Numerical Algorithms最新文献

Based on the diagonal and off-diagonal splitting (DOS) iteration scheme (Dehghan et al. Filomat 31(5), 1441–1452 2017), we offer an iteration procedure called Uzawa-DOS to solve a class of saddle-point problems (SPPs). Each iteration of this iterative method involves two subsystems with diagonal and lower triangular matrices. Due to the simple structure of involved coefficient matrices, two linear subsystems are solvable exactly, which is a notable precedence of the Uzawa-DOS method and can make it inexpensive to execute. Theoretical analysis verifies convergence of the proposed method under appropriate conditions. The suggested method is validated by numerical experiments.

In this paper, by regarding the two-subspace Kaczmarz method as an alternated inertial randomized Kaczmarz algorithm we present a better convergence rate estimate under a mild condition. Furthermore, we accelerate the alternated inertial randomized Kaczmarz algorithm and introduce a multi-step inertial randomized Kaczmarz algorithm which is proved to have a faster convergence rate. Numerical experiments support the theory results and illustrate that the multi-inertial randomized Kaczmarz algorithm significantly outperform the two-subspace Kaczmarz method in solving coherent linear systems.

The Caputo fractional derivative of a real continuous function g distinguishes from the other fractional derivative methods with the demand for the existence of its first order derivative (g'). This attribute leads to the investigation of Caputo fractional derivative of (alpha )-fractal splines rather than just a continuous non-differentiable (alpha )-fractal function. A bounded linear operator corresponding to the Caputo fractional derivative of fractal version is reported. In addition, a new family of fractal perturbations is proposed in association with the fractional derivative. Thereafter, a numerical approach is used to determine the exact Caputo fractional derivative of fractal functions in terms of Legendre polynomials.

The Galerkin method is proposed for initial value problem of auto-convolution Volterra integro-differential equation (AVIDE). The solvability of the Galerkin method is discussed, and the uniform boundedness of the numerical solution is provided by defining a discrete weighted exponential norm. In particular, it is proved that the quadrature Galerkin method obtained from the Galerkin method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the continuous piecewise polynomial collocation method. For the Galerkin approximated solution in continuous piecewise polynomial space of degree (varvec{m}), at first, the (varvec{m}) global convergence order is obtained. By defining a projection operator, the convergence is improved, and the optimal (varvec{m+1}) global convergence order is gained, as well as (varvec{2m}) local convergence order at mesh points. Furthermore, all the above analysis for uniform mesh can be extended to typical quasi-uniform meshes. Some numerical experiments are given to illustrate the theoretical results.

This paper aims to propose a new splitting mixed finite element method (MFE) for solving viscoelastic wave equations and give convergence analysis. First, by introducing two new variables (q=u_t) and (varvec{sigma }=A(x)nabla u+B(x)nabla u_t), a new system of first-order differential-integral equations is derived from the original second-order viscoelastic wave equation. Then, the semi-discrete and fully-discrete splitting MFE schemes are proposed by using the MFE spaces and the second-order time discetization. By the two schemes the approximate solutions for the unknowns u, (u_t) and (sigma ) are obtained simultaneously. It is proved that the semi-discrete and fully-discrete schemes have the optimal error estimates in (L^2)-norm. Meanwhile, it is proved that the fully-discrete SMFE scheme based on the Raviart-Thomas mixed finite element spaces and the uniform rectangular mesh partitions is super convergent. Finally, numerical experiments to compute the (L^2) errors for approximating u, q and (varvec{sigma }) and their convergence rates are presented, and the theoretical analysis on error estimates and convergence is then confirmed.

In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted by ({mathbb {H}_{n}(x;q)}_{nge 0}), which are orthogonal with respect to the following non-standard inner product involving q-differences:

where (lambda ) belongs to the set of positive real numbers, (mathscr {D}_{q}^{j}) denotes the j-th q -discrete analogue of the derivative operator, (q^jalpha in mathbb {R}backslash (-1,1)), and ((qx,-qx;q)_{infty }d_{q}(x)) denotes the orthogonality weight with its points of increase in a geometric progression. Connection formulas between these polynomials and standard q-Hermite I polynomials are deduced. The basic hypergeometric representation of (mathbb {H}_{n}(x;q)) is obtained. Moreover, for certain real values of (alpha ) satisfying the condition (q^jalpha in mathbb {R}backslash (-1,1)), we present results concerning the location of the zeros of (mathbb {H}_{n}(x;q)) and perform a comprehensive analysis of their asymptotic behavior as the parameter (lambda ) tends to infinity.

This paper introduces and analyzes some new highly efficient iterative procedures for approximating fixed points of contractive-type mappings. The stability, data dependence, strong convergence, and performance of the proposed schemes are addressed. Numerical examples demonstrate that the newly introduced schemes produce approximations of great accuracy and comparable to other similar robust schemes appeared in the literature. Nevertheless, all the schemes developed here are more efficient than other robust schemes used for comparison.

In this paper, we consider an improved iterative method for solving the monotone inclusion problem in the form of (0 in A(x) + D(x) + N_{C}(x)) in a real Hilbert space, where A is a maximally monotone operator, D and B are monotone and Lipschitz continuous, and C is the nonempty set of zeros of the operator B. We investigate the weak ergodic and strong convergence (when A is strongly monotone) of the iterates produced by our considered method. We show that the algorithmic scheme can also be applied to minimax problems. Furthermore, we discuss how to apply the method to the inclusion problem involving a finite sum of compositions of linear continuous operators by using the product space approach and employ it for convex minimization. Finally, we present a numerical experiment in TV-based image inpainting to validate the proposed theoretical theorem.

This work focuses on the numerical approximations of neutral stochastic delay differential equations with their drift and diffusion coefficients growing super-linearly with respect to both delay variables and state variables. Under generalized monotonicity conditions, we prove that the backward Euler method not only converges strongly in the mean square sense with order 1/2, but also inherit the mean square exponential stability of the original equations. As a byproduct, we obtain the same results on convergence rate and exponential stability of the backward Euler method for stochastic delay differential equations under generalized monotonicity conditions. These theoretical results are finally supported by several numerical experiments.

In this paper, we present a fully discrete finite difference scheme with efficient convolution of artificial boundary conditions for solving the Cauchy problem associated with the one-dimensional linearized Benjamin-Bona-Mahony equation. The scheme utilizes the Padé expansion of the square root function in the complex plane to implement the fast convolution, resulting in significant reduction of computational costs involved in the time convolution process. Moreover, the introduction of a constant damping term in the governing equations allows for convergence analysis under specific conditions. The theoretical analysis is complemented by numerical examples that illustrate the performance of the proposed numerical method.