Numerical Algorithms最新文献

This work is focused on developing a hybrid numerical method that combines a higher-order finite difference method and multi-dimensional Hermite wavelets to address two-dimensional multi-term time-fractional integro-partial differential equations with multi-term weakly singular kernels having bounded and unbounded time derivatives at the initial time (t=0). Specifically, the multi-term fractional operators are discretized using a higher-order approximation designed by employing different interpolation schemes based on linear, quadratic, and cubic interpolation leading to (mathcal {O}(N^{-(4-alpha _1)})) accuracy on a suitably chosen nonuniform mesh and (mathcal {O}(N^{-alpha _1})) accuracy on a uniformly distributed mesh. The weakly singular integral operators are approximated by a modified numerical quadrature, which is a combination of the composite trapezoidal approximation and the midpoint rule. The effects of the exponents of the weakly singular kernels over fractional orders are analyzed in terms of accuracy over uniform and nonuniform meshes for the solution having both bounded and unbounded time derivatives. The stability of the proposed semi-discrete scheme is derived based on (L^infty )-norm for uniformly distributed temporal mesh. Further, we employ the uniformly distributed collocation points in spatial directions to estimate the tensor-based wavelet coefficients. Moreover, the convergence analysis of the fully discrete scheme is carried out based on (L^2)-norm leading to (mathcal {O}(N^{-alpha _1})) accuracy on a uniform mesh. It also highlights the higher-order accuracy over nonuniform mesh. Additionally, we discuss the convergence analysis of the proposed scheme in the context of the multi-term time-fractional diffusion equations involving time singularity demonstrating a (mathcal {O}(N^{-(4-alpha _1)})) accuracy on a nonuniform mesh with suitably chosen grading parameter. Note that the scheme reduces to (mathcal {O}(N^{-alpha _1})) accuracy on a uniform mesh. Several tests are performed on numerous examples in (L^infty )- and (L^2)-norm to show the efficiency of the proposed method. Further, the solutions’ nature and accuracy in terms of absolute point-wise error are illustrated through several isosurface plots for different regularities of the exact solution. These experiments confirm the theoretical accuracy and guarantee the convergence of approximations to the functions having time singularity, and the higher-order accuracy for a suitably chosen nonuniform mesh.

We present a Virtual Element MATLAB solver for elliptic and parabolic, linear and semilinear Partial Differential Equations (PDEs) in two and three space dimensions, which is coined VEMcomp. Such PDEs are widely applicable to describing problems in material sciences, engineering, cellular and developmental biology, among many other applications. The library covers linear and nonlinear models posed on different simple and complex geometries, involving time-dependent bulk, surface, and bulk-surface PDEs. The solver employs the Virtual Element Method (VEM) of lowest polynomial order ({k=1}) on general polygonal and polyhedral meshes, including the Finite Element Method (FEM) of order ({k=1}) as a special case when the considered mesh is simplicial. VEMcomp has three main purposes. First, VEMcomp generates polygonal and polyhedral meshes optimized for fast matrix assembly. Triangular and tetrahedral meshes are encompassed as special cases. For surface PDEs, VEMcomp is compatible with the well-known Matlab package DistMesh for mesh generation. Second, given a mesh for the considered geometry, possibly generated with an external package, VEMcomp computes all the matrices (mass and stiffness) required by the VEM or FEM method. Third, for multiple classes of stationary and time-dependent bulk, surface and bulk-surface PDEs, VEMcomp solves the considered PDE problem with the VEM or FEM in space and IMEX Euler in time, through a user-friendly interface. As an optional post-processing, VEMcomp comes with its own functions for plotting the numerical solutions and evaluating the error when possible. An extensive set of examples illustrates the usage of the library.

In this present work, in order to solve the numerical solution of quaternion matrix equation (EY=F), the quaternion modified gradient-based algorithm (QMGI) is proposed by applying the real presentation of quaternion matrix. The proposed method can be applied to solve the quaternion solution, pure imaginary solution, and real solution of the studied equation (EY=F). If the studied equation is consistent, it is proved that the proposed algorithm converges to the exact solution for given any initial quaternion matrix under appropriate conditions. If the studied equation is not consistent, it is found that the QMGI algorithm converges to the least squares solution. And some numerical examples are examined to confirm the feasibility and efficiency of the proposed algorithms, which all indicate that the proposed QMGI algorithm is much more effective than QGI algorithm and QRGI algorithm in computational time and accuracy. Moreover, QMGl algorithm is applied to color image encryption and evaluated the encryption effectiveness from four aspects. All metrics are close to the ideal values. lt is demonstrated that the effectiveness of the encryption scheme and the accuracy of the obtained theory results in this paper.

