Showing papers in "Numerical Algorithms in 2017"
TL;DR: This operational matrix is utilized to transform the problem to a set of algebraic equations with unknown Bernoulli wavelet coefficients, and upper bound for the error of operational matrix of the fractional integration is given.
Abstract: In this research, a Bernoulli wavelet operational matrix of fractional integration is presented Bernoulli wavelets and their properties are employed for deriving a general procedure for forming this matrix The application of the proposed operational matrix for solving the fractional delay differential equations is explained Also, upper bound for the error of operational matrix of the fractional integration is given This operational matrix is utilized to transform the problem to a set of algebraic equations with unknown Bernoulli wavelet coefficients Several numerical examples are solved to demonstrate the validity and applicability of the presented technique
122 citations
TL;DR: By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, it is proved the fully discrete system is uniquely solvable.
Abstract: In this paper, a class of nonlinear Riesz space-fractional Schrodinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.
102 citations
TL;DR: The main aim of this paper is to combine the alternating direction implicit approach with the IEFG method on the distributed order time-fractional diffusion-wave equation and propose three schemes based on the trapezoidal, Simpson, and Gauss-Legendre quadrature techniques.
Abstract: In the current decade, the meshless methods have been developed for solving partial differential equations. The meshless methods may be classified in two basic parts: 1.The meshless methods based on the strong form2.The meshless methods based on the weak form
The element-free Galerkin (EFG) method is a meshless method based on the global weak form. The test and trial functions in element-free Galerkin are shape functions of moving least squares (MLS) approximation. Also, the traditional MLS shape functions have not the ź-Kronecker property. Recently, a new class of MLS shape functions has been presented. These are well-known as the interpolating MLS (IMLS) shape functions. The IMLS shape functions have the ź-Kronecker property; thus the essential boundary conditions can be applied directly. The main aim of this paper is to combine the alternating direction implicit approach with the IEFG method. To this end, we apply the mentioned technique on the distributed order time-fractional diffusion-wave equation. For comparing the numerical results, we propose three schemes based on the trapezoidal, Simpson, and Gauss-Legendre quadrature techniques. Also, we investigate the uniqueness, existence and stability analysis of the new schemes and we obtain an error estimate for the full-discrete schemes. The time-fractional derivative has been described in Caputo's sense. Numerical examples demonstrate the theoretical results and the efficiency of the proposed schemes.
81 citations
TL;DR: The aim in this paper is to study strong convergence results for L-Lipschitz continuous monotone variational inequality but L is unknown using a combination of subgradient extra-gradient method and viscosity approximation method with adoption of Armijo-like step size rule in infinite dimensional real Hilbert spaces.
Abstract: Our aim in this paper is to study strong convergence results for L-Lipschitz continuous monotone variational inequality but L is unknown using a combination of subgradient extra-gradient method and viscosity approximation method with adoption of Armijo-like step size rule in infinite dimensional real Hilbert spaces. Our results are obtained under mild conditions on the iterative parameters. We apply our result to nonlinear Hammerstein integral equations and finally provide some numerical experiments to illustrate our proposed algorithm.
67 citations
TL;DR: A 50-line MATLAB implementation of the lowest order virtual element method for the two-dimensional Poisson problem on general polygonal meshes to demonstrate how the key components of the method can be translated into code.
Abstract: We present a 50-line MATLAB implementation of the lowest order virtual element method for the two-dimensional Poisson problem on general polygonal meshes. The matrix formulation of the method is discussed, along with the structure of the overall algorithm for computing with a virtual element method. The purpose of this software is primarily educational, to demonstrate how the key components of the method can be translated into code.
61 citations
TL;DR: The results demonstrate that the proposed methodology is very useful and simple in the determination of the solution of the K-S equations of fractional order.
Abstract: In this study, we discuss the application of an analytical technique namely modified homotopy analysis transform method (MHATM) for solving coupled one- dimensional time-fractional Keller-Segel (K-S) equations. The MHATM is a new analytical technique based on homotopy polynomial. We provide a convergence analysis of MHATM and the solution obtained by the proposed method is verified through different graphical representations. The results demonstrate that the proposed methodology is very useful and simple in the determination of the solution of the K-S equations of fractional order.
60 citations
TL;DR: A derivative-free iterative scheme that uses the residual vector as search direction for solving large-scale systems of nonlinear monotone equations is presented and computational experiments show that the new algorithm is computationally efficient.
