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Formerly the geometry was based on shapes, but since the last centuries this founding mathematical science deals with transformations, projections, and mappings. Projective geometry identifies a line with a single point, like the perspective on the horizon line and, due to this fact, it requires a restructuring of the real mathematical and numerical analysis. In particular, the problem of interpolating data must be refocused. In this paper, we define a linear structure along with a metric on a projective space, and prove that the space thus constructed is complete. Then, we consider an iterated function system giving rise to a fractal interpolation function of a set of data. [ABSTRACT FROM AUTHOR]

In this paper, we provide a numerical solution for the nonlinear tempered fractional integrodifferential equation in three dimensions. We use the trapezoidal convolution rule with backward differences (BDF2) for temporal discretization, and develop an alternating direction implicit difference scheme for spatial discretization. A novel fast approximation is applied to tackle the nonlinear term. The stability and convergence analysis of the numerical scheme are analyzed. Furthermore, some numerical experiments are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The best polynomial approximation and the Chebyshev approximation are both important in numerical analysis. In tradition, the best approximation is regarded as better than the Chebyshev approximation, because it is usually considered in the uniform norm. However, it is not always superior to the latter, as noticed by Trefethen (Math. Today 47:184–188, 2011) for the function f (x) = | x - 0.25 | . Recently, Wang (arxiv, 2106.03456, 2021) observed a similar phenomenon for a function with an algebraic singularity and proved it in theory. In this paper, we find that for functions with logarithmic endpoint singularities, the pointwise errors of the Chebyshev approximation are smaller than those of the best approximations of the same degree, except only at the narrow boundary layer. The pointwise error for the Chebyshev series truncated at the degree n is O (n - κ) ( κ = min { 2 γ + 1 , 2 δ + 1 } ), but increases as O (n - (κ - 1)) at the endpoint singularities. Theorems are given to explain this effect. [ABSTRACT FROM AUTHOR]

Approximation of a multivariate function is an important theme in the field of numerical analysis and its applications, which continues to receive a constant attention. In this paper, we provide a parameterized family of self-referential (fractal) approximants for a given multivariate smooth function defined on an axis-aligned hyper-rectangle. Each element of this class preserves the smoothness of the original function and interpolates the original function at a prefixed gridded data set. As an application of this construction, we deduce a fractal methodology to approach a multivariate Hermite interpolation problem. This part of our paper extends the classical bivariate Hermite's interpolation formula by Ahlin (Math. Comp. 18:264–273, 1964) in a twofold sense: (i) records, in particular, a multivariate generalization of this bivariate interpolation theory; (ii) replaces the unicity of the Hermite interpolant with a parameterized family of self-referential Hermite interpolants which contains the multivariate analogue of Ahlin's interpolant as a particular case. Some related aspects including the approximation by multivariate self-referential functions preserving Popoviciu convexity are given, too. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical positivity, convergence and dynamical behaviors (including extinction and persistence) for stochastic SIS model. Compared with the existing numerical methods, a linearly backward Euler method with truncated Wiener process is introduced with a less computational cost and a better dynamic behavior. We discuss the numerical positivity by the truncated Wiener process, which is the basis for the investigation of convergence and dynamic behavior. The numerical dynamical behaviors (extinction and persistence) are obtained by an exponential representation for the nonlinear stochastic stability function and the large number theorem for martingale, which reproduces the existing theoretical results of exact solution. Finally, numerical examples are given to validate our numerical results for stochastic SIS model. [ABSTRACT FROM AUTHOR]

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension d by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the μ-mode product (also known as tensor-matrix product or mode-n product) and related operations, such as the Tucker operator. Their MathWorks MATLAB®/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

