In this paper, we investigate the solutions to the dual quaternion matrix equation

, including the general solution, the solution containing only the standard part, and the solution containing only the dual part. First, we present the real representation matrix of the dual quaternion matrix and analyze the properties of its vector operator. Then we transform the dual quaternion matrix equation into a real linear system and give the equivalent condition for the existence of the solutions to the dual quaternion matrix equation

AXB=C

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, which includes the general solution, the solution containing only the standard part, and the solution containing only the dual part. Furthermore, we obtain the expressions of these solutions and the minimal

{\text{F}}^R

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-norm solutions to the dual quaternion matrix equation

AXB=C

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. Finally, we propose the algorithms and conduct numerical experiments to verify the validity of our method.

This paper further extends the shift-splitting (SS) and local shift-splitting (LSS) preconditioners to solve the general block two-by-two linear systems. We demonstrate that the eigenvalues of the corresponding preconditioned matrices cluster tightly around 2 by detailed spectral property analysis. Numerical experiments not only validate the theoretical results but also show the effectiveness and superiority of the SS and LSS preconditioners by comparing them with some existing preconditioners applied to the generalized minimal residual (GMRES) method for solving the block two-by-two linear systems.

We study a greedy coordinate descent method to solve large linear least-squares problems expanding on a randomized coordinate descent method presented by Leventhal and Lewis (Math. Oper. Res. 35: 641–654, 2010). For an overdetermined system, they proved its exponential convergence, regardless of its consistency. In our work, we study a greedy selection rule for the coordinate descent method which we refer to as the two-dimensional maximal residual Gauss-Seidel (D2MRGS) method. In this method, we select two coordinates in every iteration and treat the current approximation in those directions. Convergence is analyzed for the stated method and numerical experiments are provided to demonstrate its efficiency.

The strong vertex (edge) span of a given graph G is the maximum distance that two players can maintain at all times while visiting all vertices (edges) of G and moving either to an adjacent vertex or staying in the current position independently of each other. We introduce the notions of switching walks and the triod size of a tree, which are used to determine the strong vertex and the strong edge span of an arbitrary tree. The obtained results are used in an algorithm that computes the strong vertex (edge) span of the input tree in linear time.

In this paper, the backward Euler method and the shifted Grünwald-Letnikov formulas are utilized to discretize the space-fractional diffusion equations. The discretized result is a system of linear equations with a coefficient matrix being the sum of a diagonal matrix and a non-Hermitian Toeplitz matrix. By utilizing the Hermitian and skew-Hermitian splitting of the Toeplitz matrix, we develop a two-parameter DT

S iteration method to solve the linear systems. The convergence is also discussed. A DT

_{\text{HS}}^{}

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S-

\tau (\alpha ,\gamma )

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preconditioner is proposed and the preconditioned GMRES method combined with the proposed preconditioner is applied to solve the linear systems. The spectral analysis of the DT

_{\text{HS}}^{}

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S-

\tau (\alpha ,\gamma )

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preconditioned matrix is provided. Experimental results demonstrate the effectiveness of the proposed methods in solving the space-fractional diffusion equations.

Mathematical modelling is fundamental to understanding real-world phenomena. Despite the inherent complexity in designing such models, numerical approaches and, more recently, machine learning techniques, have emerged as powerful tools in this area. This work proposes integrating the finite element method (FEM) into forecasting and introduces parallel techniques for regression problems, with a specific focus on the use of Matérn kernels on local mesh support. This approach generalises the modelling based on radial basis function kernels and offers more flexibility to control the smoothness of the modelled functions. An exhaustive study explores the impact of diverse norms and Matérn kernel variations on the performance of models, and aims to improve the computational efficiency of the model fitting and prediction processes. Furthermore, a heuristic framework is introduced to derive optimal complexity parameters for each Matérn-based FEM kernel. The proposed parallel approaches use dynamic strategies, which significantly reduce the computational time of the algorithms compared to other methods and parallel computing techniques presented in recent years. The proposed methodology is assessed in the context of bias corrections for temperature forecasts made by the Local Data Assimilation and Prediction System (LDAPS) model. A comprehensive comparative analysis which includes machine learning algorithms provides significant insights into the training process, norm selection, and kernel choice, and shows that Matérn-based methods emerge as a choice to be considered for regression problems.

In this paper, we study the a posteriori error estimates of the conforming mixed method for the modified transmission eigenvalue problem proposed by Cogar et al. (Inverse Problems 33: 055010, 2017). We give the a posteriori error estimator of the approximate eigenpair, prove the reliability and efficiency of the estimator for the approximate eigenfunction, and present the reliability of the estimator for the approximate eigenvalue. We also implement adaptive computation and exhibit the numerical experiments which show that the approximate eigenvalues obtained by the adaptive computation reach the optimal convergence order.

In this paper, a novel augmented Lagrangian preconditioner based on the global Arnoldi for accelerating the convergence of Krylov subspace methods is applied to linear systems of equations with a block three-by-three structure, and these systems typically arise from discretizing the Stokes equations using mixed finite-element methods. Spectral analyses are established for the exact versions of these preconditioners. Finally, the obtained numerical results claim that our novel approach is more efficient and robust for solving the discrete Stokes problems. The efficiency of our new approach is evaluated by measuring the computational time.