Abstract: A derivative-free iterative scheme that uses the residual vector as search direction for solving large-scale systems of nonlinear monotone equations is presented. It is closely related to two recently proposed spectral residual methods for nonlinear systems which use a nonmonotone line-search globalization strategy and a step-size based on the Barzilai-Borwein choice. The global convergence analysis is presented. In order to study the numerical behavior of the algorithm, it is included an extensive series of numerical experiments. Our computational experiments show that the new algorithm is computationally efficient.
59 citations
TL;DR: A relaxation modulus-based matrix splitting iteration method is established, which covers the known general modulus based matrix splitting iterations methods and is efficient and accelerate the convergence performance with less iteration steps and CPU times.
Abstract: In this paper, a relaxation modulus-based matrix splitting iteration method is established, which covers the known general modulus-based matrix splitting iteration methods. The convergence analysis and the strategy of the choice of the parameters are given. Numerical examples show that the proposed methods are efficient and accelerate the convergence performance with less iteration steps and CPU times.
53 citations
TL;DR: In this article, a review of the available techniques for accurate, fast, and reliable computation of confluent hypergeometric functions in different parameter and variable regimes is presented, including Taylor and asymptotic series computations, Gauss---Jacobi quadrature, numerical solution of differential equations, recurrence relations and others.
Abstract: The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss---Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide "roadmaps" with our recommendation for which methods should be used in each situation.
49 citations
TL;DR: It is proved that the scheme is unconditionally stable and parameter uniform convergent for bigger shift arguments as well as for small shift arguments, and the performance of the method is corroborated by numerical examples.
Abstract: In this work, a parameter uniform numerical method is developed to find the approximate solution of time-dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments in the space variable. The earlier work on such type of initial-interval boundary value problems is restricted to the case of small delay and advance arguments while in practical situations the shift arguments can be of arbitrary size (i.e. it may be bigger or small enough in size). The fitted mesh technique to establish parameter uniform error estimates is not extendable for the class of singularly perturbed parabolic partial differential-difference equations (SPPPDDEs) with general shift arguments in the space variable. To observe the dispersive behaviour of the solution of the problem considered in this paper, we use systematically constructed suitable denominator function for the discrete second order derivative. The motivation behind the construction of the scheme is modelling rules for non-standard finite difference methods (NSFDMs), developed by Mickens. The proposed numerical scheme is analysed for consistency and stability. It is proved that the scheme is unconditionally stable and parameter uniform convergent. The scheme is convergent for bigger shift arguments as well as for small shift arguments. The performance of the method is corroborated by numerical examples.
48 citations
TL;DR: A fourth-order implicit-explicit time-discretization scheme based on the exponential time differencing approach with aFourth-order compact scheme in space is proposed for space fractional nonlinear Schrödinger equations and it is shown that the compact scheme is fourth- order convergent in space and in time.
Abstract: A fourth-order implicit-explicit time-discretization scheme based on the exponential time differencing approach with a fourth-order compact scheme in space is proposed for space fractional nonlinear Schrodinger equations. The stability and convergence of the compact scheme are discussed. It is shown that the compact scheme is fourth-order convergent in space and in time. Numerical experiments are performed on single and coupled systems of two and four fractional nonlinear Schrodinger equations. The results demonstrate accuracy, efficiency, and reliability of the scheme. A linearly implicit conservative method with the fourth-order compact scheme in space is also considered and used on the system of space fractional nonlinear Schrodinger equations.
TL;DR: This paper studies several iterative methods for finding a solution to the multiple-sets split feasibility problem, which solves a certain variational inequality.
Abstract: In this paper, for the multiple-sets split feasibility problem, that is to find a point closest to a family of closed convex subsets in one space such that its image under a linear bounded mapping will be closest to another family of closed convex subsets in the image space, we study several iterative methods for finding a solution, which solves a certain variational inequality. We show that particular cases of our algorithms are some improvements for existing ones in literature. We also give two numerical examples for illustrating our algorithms.
TL;DR: A compact difference scheme with convergence order O(τ2 + h4) is proposed for the fourth-order fractional sub-diffusion equation, where h and τ are space and temporal step length, respectively, and the extension to the two-dimensional case is considered.
