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Mathematics
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Numerical Linear Algebra and the Applications
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Numerical Linear Algebra and the Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (30 April 2021) | Viewed by 15761

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Khalide Jbilou

E-Mail Website
Guest Editor
1. Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral Côte d’Opale, CEDEX, BP 699-62228 Calais, France
2. Laboratory MSDA, University Polytechnic Mohammed VI, Benguerir 43150, Morocco
Interests: Linear algebra; computer science; numerical analysis; Ill-posed problems
Prof. Dr. Marilena Mitrouli

E-Mail Website
Guest Editor
Department of Mathematics, National and Kapodistrian University of Athens, GR-157 84 Athens, Greece
Interests: Numerical Analysis; Numerical Linear Algebra; Matrix Analysis

Special Issue Information

Dear Colleagues,

Numerical linear algebra is a very important topic in mathematics and has important recent applications in deep learning, machine learning, image processing, applied statistics, artificial intelligence and other interesting modern applications in many fields. The purpose of this Special Issue in Mathematics is to present the latest contributions and recent developments of numerical linear algebra and applications in different real domains. We invite authors to submit original and new papers and high-quality reviews related to the following topics: applied linear algebra, linear and nonlinear systems of equations, large matrix equations, numerical tensor problems with applications, ill-posed problems and image processing, linear algebra and applied statistics, model reduction in dynamic systems and other related subjects. The submitted papers will be reviewed in line with the traditional submission process.

This Special Issue will be dedicated to the inspired mathematician Constantin Petridi, who has devoted his life to mathematics.

Prof. Dr. Khalide Jbilou
Prof. Dr. Marilena Mitrouli
Guest Editors

Manuscript Submission Information

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Keywords

  • linear algebra
  • matrix equations
  • applied statistics
  • tensor algorithms
  • image processing
  • ill-posed problems
  • model reductions

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Published Papers (7 papers)