In this paper, we consider deriving some new gradient-based iterative (GI)-like algorithms for solving a class of Sylvester tensor equations, which often arise from control systems and image processing. We first study the optimal parameter and the iteration matrix’s minimal spectral radius of the relaxed GI (RGI) algorithm proposed by Zhang and Wang (Taiwan J Math 26: 501–519, 2022) in terms of matricization of a tensor and straightening operator. Then based on the Jacobi method, by using the diagonal matrices to replace the system matrices in mode products contained in the RGI and the modified RGI (MRGI) algorithms (Taiwan J Math 26: 501–519, 2022), we design the Jacobi RGI (JRGI) and improved MRGI (IMRGI) algorithms for the Sylvester tensor equations, which require less computational load and are more efficient than the RGI and the MRGI ones, respectively. We deduce the sufficient convergence condition and quasi-optimal parameter of the JRGI algorithm, and sufficient and necessary conditions for the convergence of the IMRGI algorithm. Furthermore, we apply a new update strategy to the RGI algorithm and develop an updated RGI (URGI) algorithm for the Sylvester tensor equations, which is different from the MGI (Math Probl Eng 819479: 1–7, 2013) and the MRGI ones. The URGI algorithm takes full advantage of the latest computed results and returns better convergence behavior than RGI, MGI, and MRGI ones. Also, we prove that the proposed URGI algorithm is convergent under proper conditions. Finally, numerical experiments are performed to verify that the proposed algorithms are efficient and have advantages over some existing ones.

In this work, modified Patankar-Runge-Kutta (MPRK) schemes up to order four are considered and equipped with a dense output formula of appropriate accuracy. Since these time integrators are conservative and positivity-preserving for any time step size, we impose the same requirements on the corresponding dense output formula. In particular, we discover that there is an explicit first-order formula. However, to develop a boot-strapping technique, we propose to use implicit formulae which naturally fit into the framework of MPRK schemes. In particular, if lower-order MPRK schemes are used to construct methods of higher order, the same can be done with the dense output formulae we propose in this work. We explicitly construct formulae up to order three and demonstrate how to generalize this approach as long as the underlying Runge-Kutta method possesses a dense output formula of appropriate accuracy. We also note that even though linear systems have to be solved to compute an approximation for intermediate points in time using these higher-order dense output formulae, the overall computational effort to reach a given number of approximations is reduced compared to using the scheme with a smaller step size. We support this fact and our theoretical findings by means of numerical experiments.

In this article, for a degenerate inhomogeneous equation of even order with a fractional derivative in the sense of Caputo, a boundary value problem of the Robin problem type is studied. The solution is constructed in the form of a series of eigenfunctions of a one-dimensional spectral problem for a degenerate equation of even order. The existence and positivity of eigenvalues are shown by reducing the spectral problem to an equivalent integral equation with a symmetric kernel. Also, when constructing the posed problem, a boundary value problem for a one-dimensional fractional order equation was studied. Depending on the sign of the constant coefficient of the equation q, the necessary estimates for the solution were obtained. Sufficient conditions for the convergence of the series, which is a solution to the Robin problem, and the series obtained by differentiation are found. The uniqueness of the solution is demonstrated by the spectral method.

Energy-conserving Hermite methods for solving Maxwell’s equations in dielectric and dispersive media are described and analyzed. In three space dimensions, methods of order 2m to

require

{(m+1)}^3

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degrees-of-freedom per node for each field variable and can be explicitly marched in time with steps independent of m. We prove the stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special semi-norm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of the electromagnetic wave propagation over thousands of wavelengths.

Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates. In this paper, the immersed interface method (IIM) tailored for Navier-Stokes equations involving interfaces under the cylindrical coordinates has been developed. Note that, while the IIM has been developed for Stokes equations in the cylindrical coordinates assuming the axis-symmetry in the literature, there is a gap in dealing with Navier-Stokes equations, where the non-linear term includes an additional component involving the coordinate

, even if the geometry and force term are axis-symmetric. Solving the Navier-Stokes equations in cylindrical coordinates becomes challenging when dealing with interfaces that feature a discontinuous pressure and a non-smooth velocity, in addition to the pole singularity at

r=0

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. In the newly developed algorithm, we have derived the jump conditions under the cylindrical coordinates. The numerical algorithm is based on a finite difference discretization on a uniform and staggered grid in the cylindrical coordinates. The finite difference scheme is standard away from the interface but is modified at grid points near and on the interface. As expected, the method is shown to be second-order accurate for the velocity. The developed new IIM is applied to the solution of some related fluid dynamic problems with interfaces.

In this paper, we investigate the instability of growing tumors by employing both analytical and numerical techniques to validate previous results and extend the analytical findings presented in a prior study by Feng et al. (Z Angew Math Phys 74:107, 2023). Building upon the insights derived from the analytical reconstruction of key results in the aforementioned work in one dimension and two dimensions, we extend our analysis to three dimensions. Specifically, we focus on the determination of boundary instability using perturbation and asymptotic analysis along with spherical harmonics. Additionally, we have validated our analytical results in a two-dimensional (2D) framework by implementing the Alternating Direction Implicit (ADI) method. Our primary focus has been on ensuring that the numerical simulation of the propagation speed aligns accurately with the analytical findings. Furthermore, we have matched the simulated boundary stability with the analytical predictions derived from the evolution function, which will be defined in subsequent sections of our paper. This alignment is essential for accurately determining the stability or instability of tumor boundaries.

An upwind weak Galerkin finite element scheme was devised and analyzed in this article for convection-dominated Oseen equations. The numerical algorithm was based on the weak Galerkin method enhanced by upwind stabilization. The resulting finite element scheme uses equal-order, say k, polynomial spaces on each element for the velocity and the pressure unknowns. With finite elements of order

, the numerical solutions are proved to converge at the rate of

O(h^{k+\frac{1}{2}})

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in an energy-like norm for convection-dominated Oseen equations. Numerical results are presented to demonstrate the accuracy and effectiveness of the upwind weak Galerkin scheme.