Abstract: In the present work, a compact difference scheme with convergence order O(τ
2 + h
4) is proposed for the fourth-order fractional sub-diffusion equation, where h and τ are space and temporal step length, respectively. The method is based on applying the L2−1
σ
formula to approximate the time Caputo fractional derivative and employing compact operator to approximate the spatial fourth-order derivative. Using the special properties of L2−1
σ
formula and mathematical induction method, we obtain the unconditional stability and convergence for our scheme by discrete energy method. Furthermore, the extension to the two-dimensional case is also considered. Numerical examples are given to verify the theoretical analysis and efficiency of the new developed scheme.
TL;DR: A meshless collocation method is considered to solve the multi-term time fractional diffusion-wave equation in two dimensions using the moving least squares reproducing kernel particle approximation to construct the shape functions for spatial approximation.
Abstract: In this paper, a meshless collocation method is considered to solve the multi-term time fractional diffusion-wave equation in two dimensions. The moving least squares reproducing kernel particle approximation is employed to construct the shape functions for spatial approximation. Also, the Caputo's time fractional derivatives are approximated by a scheme of order O(ź3źź), 1<ź < 2. Stability and convergence of the proposed scheme are discussed. Some numerical examples are given to confirm the efficiency and reliability of the proposed method.
TL;DR: In this article, the authors considered distributed optimal control for the Stokes system and test the particular case when the arising linear system can be compressed after eliminating the control function, in which case a system arises in a form which enables the application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic.
Abstract: The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control becomes an optimization problem where the governing partial differential equations of the fluid flow are stated as constraints. When discretized, the optimal control of the Navier-Stokes equations leads to large sparse saddle point systems in two levels. In this paper, we consider distributed optimal control for the Stokes system and test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables the application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain conditions, the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and execution time is favorably compared with other published methods.
TL;DR: The present method is applied to obtain numerical solution of singular boundary value problems arising in various physical models, and numerical results show the advantages of the method over the existing methods.
Abstract: In this paper, we consider the following class of singular two-point boundary value problem posed on the interval x 𝜖 (0, 1]
$$\begin{array}{@{}rcl@{}} (g(x)y^{\prime})^{\prime}=g(x)f(x,y),\\ y^{\prime}(0)=0,\mu y(1)+\sigma y^{\prime}(1)=B. \end{array} $$
A recursive scheme is developed, and its convergence properties are studied. Further, the error estimation of the method is discussed. The proposed scheme is based on the integral equation formalism and optimal homotopy analysis method in which a recursive scheme is established without any undetermined coefficients. The original differential equation is transformed into an equivalent integral equation to remove the singularity. The integral equation is then made free of undetermined coefficients by imposing the boundary conditions on it. Finally, the integral equation without any undetermined coefficients is efficiently treated by using optimal homotopy analysis method for finding the numerical solution. The optimal control-convergence parameter involved in the components of the series solution is obtained by minimizing the squared residual error equation. The present method is applied to obtain numerical solution of singular boundary value problems arising in various physical models, and numerical results show the advantages of our method over the existing methods.
TL;DR: A scaled-circulant preconditioner is proposed to use to deal with Toeplitz-like discretization matrices of steady-state variable-coefficient conservative space-fractional diffusion equations and it is demonstrated that the preconditionsed Krylov subspace method converges very quickly.
Abstract: We consider the preconditioned Krylov subspace method for linear systems arising from the finite volume discretization method of steady-state variable-coefficient conservative space-fractional diffusion equations. We propose to use a scaled-circulant preconditioner to deal with such Toeplitz-like discretization matrices. We show that the difference between the scaled-circulant preconditioner and the coefficient matrix is equal to the sum of a small-norm matrix and a low-rank matrix. Numerical tests are conducted to show the effectiveness of the proposed method for one- and two-dimensional steady-state space-fractional diffusion equations and demonstrate that the preconditioned Krylov subspace method converges very quickly.
TL;DR: To solve a class of nonlinear complementarity problems, accelerated modulus-based matrix splitting iteration methods are presented and analyzed and have better performance in aspects of the number of iteration steps and CPU time.
Abstract: To solve a class of nonlinear complementarity problems, accelerated modulus-based matrix splitting iteration methods are presented and analyzed. Convergence analysis and the choice of the parameters are given when the system matrix is either positive definite or an H+-matrix. Numerical experiments further demonstrate that the proposed methods are efficient and have better performance than the existing modulus-based iteration method in aspects of the number of iteration steps and CPU time.
