Linear Algebraic System Research Articles

Developing Kaczmarz method for solving Sylvester matrix equations

The Kaczmarz method is one of the most simple and computationally efficient iterative methods for solving large-scale linear algebraic system of equations Ax=b, which at each iteration of it, only one row of coefficient matrix need to be known. The aim of this paper is to derive the matrix form of Kaczmarz method for finding the solution of Sylvester matrix equation. First, we present an iterative algorithm for finding the solution of the matrix equation AX=F by extending the Kaczmarz method and obtain the convergence of it for any initial matrix. Then, by applying the hierarchical approach, we obtain a new iterative algorithm for solving the Sylvester matrix equation AX+XB=C. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed algorithm and compare with some similar gradient-based methods.

Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach

Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.

A time-splitting local meshfree approach for time-fractional anisotropic diffusion equation: application in image denoising

Image denoising approaches based on partial differential modeling have attracted a lot of attention in image processing due to their high performance. The nonlinear anisotropic diffusion equations, specially Perona–Malik model, are powerful tools that improve the quality of the image by removing noise while preserving details and edges. In this paper, we propose a powerful and accurate local meshless algorithm to solve the time-fractional Perona–Malik model which has an adjustable fractional derivative making the control of the diffusion process more convenient than the classical one. In order to overcome the complexities of the problem, a suitable combination of the compactly supported radial basis function method and operator splitting technique is proposed to convert a complex time-fractional partial differential equation into sparse linear algebraic systems that standard solvers can solve. The numerical results of classical and fractional models are explored in different metrics to demonstrate the proposed scheme’s effectiveness. The numerical experiments confirm that the method is suitable to denoise digital images and show that the fractional derivative increases the model’s ability to remove noise in images.

Double-Preconditioning Techniques for Fractional Partial Differential Equation Solvers

This paper is devoted to the numerical computation of the solution to stationary space fractional Partial Differential Equations (PDEs). To this end, we derive some algorithms for solving fractional algebraic linear systems resulting from the discretization of these space fractional PDEs. The proposed approach is based on an efficient computation of Cauchy’s integrals allowing to estimate positive real powers of a (sparse) matrix A. A first preconditioner M is used to reduce the length of the Cauchy integral contour enclosing the spectrum of MA, hence allowing in some cases for a large reduction of the number of quadrature nodes along the integral contour. Next, ILU-factorizations are used to efficiently solve the linear systems involved in the computation of approximate Cauchy’s integrals. Therefore, the strategy is independent of the way the Cauchy integrals are computed, although in this paper, for the sake of simplicity, we use standard quadrature rules. Based on this approach, we derive some solvers for stationary deterministic or stochastic space fractional Poisson equations which are tested in a series of numerical experiments.

Meso-scale method of asymptotic analysis of elastic vibrations in periodic and non-periodic multi-structures

Summary The method of meso-scale asymptotic approximations has proved to be very effective for the analysis of models of solids containing large clusters of defects, such as small inclusions or voids. Here, we present a new avenue where the method is extended to elastic multi-structures. Geometrically, a multi-structure makes a step up in the context of overall dimensions, compared to the dimensions of its individual constituents. The main mathematical challenge comes from the analysis of the junction regions assigned to the multi-structure itself. Attention is given to problems of vibration and on the coupling of vibration modes corresponding to displacements of different orientations. The method is demonstrated through the dynamic analysis of infinite or finite multi-scale asymmetric flexural systems consisting of a heavy beam connected to a non-periodic array of massless flexural resonators within some interval. In modelling the interaction between the beam and the resonators, we derive a vectorial system of partial differential equations through which the axial and flexural motions of the heavy beam are coupled. The solution of these equations is written explicitly in terms of Green’s functions having intensities determined from a linear algebraic system. The influence of the resonators on the heavy beam is investigated within the framework of scattering and eigenvalue problems. For large collections of resonators, dynamic homogenization approximations for the medium within the location of the resonant array are derived, leading to (i) the classical Rayleigh beam for symmetric systems and (ii) a generalized Rayleigh beam for asymmetric structures that support flexural–longitudinal wave coupling. Independent numerical simulations are also presented that demonstrate the accuracy of the analytical results.