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Research

16 pages, 419 KiB  
Article
Solving High-Dimensional Problems in Statistical Modelling: A Comparative Study
by Stamatis Choudalakis, Marilena Mitrouli, Athanasios Polychronou and Paraskevi Roupa
Mathematics 2021, 9(15), 1806; https://doi.org/10.3390/math9151806 - 30 Jul 2021
Cited by 4 | Viewed by 1635
Abstract
In this work, we present numerical methods appropriate for parameter estimation in high-dimensional statistical modelling. The solution of these problems is not unique and a crucial question arises regarding the way that a solution can be found. A common choice is to keep [...] Read more.
In this work, we present numerical methods appropriate for parameter estimation in high-dimensional statistical modelling. The solution of these problems is not unique and a crucial question arises regarding the way that a solution can be found. A common choice is to keep the corresponding solution with the minimum norm. There are cases in which this solution is not adequate and regularisation techniques have to be considered. We classify specific cases for which regularisation is required or not. We present a thorough comparison among existing methods for both estimating the coefficients of the model which corresponds to design matrices with correlated covariates and for variable selection for supersaturated designs. An extensive analysis for the properties of design matrices with correlated covariates is given. Numerical results for simulated and real data are presented. Full article
(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)
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18 pages, 717 KiB  
Article
Eigenvalue Estimates via Pseudospectra
by Georgios Katsouleas, Vasiliki Panagakou and Panayiotis Psarrakos
Mathematics 2021, 9(15), 1729; https://doi.org/10.3390/math9151729 - 22 Jul 2021
Viewed by 2195
Abstract
In this note, given a matrix A∈Cn×n (or a general matrix polynomial P(z), z∈C) and an arbitrary scalar λ0∈C, we show how to define a sequence [...] Read more.
In this note, given a matrix A∈Cn×n (or a general matrix polynomial P(z), z∈C) and an arbitrary scalar λ0∈C, we show how to define a sequence μkk∈N which converges to some element of its spectrum. The scalar λ0 serves as initial term (μ0=λ0), while additional terms are constructed through a recursive procedure, exploiting the fact that each term μk of this sequence is in fact a point lying on the boundary curve of some pseudospectral set of A (or P(z)). Then, the next term in the sequence is detected in the direction which is normal to this curve at the point μk. Repeating the construction for additional initial points, it is possible to approximate peripheral eigenvalues, localize the spectrum and even obtain spectral enclosures. Hence, as a by-product of our method, a computationally cheap procedure for approximate pseudospectra computations emerges. An advantage of the proposed approach is that it does not make any assumptions on the location of the spectrum. The fact that all computations are performed on some dynamically chosen locations on the complex plane which converge to the eigenvalues, rather than on a large number of predefined points on a rigid grid, can be used to accelerate conventional grid algorithms. Parallel implementation of the method or use in conjunction with randomization techniques can lead to further computational savings when applied to large-scale matrices. Full article
(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)
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16 pages, 868 KiB  
Article
Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices
by Anna Concas, Lothar Reichel, Giuseppe Rodriguez and Yunzi Zhang
Mathematics 2021, 9(13), 1522; https://doi.org/10.3390/math9131522 - 29 Jun 2021
Cited by 1 | Viewed by 1795
Abstract
The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos [...] Read more.
The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed. Full article
(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)
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13 pages, 360 KiB  
Article
Estimating the Quadratic Form xTA−mx for Symmetric Matrices: Further Progress and Numerical Computations
by Marilena Mitrouli, Athanasios Polychronou, Paraskevi Roupa and Ondřej Turek
Mathematics 2021, 9(12), 1432; https://doi.org/10.3390/math9121432 - 19 Jun 2021
Cited by 3 | Viewed by 2121
Abstract
In this paper, we study estimates for quadratic forms of the type xTA−mx, m∈N, for symmetric matrices. We derive a general approach for estimating this type of quadratic form and we present some upper [...] Read more.
In this paper, we study estimates for quadratic forms of the type xTA−mx, m∈N, for symmetric matrices. We derive a general approach for estimating this type of quadratic form and we present some upper bounds for the corresponding absolute error. Specifically, we consider three different approaches for estimating the quadratic form xTA−mx. The first approach is based on a projection method, the second is a minimization procedure, and the last approach is heuristic. Numerical examples showing the effectiveness of the estimates are presented. Furthermore, we compare the behavior of the proposed estimates with other methods that are derived in the literature. Full article
(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)
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17 pages, 1694 KiB  
Article
A Multidimensional Principal Component Analysis via the C-Product Golub–Kahan–SVD for Classification and Face Recognition
by Mustapha Hached, Khalide Jbilou, Christos Koukouvinos and Marilena Mitrouli
Mathematics 2021, 9(11), 1249; https://doi.org/10.3390/math9111249 - 29 May 2021
Cited by 10 | Viewed by 2391
Abstract
Face recognition and identification are very important applications in machine learning. Due to the increasing amount of available data, traditional approaches based on matricization and matrix PCA methods can be difficult to implement. Moreover, the tensorial approaches are a natural choice, due to [...] Read more.
Face recognition and identification are very important applications in machine learning. Due to the increasing amount of available data, traditional approaches based on matricization and matrix PCA methods can be difficult to implement. Moreover, the tensorial approaches are a natural choice, due to the mere structure of the databases, for example in the case of color images. Nevertheless, even though various authors proposed factorization strategies for tensors, the size of the considered tensors can pose some serious issues. Indeed, the most demanding part of the computational effort in recognition or identification problems resides in the training process. When only a few features are needed to construct the projection space, there is no need to compute a SVD on the whole data. Two versions of the tensor Golub–Kahan algorithm are considered in this manuscript, as an alternative to the classical use of the tensor SVD which is based on truncated strategies. In this paper, we consider the Tensor Tubal Golub–Kahan Principal Component Analysis method which purpose it to extract the main features of images using the tensor singular value decomposition (SVD) based on the tensor cosine product that uses the discrete cosine transform. This approach is applied for classification and face recognition and numerical tests show its effectiveness. Full article
(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)
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18 pages, 419 KiB  
Article
Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation
by Vera Angelova, Mustapha Hached and Khalide Jbilou
Mathematics 2021, 9(8), 855; https://doi.org/10.3390/math9080855 - 14 Apr 2021
Cited by 2 | Viewed by 2086
Abstract
Nonsymmetric differential matrix Riccati equations arise in many problems related to science and engineering. This work is focusing on the sensitivity of the solution to perturbations in the matrix coefficients and the initial condition. Two approaches of nonlocal perturbation analysis of the symmetric [...] Read more.
Nonsymmetric differential matrix Riccati equations arise in many problems related to science and engineering. This work is focusing on the sensitivity of the solution to perturbations in the matrix coefficients and the initial condition. Two approaches of nonlocal perturbation analysis of the symmetric differential Riccati equation are extended to the nonsymmetric case. Applying the techniques of Fréchet derivatives, Lyapunov majorants and fixed-point principle, two perturbation bounds are derived: the first one is based on the integral form of the solution and the second one considers the equivalent solution to the initial value problem of the associated differential system. The first bound is derived for the nonsymmetric differential Riccati equation in its general form. The perturbation bound based on the sensitivity analysis of the associated linear differential system is formulated for the low-dimensional approximate solution to the large-scale nonsymmetric differential Riccati equation. The two bounds exploit the existing sensitivity estimates for the matrix exponential and are alternative. Full article
(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)
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14 pages, 297 KiB  
Article
An Inverse Mixed Impedance Scattering Problem in a Chiral Medium
by Evagelia S. Athanasiadou
Mathematics 2021, 9(1), 104; https://doi.org/10.3390/math9010104 - 5 Jan 2021
Cited by 5 | Viewed by 1942
Abstract
An inverse scattering problem of time-harmonic chiral electromagnetic waves for a buried partially coated object was studied. The buried object was embedded in a piecewise isotropic homogeneous background chiral material. On the boundary of the scattering object, the total electromagnetic field satisfied perfect [...] Read more.
An inverse scattering problem of time-harmonic chiral electromagnetic waves for a buried partially coated object was studied. The buried object was embedded in a piecewise isotropic homogeneous background chiral material. On the boundary of the scattering object, the total electromagnetic field satisfied perfect conductor and impedance boundary conditions. A modified linear sampling method, which originated from the chiral reciprocity gap functional, was employed for reconstruction of the shape of the buried object without requiring any a priori knowledge of the material properties of the scattering object. Furthermore, a characterization of the impedance of the object’s surface was determined. Full article
(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)
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