods include the tensor power method, tensor subspace iteration, and the tensor QR algorithm. Computed examples illustrate the performance of these methods.

math.NA↗

arXiv2023

Network analysis with the aid of the path length matrix

Let a network be represented by a simple graph $\mathcal{G}$ with $n$ vertices. A common approach to investigate properties of a network is to use the adjacency matrix $A=[a_{ij}]_{i,j=1}^n\in\R^{n\times n}$ associated with the graph $\mathcal{G}$, where $a_{ij}>0$ if there is an edge pointing from vertex $v_i$ to vertex $v_j$, and $a_{ij}=0$ otherwise. Both $A$ and its positive integer powers reveal important properties of the graph. This paper proposes to study properties of a graph $\mathcal{G}$ by also using the path length matrix for the graph. The $(ij)^{th}$ entry of the path length matrix is the length of the shortest path from vertex $v_i$ to vertex $v_j$; if there is no path between these vertices, then the value of the entry is $\infty$. Powers of the path length matrix are formed by using min-plus matrix multiplication and are important for exhibiting properties of $\mathcal{G}$. We show how several known measures of communication such as closeness centrality, harmonic centrality, and eccentricity are related to the path length matrix, and we introduce new measures of communication, such as the harmonic $K$-centrality and global $K$-efficiency, where only (short) paths made up of at most $K$ edges are taken into account. The sensitivity of the global $K$-efficiency to changes of the entries of the adjacency matrix also is considered.

math.NA↗

arXiv2023

A tensor bidiagonalization method for higher-order singular value decomposition with applications

The need to know a few singular triplets associated with the largest singular values of third-order tensors arises in data compression and extraction. This paper describes a new method for their computation using the t-product. Methods for determining a couple of singular triplets associated with the smallest singular values also are presented. The proposed methods generalize available restarted Lanczos bidiagonalization methods for computing a few of the largest or smallest singular triplets of a matrix. The methods of this paper use Ritz and harmonic Ritz lateral slices to determine accurate approximations of the largest and smallest singular triplets, respectively. Computed examples show applications to data compression and face recognition.

math.NA↗

arXiv2022

Perron communicability and sensitivity of multilayer networks

Modeling complex systems that consist of different types of objects leads to multilayer networks, where nodes in the different layers represent different kind of objects. Nodes are connected by edges, which have positive weights. A multilayer network is associated with a supra-adjacency matrix. This paper investigates the sensitivity of the communicability in a multilayer network to perturbations of the network by studying the sensitivity of the Perron root of the supra-adjacency matrix. Our analysis sheds light on which edge weights to make larger to increase the communicability of the network, and which edge weights can be made smaller or set to zero without affecting the communicability significantly.

math.NA↗

arXiv2022

Adaptive cross approximation for Tikhonov regularization in general form

Many problems in Science and Engineering give rise to linear integral equations of the first kind with a smooth kernel. Discretization of the integral operator yields a matrix, whose singular values cluster at the origin. We describe the approximation of such matrices by adaptive cross approximation, which avoids forming the entire matrix. The choice of the number of steps of adaptive cross approximation is discussed. The discretized right-hand side represents data that commonly are contaminated by measurement error. Solution of the linear system of equations so obtained is not meaningful because the matrix determined by adaptive cross approximation is rank-deficient. We remedy this difficulty by using Tikhonov regularization and discuss how a fairly general regularization matrix can be used. Computed examples illustrate that the use of a regularization matrix different from the identity can improve the quality of the computed approximate solutions significantly.

math.NA↗