Novel (2+1) and (3+1) forms of the Biswas–Milovic equation and optical soliton solutions via two efficient techniques

Purpose:The aim of this manuscript is to present the novel (2+1) and (3+1) forms of the Biswas–Milovic equation, which was introduced to the literature during 2010. The Biswas–Milovic equation is one of the important equation to model the soliton propagation through optical waveguides and it has been the subject of many researches. In order to obtain basic optical soliton solutions (bright, dark, singular) of the introduced equations the efficient the new Kudryashov and the unified Riccati equation expansion methods have been utilized. Methodology:After, introducing the nonlinear partial differential equations (NLPDEs) of the (2+1) and (3+1)-dimensional Biswas–Milovic equations ((2+1)-BME and (3+1)-BME), firstly, the necessary complex wave transformations are defined. Then nonlinear ordinary differential (NODE) forms of the BMEs are obtained by applying the complex wave transformations. Later, the algorithms of the new Kudryashov (nKM) and the unified Riccati equation expansion method (UREEM) are given and applied to this NODEs form. The polynomial form, the linear algebraic system, and the appropriate solution sets are obtained, respectively. Finally, optical soliton solutions of the introduced BME equations are gained by combining the serial form, solution function, solution sets, wave transform and the applied method. Findings:It has been shown that the introduced (2+1)-BME and (3+1)-BME equations produce optical soliton solutions, and the basic optical solitons are obtained by the proposed methods and presented graphically. Moreover, it has been seen that the BME equation is an effective equation not only in (1+1)-dimension but also in modeling (2+1) and (3+1)-dimensional optical soliton behavior in fiber. Originality:The (2+1)-BME and (3+1)-BME equations presented in the study have been introduced for the first time in this article.

On the examination of optical soliton pulses of Manakov system with auxiliary equation technique

Purpose:This manuscript addresses to investigate the optical soliton solution of the Manakov model which governs soliton transmission technology by utilizing the auxiliary equation technique. Methodology:After obtaining the nonlinear ordinary differential equation form (NODE) of investigated nonlinear partial differential equation problem (NLPDE) with the complex wave transform (cWT), the serial form is suggested in accordance with the utilized auxiliary method, and after written in the NODE form, the linear algebraic system is formed and by aid of the any symbolic calculation package the unknown parameters are obtained. In order to gain the soliton solutions, the proposed series form, the solution functions of the method, and the cWT are brought together and it is checked that the obtained solutions satisfy the main equation. By considering the functions that satisfy the main equation as solutions, graphical presentations are made and interpreted in order to better understand the obtained results. Findings:By applying the effective auxiliary equation method, optical soliton pulse solutions such as bright, singular, V-shaped, periodic singular, periodic bright, periodic bright–dark have been obtained, and it has been shown that the method can be applied effectively for such equations. Originality:The auxiliary equation method has not been applied to the Manakov model and the V-shaped, periodic singular, periodic bright soliton pulse solutions have not been reported before.

Mode-matching analysis of flexural trifurcated waveguide with porosity effects

The strong potential of porous media to absorb and change an incident sound wave interacting with a fluid that fills the pores of the material has sparked interest in their acoustic properties. Therefore, the current study describes the energy distribution and absorption mechanism of a dominating acoustics wave mode traveling out of the mouth of a semi-rigid waveguide. The underlying structure is tuned due to a trifurcated inlet region with flexible outer boundaries. In the outlet region of the duct, porosity effects for different porous materials are encountered. The continuity of pressure and velocity through the interface is used to match the scattering orthogonal and non-orthogonal duct modes. Mode matching techniques are used to compute the reflection and transmission coefficients. The resulting infinite linear algebraic systems are truncated and numerically solved. Of particular interest, the effects of considering different porous materials are observed and analyzed thoroughly. The obtained solution is confirmed to satisfy all geometrical conditions as well as the law of conservation of energy by mathematical and graphical representation.

On randomized partial block Kaczmarz method for solving huge linear algebraic systems

On randomized partial block Kaczmarz method for solving huge linear algebraic systems

On Non-Hermitian Positive (Semi)Definite Linear Algebraic Systems Arising from Dissipative Hamiltonian DAEs

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based on efficient three-term recurrences. We illustrate the performance of these iterative methods on several examples. The semidefinite case can be challenging and requires additional techniques to deal with the “singular part,” while the “positive definite part” can still be treated with the three-term recurrence methods.