Exponential integrators, due to their robust stability properties, have been considered as reliable schemes for numerical solutions of stiff systems. In this paper, we propose generalized exponential time differencing (GETD) schemes for nonlinear fractional differential equations of order α ∈ (0,1). First, we improve the suboptimal performance of the multistep GETD schemes. Using graded mesh, uniform optimal convergence rates under no additional smoothness requirements are obtained. Second, we develop and analyze novel second-order and third-order accurate predictor-corrector type GETD schemes. Using linear stability analysis and numerical illustrations, we demonstrate that the newly introduced schemes have better stability properties than the multistep GETD schemes. Partial fraction decompositions of global Padé approximations for Mittag-Leffler function are used for efficient implementation. Numerical examples involving nonlinear scalar equations and stiff systems are provided to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider ℓ2,p(0 < p < 1) regularized total variation with overlapping group sparsity prior for image restoration with impulse noise. The proposed prior inherits the advantages of a total variation regularizer to preserve edges and reduce the staircase effect, meanwhile promoting group-level sparseness. Since the new model is nonsmooth and nonconvex, we utilize the proximal alternate minimization method to solve it by drawing support from the half-quadratic scheme to deal with the constituted ℓ2-ℓ2,p subproblems. In addition, we also provide the convergence analysis for the used numerical methods. Experimental results demonstrate that the new approach outperforms representative gradient-based methods in terms of both visual perception and numerical indexes. [ABSTRACT FROM AUTHOR]

Zeroing dynamics (ZD) has shown great performance to solve various time-varying problems. In this paper, the problem of time-varying Sylvester-transpose matrix inequality is first investigated. Since it is difficult to solve a matrix inequality with a matrix variable and its transpose by traditional methods, this paper proposes a continuous ZD (CZD) model by employing ZD design formula and JMP function array to solve this challenging problem. Furthermore, for better implementation on digital computers, three discrete ZD (DZD) models are proposed by using three different discretization formulas with different precision, i.e., the Euler-forward formula, the 6-instant Zhang et al discretization (ZeaD) formula and the 7-instant ZeaD formula. What is more, theoretical truncation error analyses and numerical experiments substantiate the convergence, efficacy and superiority of the DZD models for solving time-varying Sylvester-transpose matrix inequality. [ABSTRACT FROM AUTHOR]

This paper deals with the design and analysis of a high-order numerical scheme for the two-dimensional time-fractional reaction-subdiffusion equation. The time-fractional derivative of order α, (α ∈ (0,1)) in the model problem is approximated by means of the L2 − 1σ scheme and the space derivatives are discretized by means of a compact alternating direction implicit (ADI) finite difference scheme. Convergence and stability of the method are analyzed. Five numerical examples are provided to demonstrate the applicability and accuracy of the method. It is shown that the method is unconditionally stable and is of O (Δ t 1 + α + h x 4 + h y 4) accuracy, where Δt is the temporal step size and hx and hy are the spatial step sizes. The computed results are in good agreement with the theoretical analysis. Moreover, the comparison with the corresponding results of existing methods demonstrates that our method has advantages in convergence order and error improvement when solving the time-fractional reaction-subdiffusion equation. The CPU time consumed by the proposed method is provided. [ABSTRACT FROM AUTHOR]

In this paper, we present a global-in-time non-overlapping Schwarz method applied to the Ekman boundary layer problem. Such a coupled problem is representative of large-scale atmospheric and oceanic flows in the vicinity of the air-sea interface. Schwarz waveform relaxation (SWR) algorithms provide attractive methods for ensuring a "tight coupling" between the ocean and the atmosphere. However, the convergence study of such algorithms in this context raises a number of challenges. Numerous convergence studies of Schwarz methods have been carried out in idealized settings, but the underlying assumptions to make these studies tractable may prohibit them to be directly extended to the complexity of climate models. We illustrate this aspect on the coupled Ekman problem, which includes several essential features inherent to climate modeling while being simple enough for analytical results to be derived. We investigate its well-posedness and derive an appropriate SWR algorithm. Sufficient conditions for ensuring its convergence for different viscosity profiles are then established. Finally, we illustrate the relevance of our theoretical analysis with numerical results and suggest ways to improve the computational cost of the coupling. Our study emphasizes the fact that the convergence properties can be highly sensitive to some model characteristics such as the geometry of the problem and the use of continuously variable viscosity coefficients. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up behavior of numerical solutions to nonlinear fractional ordinary differential equations with a dissipative term. Based on the positivity preservation of the explicit L1-scheme, it is shown that for sufficiently small initial values, numerical solutions exist globally. Whereas for large initial values, numerical solutions with a suitable adaptive step strategy blow up in finite time. Finally, some numerical experiments are provided for verifying the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is devoted to the development of an innovative Matlab software, dedicated to the numerical analysis of two-dimensional elliptic problems, by means of the probabilistic approach. This approach combines features of the Monte Carlo random walk method with discretization and approximation techniques, typical for meshless methods. It allows for determination of an approximate solution of elliptic equations at the specified point (or group of points), without a necessity to generate large system of equations for the entire problem domain. While the procedure is simple and fast, the final solution may suffer from both stochastic and discretization errors. The attached Matlab software is based on several original author's concepts. It permits the use of arbitrarily irregular clouds of nodes, non-homogeneous right-hand side functions, mixed type of boundary conditions as well as variable material coefficients (of anisotropic materials). The paper is illustrated with results of analysis of selected elliptic problems, obtained by means of this software. [ABSTRACT FROM AUTHOR]

