Special Issue "Modern Applications of Numerical Linear Algebra"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 January 2022) | Viewed by 3053

Special Issue Editor

Prof. Dr. Ivan Slapničar
E-Mail Website
Guest Editor
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, R. Boškovića 32, HR-21000 Split, Croatia
Interests: numerical linear algebra; structured matrices; matrix eigenvalue computation; nonlinear equations; polynomial rootfinding; data mining; image processing; inverse problems; operator theory; vibrational systems; evolutionary models

Special Issue Information

Dear Colleagues,

Matrix computations are ubiquitous in solving many modern applications, such as data clustering, signal processing, image analysis, problems on graphs, sparse data extraction, vibrational systems, as well as optimization and data mining problems involving huge matrices which lead to randomized algorithms, to name just a few.

Often there is a need to deal with more than one matrix, as in the analysis of vibrational systems with highly structured matrices, or with matrices over non-standard number fields like quaternions. In some cases, matrix problems are related to zeros of polynomials.

We invite you to submit your latest research in any of the areas related to applications of matrix computations including, but not limited to, the ones listed above.

The goal of this Special Issue is to present a plethora of interesting problem-oriented methods, preferably accompanied by their analysis with respect to perturbation theory and numerical stability.

Prof. Dr. Ivan Slapničar
Guest Editor

Manuscript Submission Information

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Keywords

  • linear algebra
  • eigenvalue decomposition
  • singular value decomposition
  • matrix equations
  • matrix pencils
  • matrices of quaternions
  • tensor computation
  • least squares
  • functions of matrices
  • sparse matrices
  • image processing
  • ill-posed problems
  • model reductions
  • graph algorithms
  • vibrational systems
  • signal decomposition
  • polynomials

Published Papers (3 papers)

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Research

Article
Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems
Mathematics 2022, 10(5), 790; https://doi.org/10.3390/math10050790 - 02 Mar 2022
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Abstract
We propose a fast algorithm for computing optimal viscosities of dampers of a linear vibrational system. We are using a standard approach where the vibrational system is first modeled using the second-order structure. This structure yields a quadratic eigenvalue problem which is then [...] Read more.
We propose a fast algorithm for computing optimal viscosities of dampers of a linear vibrational system. We are using a standard approach where the vibrational system is first modeled using the second-order structure. This structure yields a quadratic eigenvalue problem which is then linearized. Optimal viscosities are those for which the trace of the solution of the Lyapunov equation with the linearized matrix is minimal. Here, the free term of the Lyapunov equation is a low-rank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires O(n3) floating-point operations. In our approach, we transform the linearized matrix into an eigenvalue problem of a diagonal-plus-low-rank matrix whose eigenvectors have a Cauchy-like structure. Our algorithm is based on a new fast eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast multiplication of linked Cauchy-like matrices, yielding computation of optimal viscosities for each choice of external dampers in O(kn2) operations, k being the number of dampers. The accuracy of our algorithm is compatible with the accuracy of the standard approach. Full article
(This article belongs to the Special Issue Modern Applications of Numerical Linear Algebra)
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Article
When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System = A(t)x?
Mathematics 2022, 10(1), 141; https://doi.org/10.3390/math10010141 - 04 Jan 2022
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Abstract
In this paper, the class of matrix functions A(t) is determined for which the condition that the pointwise spectrum σ(A(t))zC;zα for all [...] Read more.
In this paper, the class of matrix functions A(t) is determined for which the condition that the pointwise spectrum σ(A(t))zC;zα for all tt0 and some α>0 is sufficient for uniform asymptotic stability of the linear time-varying system x˙=A(t)x. We prove that this class contains as a proper subset the matrix functions with the values in the special orthogonal group SO(n). Full article
(This article belongs to the Special Issue Modern Applications of Numerical Linear Algebra)
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Article
Acceleration and Parallelization of a Linear Equation Solver for Crack Growth Simulation Based on the Phase Field Model
Mathematics 2021, 9(18), 2248; https://doi.org/10.3390/math9182248 - 13 Sep 2021
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Abstract
We aim to accelerate the linear equation solver for crack growth simulation based on the phase field model. As a first step, we analyze the properties of the coefficient matrices and prove that they are symmetric positive definite. This justifies the use of [...] Read more.
We aim to accelerate the linear equation solver for crack growth simulation based on the phase field model. As a first step, we analyze the properties of the coefficient matrices and prove that they are symmetric positive definite. This justifies the use of the conjugate gradient method with the efficient incomplete Cholesky preconditioner. We then parallelize this preconditioner using so-called block multi-color ordering and evaluate its performance on multicore processors. The experimental results show that our solver scales well and achieves an acceleration of several times over the original solver based on the diagonally scaled CG method. Full article
(This article belongs to the Special Issue Modern Applications of Numerical Linear Algebra)
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