UGKS-Based Implicit Iterative Method for Multiscale Nonequilibrium Flow Simulations

The unified gas-kinetic scheme (UGKS) has been developed for rarefied and continuum flow simulations. To further enhance the computational efficiency of the implicit UGKS (IUGKS) [Y. Zhu, C. Zhong, and K. Xu, J. Comput. Phys., 315 (2016), pp. 16--38] for the steady-state solution, a two-step IUGKS is proposed in this paper. The multiscale solution of the UGKS is determined by the integral solution of the kinetic model equation, which is composed of the Lagrangian integration of the equilibrium and the free particle transport of the nonequilibrium state. With the implicit evaluation of the macroscopic variables in the first iterative step, the integration of the equilibrium can be directly used in the flux calculation of macroscopic flow variables in the second iterative step. This is equivalent to including the viscous flux in the implicit scheme to accelerate the convergence of the solution instead of using the Euler flux in the iterative process of the original IUGKS. In the present IUGKS, the update of macroscopic flow variables is closely coupled with the implicit evolution of the gas distribution function. At the same time, to get a more accurate physical solution the full Boltzmann collision term has been incorporated into the current scheme through the penalty method. Different iterative techniques, such as lower-upper symmetric Gauss--Seidel and multigrid, are used for solving the linear algebraic system of coupled macroscopic and microscopic equations. The efficiency of the current IUGKS has reached an outstanding level among all implicit schemes for the kinetic equations in the literature. Several numerical examples are used to validate the performance of the IUGKS. Accurate solutions have been obtained efficiently in all cases from rarefied to continuum regimes and from low to hypersonic speed.

A highly accurate matrix method for solving a class of strongly nonlinear BVP arising in modeling of human shape corneal

This research study presents a novel highly accurate matrix approach for numerical treatments of a strongly nonlinear boundary value problem occurring in the modeling of the corneal shape of the human eye. Using the technique of quasilinearization, the nonlinear model is reduced into a sequence of linearized problems. Then, a spectral collocation procedure based on novel shifted Vieta‐Fibonacci (SVF) is applied to transform each subproblem into a linear algebraic system of equations. An upper bound for the error is provided, and convergence analysis of the SVF series solution in the weighted norm is discussed. The technique of error correction is used to improve the SVF polynomial solutions by means of the residual error function. Various numerical tests are provided to demonstrate the accuracy and efficiency of the presented collocation algorithm. The validation of the proposed approach is shown by comparison with available existing numerical solutions.

Least Squares Method for Solving Fuzzy LR Interval Algebraic Linear Systems

We first investigate the solvability conditions of fuzzy LR interval algebraic linear systems with fuzzy LR interval coefficient matrix and fuzzy LR interval hand-right vector. Next, we compute the solution of FLRIALS (fuzzy LR interval algebraic linear systems) using the least squares approach. We show that solving the least squares problem is equivalent to solving a quadratic programming problem. Indeed, instead of the FLRIALS, this quadratic programming has a non-convex constraint. Thus, we relax this constraint to have a convex search space. Then, from the solution obtained from quadratic programming on convex search space, we give an approximate solution for FLRIALS.

Анализ функции, формирующей алгебраическую систему в параметрической идентификации

Problems formulation for modern industrial systems is completely or partially reduced to the problem of parametric identification of dynamic objects. Success of solving such problems largely depends on the availability and volume of a priori information, such as input and output signals of the control object measured with noise. However, the solution of identification problem also requires derivatives of measured signals, and the obtaining of its values by numerical differentiation is an ill-posed problem. This paper considers the parametric identification problem, in which parameters estimation of a mathematical model of a linear dynamic object from experimentally obtained values of input and output signals is reduced to solving a linear algebraic system formed by integral convolution operators with analytically given forming functions. An approach to solve the numerical differentiation problem by using integration by parts in a system of linear algebraic equations forming is proposed. However, for this operation to be correct from the point of view of identification it is necessary that the forming functions should satisfy certain requirements of behavior in both the time and frequency domains. Thus, key task in the formation of a linear algebraic equation system is the choice of linearly independent functions that form this system. The paper proposes such a forming function. A detailed analysis of its properties and properties of its derivatives is presented. Experimental results obtained illustrate the correctness of using an operation of integration by parts instead of numerical differentiation of measured signals relating to the identification problem. The aim of the work is to study features of the formation of an algebraic system of equations, to analyze properties of the functions that form this system in detail, including the impact of correcting parameters of these functions and also to produce recommendations for their choice in parametric identification.

The small parameter method in the optimisation of a quasi-linear dynamical system problem

We consider an optimisation problem for the transient process in a quasi-linear dynamical system (contains a small parameter at non-linearities) with a performance index that is a linear combination of energy costs and the duration of the process. An algorithm for constructing asymptotic approximations of a given order to the solution of this problem is proposed. The algorithm is based on the asymptotic decomposition by integer powers of a small parameter of the initial values of adjoint variables and the duration of the process that are finite-dimensional elements, according to which the solution of the problem is easily restored. The computational procedure of the algorithm includes solving the problem of optimising the transient process in a linear dynamical system, integrating systems of linear differential equations, and finding the roots of non-degenerate linear algebraic systems. We also show how the constructed asymptotic approximations can be used to construct optimal control in the problem under consideration for a given value of a small parameter.