In this paper, four linearly energy-preserving finite difference methods (EP-FDMs) are designed for two-dimensional (2D) nonlinear coupled sine-Gordon equations (CSGEs) and coupled Klein-Gordon equations (CKGEs) using the invariant energy quadratization method (IEQM). The 1st EP-FDM is designed by first introducing two auxiliary functions to rewrite the original problems into the new system only including the 1st-order temporal derivatives, and then applying Crank-Nicolson (C-N) method and 2nd-order centered difference methods for the discretizations of temporal and spatial derivatives, respectively. The 2nd EP-FDM is directly devised based on the uses of 2nd-order centered difference methods to approximate 2nd-order temporal and spatial derivatives. The 1st and 2nd EP-FDMs need numerical solutions of the algebraic system with variable coefficient matrices at each time level. By modifying the 2nd EP-FDM, the 3rd EP-FDM, which is implemented by computing the system of algebraic equations with constant coefficient matrices at each time level, is developed. Finally, an energy-preserving alternating direction implicit (ADI) finite difference method (EP-ADI-FDM) is established by a combination of ADI method with the 3rd EP-FDM. By using the discrete energy method, it is shown that they are all uniquely solvable, and their solutions have a convergent rate of 𝒪 (Δ t 2 + h x 2 + h y 2) in H1-norm and satisfy the discrete conservative laws. Numerical results show the efficiency and accuracy of them. [ABSTRACT FROM AUTHOR]

In a previous publication, we have used the discrete Fourier transform to compute the binomial coefficients. In the present paper, we extend the previously proposed method to compute the multinomial coefficients, analyse its precision and performance. The other methods, analysed in our previous publication, are also extended to the multinomial case. The FFT method presents the best performance to compute all multinomial coefficients at a given level. [ABSTRACT FROM AUTHOR]

In this paper, based on two-grid discretizations, two kinds of local and parallel finite element methods are proposed and investigated for the coupled Stokes/Darcy model. Following the idea presented in Xu and Zhou (Math. Comput. 69, 881–909 1999). a classical local and parallel finite element method is proposed and investigated. To derive global continuous approximations, a new local and parallel finite element method is devised by combining the partition of unity. We theoretically analyze the resulting formulations and derive optimal error estimates. Numerical experiments are reported to assess the theoretical results. [ABSTRACT FROM AUTHOR]

Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula. [ABSTRACT FROM AUTHOR]

In this paper, we trace back the genesis of Aitken's Δ2 process and Shanks' sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks' transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the modified proximal point algorithm for common fixed points of asymptotically quasi-nonexpansive mappings in CAT(0) spaces and also prove some convergence theorems of the proposed algorithm to a common fixed point of asymptotically quasi-nonexpansive mappings and a minimizer of a convex function. The main results in this paper improve and generalize the corresponding results given by some authors. Moreover, we then give numerical examples to illustrate and show efficiency of the proposed algorithm for supporting our main results. [ABSTRACT FROM AUTHOR]

