Numerical Algorithms最新文献

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.

Complex networks are made up of vertices and edges. The latter connect the vertices. There are several ways to measure the importance of the vertices, e.g., by counting the number of edges that start or end at each vertex, or by using the subgraph centrality of the vertices. It is more difficult to assess the importance of the edges. One approach is to consider the line graph associated with the given network and determine the importance of the vertices of the line graph, but this is fairly complicated except for small networks. This paper compares two approaches to estimate the importance of edges of medium-sized to large networks. One approach computes partial derivatives of the total communicability of the weights of the edges, where a partial derivative of large magnitude indicates that the corresponding edge may be important. Our second approach computes the Perron sensitivity of the edges. A high sensitivity signals that the edge may be important. The performance of these methods and some computational aspects are discussed. Applications of interest include to determine whether a network can be replaced by a network with fewer edges with about the same communicability.

We consider a loosely coupled, non-iterative Robin-Robin coupling method proposed and analyzed in Burman et al. (J. Numer. Math. 31(1):59–77, 2023) for a parabolic-parabolic interface problem and prove estimates for the discrete time derivatives of the scalar field in different norms. When the interface is flat and perpendicular to two of the edges of the domain we prove error estimates in the (H^2)-norm. Such estimates are key ingredients to analyze a defect correction method for the parabolic-parabolic interface problem. Numerical results are shown to support our findings.

In this paper, we plan to show an eigenvalue algorithm for block Hessenberg matrices by using the idea of non-commutative integrable systems and matrix-valued orthogonal polynomials. We introduce adjacent families of matrix-valued (theta )-deformed bi-orthogonal polynomials, and derive corresponding discrete non-commutative hungry Toda lattice from discrete spectral transformations for polynomials. It is shown that this discrete system can be used as a pre-precessing algorithm for block Hessenberg matrices. Besides, some convergence analysis and numerical examples of this algorithm are presented.

In this paper, we introduce a method of successive approximations for moment functions of logistic stochastic differential equations. We first reduce the system of the corresponding moment functions to an infinite system of linear ordinary differential equations. Then, we determine certain upper and lower bounds on the moment functions, and utilize these bounds to solve the resulting systems approximately via suitable truncations, iterations and a local improvement step. After obtaining some general theoretical results on the error norms and describing a general algorithm for logistic SDE, we focus on stochastic Verhulst systems in numerical implementations. We compare their moment approximations with numerical solutions via simulation-based methods that include discretizations of the pathwise solutions as well as other convergent numerical procedures like semi-implicit split-step Euler methods.

In this article, we present a fourth-order accurate numerical method for solving generalized Black-Scholes PDE describing European and Asian options. Initially, we discretize the time derivative by the Crank-Nicolson scheme, and then the resultant semi-discrete problem by the central difference scheme on uniform meshes. In order to enhance the order of convergence of the proposed scheme, we employ the Richardson extrapolation method, by using two different meshes to solve the fully discrete problem. The stability and convergence are studied. To validate the proposed technique, several numerical experiments are carried out.

This paper focuses on the convergence analysis of the augmented Lagrangian method (ALM) for (varvec{ell }_{varvec{p}})-norm cone optimization problems. We investigate some properties of the augmented Lagrangian function and (varvec{ell }_{varvec{p}})-norm cone. Moreover, under the Jacobian uniqueness conditions, we prove that the local convergence rate of ALM for solving (varvec{ell }_{varvec{p}})-norm cone optimization problems with (varvec{p} varvec{ge } varvec{2}) is proportional to (varvec{1}varvec{/}varvec{r}), where the penalty parameter (varvec{r}) is not less than a threshold (varvec{hat{r}}). In numerical simulations, we successfully validate the effectiveness and convergence properties of ALM.

In this paper, we describe a new hybrid algorithm for computing all singular triplets above a given threshold and provide its implementation in MATLAB/Octave and R. The high performance of our codes and ease at which they can be used, either independently or within a larger numerical scheme, are illustrated through several numerical examples with applications to matrix completion and image compression. Well-documented MATLAB and R codes are provided for public use.