Dispersive optical solitons of Biswas–Arshed equation with a couple of novel approaches

Purpose:The importance of efficient analytical methods in solving nonlinear partial differential equations and nonlinear optical differential equations cannot be underestimated. Because of this importance, to solve efficiently above mentioned partial differential equations, a method has been suggested to use the improved Kudryashov and a sub equation form of new extended auxiliary equation methods. In addition the modified form of method has also been given. With this choice, it has been aimed to benefit from the advantages of improved Kudryashov and sub equation of extended auxiliary equation methods at the same time without spending additional processing and time in problem solving, to obtain more solution functions and soliton types offered by the methods. Methodology:As a first step to implement the proposed method, the nonlinear ordinary differential form (NODE) of the nonlinear partial differential equation has been obtained, the algorithms of the approached methods have been presented and applied to the Biswas–Arshed problem, which is an important equation in optic. Then, the polynomial form of the problem has been obtained, the solution function of the examined problem has been formed by choosing the appropriate sets and their substitutions in the proposed solution functions. It has been checked that the solution functions satisfy the Biswas–Arshed equation and then the graphical representation of the obtained results have been given. Findings:The application of the methods has carried out successfully, and both the linear algebraic equations system, the possible solution sets and the solution functions of the main equation, have been obtained seamlessly. It has been observed that both methods can be applied successfully and effectively and more results can be obtained at the same time, without spending an additional procedure and time. It can explicitly seen from both the obtained solution functions and the graphical results that the advantages of both methods can be used at the same time and can be applied to many nonlinear problems. Originality:In summary, the originality of this work is that the Biswas–Arshed equation has not been solved with both methods separately before, and it has not been presented in a study showing that both methods can be used together.

Dynamic responses of a plate with two circular cavities subjected to SH waves

This paper focuses on the scattering of SH waves by two circular cavities in an elastic plate. In the course of this study, wave function unfolding combined with the complex function, the mirror method, and the multi-polar coordinate translation technique are used to derive the incident and scattered conductive waves that fit the free conditions of stress on the upper and lower surface of the band domain. Then according to the stress free conditions of the edge of cylindrical hole, the infinite linear algebraic equations system is obtained. The unknown coefficients are finally calculated by intercepting the finite terms. The results show that when two cylindrical cavities of the same size are present within the plate, the presence of cavity 2, in most cases, leads to an increase in the concentration of kinetic stress along the edges of cavity 1; When two cylindrical cavity sizes are different, at medium and high frequency incidence, the enlargement of cavity 2 will lead to an increase in the kinetic stress concentration of cavity 1.

Existence and Uniqueness Theorems in the Inverse Problem of Recovering Surface Fluxes from Pointwise Measurements

Inverse problems of recovering surface fluxes on the boundary of a domain from pointwise observations are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness conditions on the data, geometric conditions on the location of measurement points, and the boundary of a domain. The proof relies on asymptotics of fundamental solutions to the corresponding elliptic problems and the Laplace transform. The inverse problem is reduced to a linear algebraic system with a nondegerate matrix.

A Parallel Strategy for Solving Sparse Linear Systems Over Finite Fields

In this paper we describe a number of parallel techniques that were applied to the problem of finding the null-spaces of thousands of large sparse matrices. This collection of matrices were derived from the discrete logarithm problem attack over the finite field $\F_{3^{6\cdot 509}}$ recently carried out by Adj et al. in~\cite{Adj2017}. Our software library was mainly executed in the supercomputer ABACUS~\cite{abacus}, where in total $21,870$ large sparse linear algebra systems were processed. Solving those linear algebra problems involved a computational effort of over $138$ core-years, requiring a memory space of over 645 gigabytes to store the corresponding vector solutions.

The method of fundamental solutions for scattering of electromagnetic waves by a chiral object

The scattering of a time-harmonic electromagnetic wave by a penetrable chiral obstacle in an achiral environment is considered. The method of fundamental solutions is employed in order to obtain numerically the solution of the problem using fundamental solutions in dyadic form. Surface vector potentials in terms of dyadic fundamental solutions together with the associated boundary integral operators are defined and their regularity properties are presented. Based on the dependence of the solution to the boundary data, appropriate systems of functions containing elements of dyadic fundamental solutions on the surface of the scatterer are constructed. Completeness and linear independence for these systems are proved with the usage of surface vector potentials. Using the transmission conditions, the scattering problem is transformed into a linear algebraic system with a coefficient matrix which consists of chiral and achiral blocks.