The Schröder iterative families of the first and second kind are of great importance in the theory and practice of iterative processes for solving nonlinear equations f(x) = 0. In both cases, the methods Er (first kind) and Sr (second kind) converge locally to a zero α of f as O(|xk − α|r). Although characteristics of these families have been studied in many papers, their dynamic and chaotic behavior has not been completely investigated. In this paper, we compare convergence properties of both iterative schemes using the two methodologies: (i) comparison by numerical examples and (ii) comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. Apart from the visualization of iterative processes, basins of attraction reveal very useful features on iterations such as consumed CPU time and average number of iterations, both as functions of starting points. We demonstrate by several examples that the Schröder family of the second kind Sr possesses better convergence characteristics than the Schröder family of the first kind Er. [ABSTRACT FROM AUTHOR]

Efficient and accurate numerical schemes for solving the Helmholtz equation are critical to the success of various wave propagation–related inverse problems, for instance, the full-waveform inversion problem. However, the numerical solution to a multi-dimensional Helmholtz equation is notoriously difficult, especially when a perfectly matched layer (PML) boundary condition is incorporated. In this paper, an optimal 13-point finite difference scheme for the Helmholtz equation with a PML in the two-dimensional domain is presented. An error analysis for the numerical approximation of the exact wavenumber is provided. Based on error analysis, the optimal 13-point finite difference scheme is developed so that the numerical dispersion is minimized. Two practical strategies for selecting optimal parameters are presented. Several numerical examples are solved by the new method to illustrate its accuracy and effectiveness in reducing numerical dispersion. [ABSTRACT FROM AUTHOR]

One problem that arises in computation involving large numbers is precision. In certain situations, the result might be represented by the standard data type, but arithmetic precision might be compromised when dealing with large numbers in the course to the result. Binomial coefficients are an example that suffer from this torment. In the present paper, we review numerical methods to compute the binomial coefficients: Pascal's recursive method; prime factorization to cancel out terms; gamma function approximation; and a simplified iterative method that avoids loss in precision. Acknowledging that the binomial coefficients might be obtained by a successive convolution of a simple discrete rectangular function, we propose a different approach based on the discrete Fourier transform and another based on its fast implementation. We analyze and compare performance and precision of all of them. The proposed methods overcome the previous ones when computing all coefficients for a given level n, attaining better precision for large values of n. [ABSTRACT FROM AUTHOR]

In this paper, we consider product integration method based on orthogonal polynomials to solve mixed system of Volterra integral equations of the first and second kind with weakly singular kernels. For investigation of the theoretical and numerical analysis of the mixed systems, the notions of the tractability index and ν-smoothing property are extended for a weakly singular Volterra integral operator. Convergence analysis of the product integration method is derived. Finally, the proposed method is illustrated by two examples, which confirm the theoretical prediction of the error estimation. [ABSTRACT FROM AUTHOR]