TL;DR: It is obvious from these results that there is a significant advantage, with respect to convergence rate of accuracy, to use the proposed trust region approach in comparison with using point-to-point distance minimization and a Newton- type update without any step size control.
Abstract: The problem of finding a rigid body transformation, which aligns a set of data points with a given surface, using a robust M-estimation technique is considered. A refined iterative closest point (ICP) algorithm is described where a minimization problem of point-to-plane distances with a proposed constraint is solved in each iteration to find an updating transformation. The constraint is derived from a sum of weighted squared point-to-point distances and forms a natural trust region, which ensures convergence. Only a minor number of additional computations are required to use it. Two alternative trust regions are introduced and analyzed. Finally, numerical results for some test problems are presented. It is obvious from these results that there is a significant advantage, with respect to convergence rate of accuracy, to use the proposed trust region approach in comparison with using point-to-point distance minimization as well as using point-to-plane distance minimization and a Newton- type update without any step size control.
TL;DR: This paper studies the condition estimation of the total least squares (TLS) problem based on small sample condition estimation (SCE), which can be incorporated into the direct solver for the TLS problem via the singular value decomposition (SVD) of the augmented matrix [A, b].
Abstract: In this paper, under the genericity condition, we study the condition estimation of the total least squares (TLS) problem based on small sample condition estimation (SCE), which can be incorporated into the direct solver for the TLS problem via the singular value decomposition (SVD) of the augmented matrix [A, b]. Our proposed condition estimation algorithms are efficient for the small and medium size TLS problem because they utilize the computed SVD of [A, b] during the numerical solution to the TLS problem. Numerical examples illustrate the reliability of the algorithms. Both normwise and componentwise perturbations are considered. Moreover, structured condition estimations are investigated for the structured TLS problem.
TL;DR: An efficient local extrapolation of the exponential operator splitting scheme is introduced to solve the multi-dimensional space-fractional nonlinear Schrödinger equations to model optical solitons in graded-index fibers.
Abstract: An efficient local extrapolation of the exponential operator splitting scheme is introduced to solve the multi-dimensional space-fractional nonlinear Schrodinger equations. Stability of the scheme is examined by investigating its amplification factor and by plotting the boundaries of the stability regions. Empirical convergence analysis and calculation of the local truncation error exhibit the second-order accuracy of the proposed scheme. The performance and reliability of the proposed scheme are tested by implementing it on two- and three-dimensional space-fractional nonlinear Schrodinger equations including the space-fractional Gross-Pitaevskii equation, which is used to model optical solitons in graded-index fibers.
TL;DR: Many of the eighth-order schemes scattered in the literature are collected and a quantitative comparison is presented, finding that the best method based on the three criteria is SA8 due to Sharma and Arora.
Abstract: Recently, there were many papers discussing the basins of attraction of various methods and ideas how to choose the parameters appearing in families of methods and weight functions used. Here, we collected many of the eighth-order schemes scattered in the literature and presented a quantitative comparison. We have used the average number of function evaluations per point, the CPU time, and the number of black points to compare the methods. Based on seven examples, we found that the best method based on the three criteria is SA8 due to Sharma and Arora.
TL;DR: A novel time-stepping scheme, called transformed jump-adapted backward Euler method, is developed in this paper to simulate a class of jump-extended CIR and CEV models and is able to preserve the positivity of the underlying problems.
Abstract: A novel time-stepping scheme, called transformed jump-adapted backward Euler method, is developed in this paper to simulate a class of jump-extended CIR and CEV models. The proposed scheme is able to preserve the positivity of the underlying problems. Furthermore, its strong convergence rate of order one is recovered for the considered models with non-Lipschitz diffusion coefficients. Numerical examples are finally reported to confirm our theoretical findings.
TL;DR: This paper proposes the fast predictor-corrector approach for the tempered fractional ordinary differential equations by digging out the potential ‘very’ short memory principle and algorithms based on the idea of equidistributing are detailedly described.
Abstract: The tempered evolution equation describes the trapped dynamics, widely appearing in nature, e.g., the motion of living particles in viscous liquid. This paper proposes the fast predictor-corrector approach for the tempered fractional ordinary differential equations by digging out the potential `very' short memory principle. Algorithms based on the idea of equidistributing are detailedly described. Error estimates for the proposed schemes are derived; and the effectiveness and low computation cost, being linearly increasing with time t, are numerically demonstrated.