In this paper, we propose a least-change secant algorithm to solve the generalized complementarity problem indirectly trough its reformulation as a nonsmooth system of nonlinear equations using a one-parametric family of complementarity functions. We present local and superlinear convergence results of new algorithm and analyze its numerical performance.

In this paper, we present a novel (L^{1})-(L^{2})-TV model for image deblurring that incorporates spatially varying regularization parameters, addressing the challenge of mixed Gaussian and impulse noise. The traditional Total Variation (TV) model with (L^{1}) and (L^{2}) fidelity terms is well-recognized for its effectiveness in such scenarios, but our proposed approach enhances this by allowing the regularization parameters to adapt based on local image characteristics. This ensures that fine details are better preserved while maintaining smoothness in homogeneous areas. The spatially dependent regularization parameters are automatically determined using local discrepancy functions. The discrete minimization problem that arises from this model is efficiently solved using the inexact alternating direction method (IADM). Our numerical experiments show that the proposed algorithm significantly improves the peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) by enhancing detailed regions and effectively removing both types of noise.

An efficient finite difference method for the multi-dimensional differential equation with variable-order Riemann-Liouville derivative is studied. Firstly, we construct an efficient discrete approximation for the multi-dimensional variable-order Riemann-Liouville derivative by the generating functions approximation theory. The convergence of the discrete operator in the Barron space is analyzed. Based on it, we present the finite difference method for the elliptic equation with variable-order Riemann-Liouville derivative. The stability and convergence of the method are proven by the maximum principle. Moreover, a fast solver is presented in the computation based on the fast Fourier transform and the multigrid algorithm in order to reduce the storage and speed up the BiCGSTAB method, respectively. We extend this method to time-dependent problems and several numerical examples show that the proposed schemes and the fast solver are efficient.

The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix approximation is sometimes considerable due to the computation of the quaternion singular value decomposition (QSVD), which limits their application to real large-scale data. To address this deficiency, an efficient quaternion matrix CUR (QMCUR) method for low-rank approximation is suggested, which provides significant acceleration in color image processing. We first explore the QMCUR approximation method, which uses actual columns and rows of the given quaternion matrix, instead of the costly QSVD. Additionally, two different sampling strategies are used to sample the above-selected columns and rows. Then, the perturbation analysis is performed on the QMCUR approximation of noisy versions of low-rank quaternion matrices. And we also employ the proposed QMCUR method to color image recovery problem. Extensive experiments on both synthetic and real data further reveal the superiority of the proposed algorithm compared with other algorithms for getting low-rank approximation, in terms of both efficiency and accuracy.

In this work, we present the a priori and a posteriori error analysis of a hybrid difference scheme for integral boundary value problems of nonlinear singularly perturbed parameterized form. The discretization for the nonlinear parameterized equation constitutes a hybrid difference scheme which is based on a suitable combination of the trapezoidal scheme and the backward difference scheme. Further, we employ the composite trapezoidal scheme for the discretization of the nonlocal boundary condition. A priori error estimation is provided for the proposed hybrid scheme, which leads to second-order uniform convergence on various a priori defined meshes. Moreover, a detailed a posteriori error analysis is carried out for the present hybrid scheme which provides a proper discretization of the error equidistribution at each partition. Numerical results strongly validate the theoretical findings for nonlinear problems with integral boundary conditions.

In this article, we study and examine an efficient numerical approach to obtain approximate solutions of the two-dimensional fractional cable model involving the time-fractional operator of distributed order. A hybrid numerical approach is used to approximate the proposed fractional model. For approximating the integral part of the distributed order including Caputo fractional derivative, the combination of Gauss quadrature rule and finite difference are used. As well as, for the integral part of the distributed order including Riemann Liouville fractional derivatives, from the mid-point quadrature rule and shifted Grünwald estimation are applied. Also, to approximate the proposed model in the space direction, the Legendre spectral numerical approach is used in order to calculate the full-discrete numerical approach. In this work, error analysis and convergence are studied. In the end, to show the effectiveness of the proposed approach, two numerical examples are stated and checked.

In this work, we propose a novel implementation of Newton’s method for solving semi-discrete optimal transport (OT) problems for cost functions which are a positive combination of p-norms, (1<p<infty ). It is well understood that the solution of a semi-discrete OT problem is equivalent to finding a partition of a bounded region in Laguerre cells, and we prove that the Laguerre cells are star-shaped with respect to the target points. By exploiting the geometry of the Laguerre cells, we obtain an efficient and reliable implementation of Newton’s method to find the sought network structure. We provide implementation details and extensive results in support of our technique in 2-d problems, as well as comparison with other approaches used in the literature.