In the paper, a space fractional Benjamin-Bona-Mahony (BBM) equation is proposed. For the direct problem, we develop the Chebyshev-Legendre spectral scheme for the proposed equation. The given approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the presented method, the computational complexity is reduced and both accuracy and efficiency are achieved compared with the Legendre spectral method. Stability and convergence analysis of the numerical method are proven. For the inverse problem, the Bayesian method is developed to estimate some relevant parameters based on the spectral format of the direct problem. The convergence analysis on the Kullback-Leibler distance between the true posterior distribution and the approximation is derived. Some numerical experiments are included to demonstrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper introduces an efficient unconditionally stable fourth-order method for solving nonlinear space-fractional reaction-diffusion systems with nonhomogeneous Dirichlet boundary conditions on bounded domains. The proposed method is based on a compact improved matrix transformed technique for fourth-order spatial approximation and exponential time differencing approximation for fourth-order time integration. The main advantage of the improved matrix transfer technique is that it leads to a system of ordinary differential equations with spatial discretization matrix raised to the desired fractional order. The key benefit of the fourth-order exponential integrator is that it can be implemented with essentially the same computational complexity as the backward Euler method by utilizing a partial fraction splitting technique in which it is just required to solve two backward Euler-type well-conditioned linear systems at each time step by computing LU decomposition of spatial discretization matrix once outside the time loop. Linear stability analysis and various numerical experiments are also performed to demonstrate stability and accuracy of the proposed method. Moreover, calculation of local truncation error and an empirical convergence analysis show the fourth-order accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present dynamical viewpoints under the Möbius conjugacy map on the Riemann sphere for a uniparametric family of optimal eighth-order multiple-root finders. Various conjugacy properties are investigated including the invariance of the fixed point and its multiplier, which enables the family of iterative methods to trace along the consistent orbits in position. The parameter spaces and dynamical planes are studied and illustrated to visualize the periodic components and their geometric properties. Both theoretical and computational analyses along with a numerical algorithm are carried out regarding the bifurcation points of satellite and primitive components. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution of a nonlinear first-order singularly perturbed differential equation with integral boundary condition is considered. The discrete method is generated by a backward Euler formula and the grid is obtained by equidistributing a monitor function based on arc-length. We first give a rigorous error analysis for the numerical method of this problem on a grid that is constructed adaptively from a knowledge of the exact solution. A first-order rate of convergence, independent of the perturbation parameter, is established. Then, an a posteriori error bound and the corresponding convergence result are derived for the presented numerical scheme on an adaptive grid, which is constructed adaptively from a discrete approximation of the exact solution. At last, numerical experiments are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We describe an algorithm to evaluate all the complex branches of the Lambert W function with rigorous error bounds in arbitrary-precision interval arithmetic or ball arithmetic. The classic 1996 paper on the Lambert W function by Corless et al. provides a thorough but partly heuristic numerical analysis of the Lambert W function which needs to be complemented with some explicit inequalities and practical observations about managing precision and branch cuts. An implementation is provided in the Arb library. [ABSTRACT FROM AUTHOR]

In this paper, an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem, we provide the optimal error estimate. [ABSTRACT FROM AUTHOR]

In Andreani et al. (Numer. Algorithms 57:457-485, 2011), an interior point method for the horizontal nonlinear complementarity problem was introduced. This method was based on inexact Newton directions and safeguarding projected gradient iterations. Global convergence, in the sense that every cluster point is stationary, was proved in Andreani et al. (Numer. Algorithms 57:457-485, 2011). In Andreani et al. (Eur. J. Oper. Res. 249:41-54, 2016), local fast convergence was proved for the underdetermined problem in the case that the Newtonian directions are computed exactly. In the present paper, it will be proved that the method introduced in Andreani et al. (Numer. Algorithms 57:457-485, 2011) enjoys fast (linear, superlinear, or quadratic) convergence in the case of truly inexact Newton computations. Some numerical experiments will illustrate the accuracy of the convergence theory. [ABSTRACT FROM AUTHOR]

This short paper concludes a general waveform relaxation (WR) method at the PDE level for semi-linear reaction-diffusion equations. For the case of multiple coupled PDE(s), new Jacobi WR and Gauss-Seidel WR are provided to accelerate the convergence result of classical WR. The convergence conditions are proved based on energy estimate. Numerical experiments are demonstrated with several WR methods in parallel to verify the effectiveness of the general WR method. [ABSTRACT FROM AUTHOR]

In this paper, a novel technique incorporated the homotopy analysis method (HAM) with Coiflets is developed to obtain highly accurate solutions of the Föppl-von Kármán equations for large bending deflection. The characteristic scale transformation is introduced to nondimensionalize the governing equations. The results are obtained for the transformed nondimensional equations, which are in very excellent agreement with analytical ones or numerical benchmarks performing good efficiency and validity. Besides, we notice the nonlinearity of the Föppl-von Kármán equations is closely connected with the load and length-width ratio of the plate. For the case of the plate suffering tremendous loads, the traditional linear theory does not work, while our Coiflets solutions are still very accurate. It is expected that our proposed approach not only keeps the outstanding merits of the HAM technique for handling strong nonlinearity, but also improves on the computational efficiency to a great extent. [ABSTRACT FROM AUTHOR]

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem Ax = λBx, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begović Kovač (Trans. Numer. Anal. (ETNA) 47, 107-147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of ΔAAΔAand ΔBBΔBare small, for some nonsingular diagonal matrices ΔAand ΔB. [ABSTRACT FROM AUTHOR]

