Numerical Algorithms最新文献

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.

This work focuses on the numerical approximations of random periodic solutions of stochastic differential equations (SDEs). Under non-globally Lipschitz conditions, we prove the existence and uniqueness of random periodic solutions for the considered equations and its numerical approximations generated by the stochastic theta (ST) methods with (theta in (1/2,1]). It is shown that the random periodic solution of each ST method converges strongly in the mean square sense to that of SDEs. More precisely, the mean square convergence order is 1/2 for SDEs with multiplicative noise and 1 for SDEs with additive noise, respectively. Numerical results are finally reported to confirm these theoretical findings.

The study considers the robust and minimum norm problems for stabilization using partial eigenvalue assignment technique in nonsingular vibration system with time delay via the acceleration-velocity-displacement active controller. The new gains expressions of active controller are derived by orthogonality relations, which keeps the no spill-over property of the vibration system. To discuss the stabilization problem using eigenvalues assignment technique, the linear equation is solved by constructing a special matrix which is proved to be nonsingular. Solving algorithm is proposed to obtain the parametric expressions of active controller. A new gradient-based optimization method is proposed to discuss the robust and minimum norm controller design by establishing the gradient formulas of cost functions. The optimization algorithm is proposed to discuss the robust and minimum norm stabilization of closed-loop eigenvalues in vibration system with time delay. The presented algorithms are feasible to the case of time delay between measurements of state and actuation of control. Numerical examples show the effectiveness of the method.

A new Krylov subspace method is designed in the computation of partial quaternion generalized singular value decomposition (QGSVD) of a large-scale quaternion matrix pair ({textbf{A}, textbf{B}}). Explicitly, we present the structure-preserving joint Lanczos bidiagonalization method to reduce (textbf{A}) and (textbf{B}) to lower and upper real bidiagonal matrices, respectively. We carry out the thick-restarted technique with the combination of a robust selective reorthogonalization strategy in the structure-preserving joint Lanczos bidiagonalization process. In the iteration process we avoid performing the explicit QR decomposition of the quaternion matrix pair. Numerical experiments illustrate the effectiveness of the proposed method.