Numerical Algorithms最新文献

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.

An efficient extrapolation cascadic multigird (EXCMG) method is developed to solve large linear systems resulting from edge element discretizations of 3D (H(textbf{curl})) problems on rectangular meshes. By treating edge unknowns as defined on the midpoints of edges, following the similar idea of the nodal EXCMG method, we design a new prolongation operator for 3D edge-based discretizations, which is used to construct a high-order approximation to the finite element solution on the refined grid. This good initial guess greatly reduces the number of iterations required by the multigrid smoother. Furthermore, the divergence correction technique is employed to further speed up the convergence of the multigrid method. Numerical examples including problems with high-contrast discontinuous coefficients are presented to validate the effectiveness of the proposed EXCMG method. The edge-based EXCMG method is more efficient than the auxiliary-space Maxwell solver (AMS) for definite problems in the considered geometrical configuration, and it can also efficiently solve large-scale indefinite problems encountered in engineering and scientific fields.

In this paper, we present several unconditionally energy-stable invariant energy quadratization (IEQ) finite element methods (FEMs) with linear, first- and second-order accuracy for solving both the Cahn-Hilliard equation and the Allen-Cahn equation. For time discretization, we compare three distinct IEQ-FEM schemes that position the intermediate function introduced by the IEQ approach in different function spaces: finite element space, continuous function space, or a combination of these spaces. Rigorous proofs establishing the existence and uniqueness of the numerical solution, along with analyses of energy dissipation for both equations and mass conservation for the Cahn-Hilliard equation, are provided. The proposed schemes’ accuracy, efficiency, and solution properties are demonstrated through numerical experiments.

A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica, Matlab, and Python as examples) do not offer a solution, is to find all the real solutions of a system of n nonlinear equations in a certain finite domain of the n-dimensional space of variables. We present two similar algorithms of minimum length and computational weight to solve this problem, in which one resembles a graphical tool of edge detection in an image extended to n dimensions. To do this, we discretize the n-dimensional space sector in which the solutions are sought. Once the discretized hypersurfaces (edges) defined by each nonlinear equation of the n-dimensional system have been identified in a single, simultaneous step, the coincidence of the hypersurfaces in each n-dimensional tile or cell containing at least one solution marks the approximate locations of all the hyperpoints that constitute the solutions. This makes the final Newton-Raphson step rapidly convergent to all the existent solutions in the predefined space sector with the desired degree of accuracy.