The paper presents explicit interval multistep methods of Milne type, which may be considered as alternative methods to other known explicit interval multistep methods (of Adams-Bashforth and Nyström). It is proved that enclosures of solutions (in the form of intervals) obtained by these methods contain the exact solutions of the initial value problem. Numerical examples show that the widths of intervals obtained by proposed methods are smaller than those obtained by explicit interval multistep methods known so far. [ABSTRACT FROM AUTHOR]

In this paper, a fractional temporal SEIR measles model is considered. The model consists of four coupled time fractional ordinary differential equations. The time-fractional derivative is defined in the Caputo sense. Firstly, we solve this model by solving an approximate model that linearizes the four time fractional ordinary differential equations (TFODE) at each time step. Secondly, we derive an analytical solution of the single TFODE. Then, we can obtain analytical solutions of the four coupled TFODE at each time step, respectively. Thirdly, a computationally effective fractional Predictor-Corrector method (FPCM) is proposed for simulating the single TFODE. And the error analysis for the fractional predictor-corrector method is also given. It can be shown that the fractional model provides an interesting technique to describe measles spreading dynamics. We conclude that the analytical and Predictor-Corrector schemes derived are easy to implement and can be extended to other fractional models. Fourthly, for demonstrating the accuracy of analytical solution for fractional decoupled measles model, we applied GMMP Scheme (Gorenflo-Mainardi-Moretti-Paradisi) to the original fractional equations. The comparison of the numerical simulations indicates that the solution of the decoupled and linearized system is close enough to the solution of the original system. And it also indicates that the linearizing technique is correct and effective. [ABSTRACT FROM AUTHOR]

A large number of free boundary problems can be formulated as linear-complementarity problems. In this paper, we propose an inexact alternating direction method of multipliers for solving linear complementarity problem arising from free boundary problems by using the special structure of these problems. The convergence of our proposed method is proved. Numerical results show that the proposed method is feasible and effective, and it is significantly faster than modified alternating direction implicit algorithm and many other methods, especially when dimension of the problem being solved is large. [ABSTRACT FROM AUTHOR]

Quasi-Monte Carlo (QMC) rules are equal weight quadrature rules for approximating integrals over the s-dimensional unit cube [0, 1]. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on [0, 1] with square integrable partial mixed derivatives of order α. Using Parseval's identity, this smoothness can be defined for all real numbers α > 1/2. In this setting, the condition α > 1/2 is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with α arbitrarily close to 1/2. To obtain such estimates we introduce a log-scale for function spaces with smoothness close to 1/2, which we call log-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order $\mathcal {O}(N^{-1/2} (\log N)^{-\mu (1-\lambda )/2})$ for any 1/ μ < λ ≤ 1, where μ > 1 is a certain power in the log-scale. A second result is concerned with tractability of numerical integration for weighted Korobov spaces with product weights $(\gamma _{j})_{j \in \mathbb {N}}$. Previous results have shown that if ${\sum }_{j=1}^{\infty } \gamma _{j}^{\tau } < \infty $ for some 1/(2 α) < τ ≤ 1 one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where τ is close to 1/(2 α), namely we show dimension independent error bounds under the condition that ${\sum }_{j=1}^{\infty } \gamma _{j} \max \{1, \log \gamma _{j}^{-1}\}^{\mu (1-\lambda )} < \infty $ for some 1/ μ < λ ≤ 1. The essential tool in our analysis is a log-scale Jensen's inequality.The results described above also apply to integration in log-cosine spaces using tent-transformed lattice rules. [ABSTRACT FROM AUTHOR]

