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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

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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

Open AccessArticle

Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems

Nevena Jakovčević Stor

Ivan Slapničar

and

Zoran Tomljanović

Mathematics 2022, 10(5), 790; https://doi.org/10.3390/math10050790 - 02 Mar 2022

Cited by 1 | Viewed by 1457

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 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 (n^{3})

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 (k n^{2})

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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Open AccessFeature PaperArticle

When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?

Robert Vrabel

Mathematics 2022, 10(1), 141; https://doi.org/10.3390/math10010141 - 04 Jan 2022

Viewed by 945

Abstract

In this paper, the class of matrix functions

A (t)

is determined for which the condition that the pointwise spectrum

σ (A (t)) \subset \{z \in C; ℜ z \leq - α\}

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)) \subset \{z \in C; ℜ z \leq - α\}

for all

t \geq t_{0}

and some

α > 0

is sufficient for uniform asymptotic stability of the linear time-varying system

\dot{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

S O (n) .

Full article

(This article belongs to the Special Issue Modern Applications of Numerical Linear Algebra)

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Figure 1

Open AccessArticle

Acceleration and Parallelization of a Linear Equation Solver for Crack Growth Simulation Based on the Phase Field Model

Gaku Ishii

Yusaku Yamamoto

and

Takeshi Takaishi

Mathematics 2021, 9(18), 2248; https://doi.org/10.3390/math9182248 - 13 Sep 2021

Viewed by 1590

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 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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Figure 1

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Published Papers (3 papers)

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