TL;DR: A fast numerical method is developed for solving the linear system that arises from compact finite difference scheme for time-space fractional diffusion equations with significant speedup.
Abstract: Based on the circulant-and-skew-circulant representation of Toeplitz matrix inversion and the divide-and-conquer technique, a fast numerical method is developed for solving N-by-N block lower triangular Toeplitz with M-by-M dense Toeplitz blocks system with \(\mathcal {O}(MN\log N(\log N+\log M))\) complexity and \(\mathcal {O}(NM)\) storage. Moreover, the method is employed for solving the linear system that arises from compact finite difference scheme for time-space fractional diffusion equations with significant speedup. Numerical examples are given to show the efficiency of the proposed method.
TL;DR: A novel efficient method for a fourth-order nonlinear boundary value problem which models a statistically bending elastic beam is proposed, which guarantees the existence and uniqueness of a solution of the problem and the convergence of an iterative method for finding it.
Abstract: In this paper, we propose a novel efficient method for a fourth-order nonlinear boundary value problem which models a statistically bending elastic beam. Differently from other authors, we reduce the problem to an operator equation for the right-hand side function. Under some easily verified conditions on this function in a specified bounded domain, we prove the contraction of the operator. This guarantees the existence and uniqueness of a solution of the problem and the convergence of an iterative method for finding it. The positivity of the solution and the monotony of iterations are also considered. We show that the examples of some other authors satisfy our conditions; therefore, they have a unique solution, while these authors only could prove the existence of a solution. Numerical experiments on these and other examples show the fast convergence of the iterative method.
TL;DR: Using the S.L.Sobolev’s method, new optimal quadrature formulas of such type for N+1≥m are obtained in the L2(m)(0,1)$L_{2}^{(m)}(0, 1)$ space for numerical calculation of Fourier coefficients.
Abstract: This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the L2(m)(0,1)$L_{2}^{(m)}(0,1)$ space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev's method, we obtain new optimal quadrature formulas of such type for N+1źm, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the L2(m)(0,1)$L_{2}^{(m)}(0,1)$ space are exact for Pmź1(x), where Pmź1(x) is a polynomial of degree mź1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.
TL;DR: Algebraic and exponential optimal h- and r-error rates are numerically validated for high-order edge elements on the problem of Maxwell’s eigenvalues in a square domain.
Abstract: Explicit generators for high-order (r>1) scalar and vector finite element spaces generally used in numerical electromagnetism are presented and classical degrees of freedom, the so-called moments, revisited. Properties of these generators on simplicial meshes are investigated, and a general technique to restore duality between moments and generators is proposed. Algebraic and exponential optimal h- and r-error rates are numerically validated for high-order edge elements on the problem of Maxwell's eigenvalues in a square domain.
TL;DR: An algorithm for identifying the strong ℋ$\mathcal {H}$-tensors based on several simple practical criteria that only depend on the elements of the tensors is proposed.
Abstract: Strong ź$\mathcal {H}$-tensors play an important role in identifying positive semidefiniteness of even-order real symmetric tensors. We provide several simple practical criteria for identifying strong ź$\mathcal {H}$-tensors. These criteria only depend on the elements of the tensors; therefore, they are easy to be verified. Meanwhile, a sufficient and necessary condition of strong ź$\mathcal {H}$-tensors is obtained. We also propose an algorithm for identifying the strong ź$\mathcal {H}$-tensors based on these criterions. Some numerical results show the feasibility and effectiveness of the algorithm.
TL;DR: Two splitting extragradient-like algorithms for solving strongly pseudomonotone equilibrium problems given by a sum of two bifunctions are proposed and the R-linear rate of convergence under suitable assumptions on bifunction is established.
Abstract: In this paper, two splitting extragradient-like algorithms for solving strongly pseudomonotone equilibrium problems given by a sum of two bifunctions are proposed. The convergence of the proposed methods is analyzed and the R-linear rate of convergence under suitable assumptions on bifunctions is established. Moreover, a noisy data case, when a part of the bifunction is contaminated by errors, is studied. Finally, some numerical experiments are given to demonstrate the efficiency of our algorithms.