Krylov subspace methods for approximating a matrix function f( A) times a vector v are analyzed in this paper. For the Arnoldi approximation to e v, two reliable a posteriori error estimates are derived from the new bounds and generalized error expansion we establish. One of them is similar to the residual norm of an approximate solution of the linear system, and the other one is determined critically by the first term of the error expansion of the Arnoldi approximation to e v due to Saad. We prove that each of the two estimates is reliable to measure the true error norm, and the second one theoretically justifies an empirical claim by Saad. In the paper, by introducing certain functions ϕ( z) defined recursively by the given function f( z) for certain nodes, we obtain the error expansion of the Krylov-like approximation for f( z) sufficiently smooth, which generalizes Saad's result on the Arnoldi approximation to e v. Similarly, it is shown that the first term of the generalized error expansion can be used as a reliable a posteriori estimate for the Krylov-like approximation to some other matrix functions times v. Numerical examples are reported to demonstrate the effectiveness of the a posteriori error estimates for the Krylov-like approximations to e v, cos( A) v and sin( A) v. [ABSTRACT FROM AUTHOR]

Solid wall boundary conditions have been an area of active research within the context of Smoothed Particle Hydrodynamics (SPH) for quite a while. Ferrand et al. (Int. J. Numer. Methods Fluids 71(4), 446-472, 2012) presented a novel approach using a renormalization factor in the SPH approximation. The computation of this factor depends on an integral along the boundary of the domain and in their original paper Ferrand et al. gave an analytical formulation for the 2-D case using the Wendland kernel. In this paper the formulation will be extended to 3-D, again providing analytical formulae. Due to the boundary being two dimensional a domain decomposition algorithm needs to be employed in order to obtain special integration domains. For these the analytical formulae will be presented when using the Wendland kernel. The algorithm presented within this paper is applied to several academic test-cases for which either analytical results or simulations with other methods are available. It will be shown that the present formulation produces accurate results and provides a significant improvement compared to approximative methods. [ABSTRACT FROM AUTHOR]

For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSS) preconditioner as well as the MGSS iteration method is derived in this paper, which generalize the modified shift-splitting (MSS) preconditioner and the MSS iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 317:535-546, 2017), respectively. The convergent and semi-convergent analyses of the MGSS iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. Meanwhile, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSS preconditioner and the MGSS iteration method and also illustrate that the MGSS iteration method outperforms the generalized shift-splitting (GSS) and the generalized modified shift-splitting (GMSS) iteration methods, and the MGSS preconditioner is superior to the shift-splitting (SS), GSS, modified SS (M-SS), GMSS and MSS preconditioners for the generalized minimal residual (GMRES) method for solving the nonsymmetric saddle point problems. [ABSTRACT FROM AUTHOR]

In this paper, we mainly introduce a partitioned scheme based on Gauge-Uzawa finite element method for the 2D time-dependent incompressible magnetohydrodynamics (MHD) equations. It is a fully decoupled projection method which combines the Gauge and Uzawa methods within a variational formulation. Firstly, the temporal discretization is based on backward Euler technique for the linear term and semi-implicit scheme for the nonlinear term. Secondly, the spatial approximation of fluid velocity, hydrodynamic pressure, and magnetic field apply the mixed element method. Finally, the validity, reliability, and accuracy of the proposed algorithms are supported by numerical experiments. [ABSTRACT FROM AUTHOR]

The multiple-sets split equality problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domains of two linear operators as well as in the operators’ ranges simultaneously. Although, for the split equality problem, there exist many algorithms, there are but few algorithms for the multiple-sets split equality problem. Hence, in this paper, we present a relaxed two points projection method to solve the problem; under some suitable conditions, we show the weak convergence and give a remark for the strong convergence method in the Hilbert space. The interest of our algorithm is that we transfer the problem to an optimization problem, then, based on the model, we present a modified gradient projection algorithm by selecting two different initial points in different sets for the problem (we call the algorithm as two points algorithm). During the process of iteration, we employ subgradient projections, not use the orthogonal projection, which makes the method implementable. Numerical experiments manifest the algorithm is efficient. [ABSTRACT FROM AUTHOR]

A method to define trivariate spline quasi-interpolation operators (QIOs) is developed by blending univariate and bivariate operators whose linear functionals allow oversampling. In this paper, we construct new operators based on univariate B-splines and bivariate box splines, exact on appropriate spaces of polynomials and having small infinity norms. An upper bound of the infinity norm for a general blending trivariate spline QIO is derived from the Bernstein-Bézier coefficients of the fundamental functions associated with the operators involved in the construction. The minimization of the resulting upper bound is then proposed and the existence of a solution is proved. The quadratic and quartic cases are completely worked out and their exact solutions are explicitly calculated. [ABSTRACT FROM AUTHOR]

In this paper, we propose two iterative algorithms for finding the minimum-norm solution of a split minimization problem. We prove strong convergence of the sequences generated by the proposed algorithms. The iterative schemes are proposed in such a way that the selection of the step-sizes does not need any prior information about the operator norm. We further give some examples to numerically verify the efficiency and implementation of our new methods and compare the two algorithms presented. Our results act as supplements to several recent important results in this area. [ABSTRACT FROM AUTHOR]

This paper is devoted to designing a practical algorithm to invert the Laplace transform by assuming that the transform possesses the Puiseux expansion at infinity. First, the general asymptotic expansion of the inverse function at zero is derived, which can be used to approximate the inverse function when the variable is small. Second, an inversion algorithm is formulated by splitting the Bromwich integral into two parts. One is the main weakly oscillatory part, which is evaluated by a composite Gauss-Legendre rule and its Kronrod extension, and the other is the remaining strongly oscillatory part, which is integrated analytically using the Puiseux expansion of the transform at infinity. Finally, some typical tests show that the algorithm can be used to invert a wide range of Laplace transforms automatically with high accuracy and the output error estimator matches well with the true error. [ABSTRACT FROM AUTHOR]

In this paper, we first present a new finite difference scheme to approximate the time fractional derivatives, which is defined in the sense of Caputo, and give a semidiscrete scheme in time with the truncation error O((Δt)3-α), where Δt is the time step size. Then a fully discrete scheme based on the semidiscrete scheme for the fractional Cattaneo equation in which the space direction is approximated by a local discontinuous Galerkin method is presented and analyzed. We prove that the method is unconditionally stable and convergent with order O(hk+1 + (Δt)3-α), where k is the degree of piecewise polynomial. Numerical examples are also given to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in the last section, we derive an asymptotic expansion of the quadrature error. [ABSTRACT FROM AUTHOR]

In this paper, we apply the Galerkin method with sinc bases for solving the boundary-value problems involving nonhomogeneous heat, convection diffusion, wave, and telegraph equations. The accuracy of the method for equation is $ \mathrm {O} (({\Delta } t)+e^{-k\sqrt {N} })$ . Error analysis is included. Several examples are given to illustrate the efficiency and implementation of the sinc-Galerkin method. Comparisons are made to confirm the reliability of the method. [ABSTRACT FROM AUTHOR]

In this paper, we study in detail the phase properties and stability of numerical methods for general oscillatory second-order initial value problems whose right-hand side functions depend on both the position y and velocity y . In order to analyze comprehensively the numerical stability of integrators for oscillatory systems, we introduce a novel linear test model y ( t) + ? y( t) + µ y ( t)=0 with µ<2 ?. Based on the new model, further discussions and analysis on the phase properties and stability of numerical methods are presented for general oscillatory problems. We give the new definitions of dispersion and dissipation which can be viewed as an essential extension of the traditional ones based on the linear test model y ( t) + ? y( t)=0. The numerical experiments are carried out, and the numerical results showthatthe analysisofphase properties and stability presentedinthispaper ismoresuitableforthenumericalmethodswhentheyareappliedtothe generaloscillatory second-order initial value problem involving both the position and velocity. [ABSTRACT FROM AUTHOR]

Nonlinear minimization, as a subcase of nonlinear optimization, is an important issue in the research of various intelligent systems. Recently, Zhang et al. developed the continuous-time and discrete-time forms of Zhang dynamics (ZD) for time-varying nonlinear minimization. Based on this previous work, another two discrete-time ZD (DTZD) algorithms are proposed and investigated in this paper. Specifically, the resultant DTZD algorithms are developed for time-varying nonlinear minimization by utilizing two different types of Taylor-type difference rules. Theoretically, each steady-state residual error in the DTZD algorithm changes in an O( τ ) manner with τ being the sampling gap. Comparative numerical results are presented to further substantiate the efficacy and superiority of the proposed DTZD algorithms for time-varying nonlinear minimization. [ABSTRACT FROM AUTHOR]