Numerical Algorithms最新文献

We provide a new approach to obtain solutions of linear differential problems set in a Banach space and equipped with nonlocal boundary conditions. From this approach we derive a family of numerical schemes for the approximation of the solutions. We show by numerical tests that these schemes are numerically robust and computationally efficient.

In this paper, the a posteriori error estimates of the exponential midpoint method for time discretization are established for linear and semilinear parabolic equations. Using the exponential midpoint approximation defined by a continuous and piecewise linear interpolation of nodal values yields suboptimal order estimates. Based on the property of the entire function, we introduce a continuous and piecewise quadratic time reconstruction of the exponential midpoint method to derive optimal order estimates; the error bounds solely depend on the discretization parameters, the data of the problem, and the approximation of the entire function. Several numerical examples are implemented to illustrate the theoretical results.

The motion of glaciers can be simulated with the (varvec{p})-Stokes equations. Up to now, Newton’s method to solve these equations has been analyzed in finite-dimensional settings only. We analyze the problem in infinite dimensions to gain a new viewpoint. We do that by proving global convergence of the infinite-dimensional Newton’s method with Armijo step sizes to the solution of these equations. We only have to add an arbitrarily small diffusion term for this convergence result. We prove that the additional diffusion term only causes minor differences in the solution compared to the original (varvec{p})-Stokes equations under the assumption of some regularity. Finally, we test our algorithms on two experiments: A reformulation of the experiment ISMIP-HOM (varvec{B}) without sliding and a block with sliding. For the former, the approximation of exact step sizes for the Picard iteration and exact step sizes and Armijo step sizes for Newton’s method are superior in the experiment compared to the Picard iteration. For the latter experiment, Newton’s method with Armijo step sizes needs many iterations until it converges fast to the solution. Thus, Newton’s method with approximately exact step sizes is better than Armijo step sizes in this experiment.

Complex networks are made up of vertices and edges. The edges, which may be directed or undirected, are equipped with positive weights. Modeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices in distinct layers represent different kinds of objects. Multiplex networks are special vertex-aligned multilayer networks, in which vertices in distinct layers are identified with each other and inter-layer edges connect each vertex with its copy in other layers and have a fixed weight (gamma >0) associated with the ease of communication between layers. This paper discusses two different approaches to analyze communication in a multiplex. One approach focuses on the multiplex global efficiency by using the multiplex path length matrix, the other approach considers the multiplex total communicability. The sensitivity of both the multiplex global efficiency and the multiplex total communicability to structural perturbations in the network is investigated to help to identify intra-layer edges that should be strengthened to enhance communicability.

In this paper, we present a rigorous convergence analysis of a homogeneous multigrid method for the hybridizable interior penalty (IP-H) method. In particular, we use the injection operator developed in (Lu et al. SIAM J. Numer. Anal. 60, 2293–2317 2022). It is shown that when the penalization parameter is sufficiently large, our proposed multigrid method is optimal under the assumption of elliptic regularity. Numerical experiments are provided to confirm the theoretical results.

The paper is concerned with a generalization of Floater–Hormann (briefly FH) rational interpolation recently introduced by the authors. Compared with the original FH interpolants, the generalized ones depend on an additional integer parameter (gamma >1), that, in the limit case (gamma =1) returns the classical FH definition. Here we focus on the general case of an arbitrary distribution of nodes and, for any (gamma >1), we estimate the sup norm of the error in terms of the maximum (h) and minimum ((h^*)) distance between two consecutive nodes. In the special case of equidistant ((h=h^*)) or quasi–equidistant ((happrox h^*)) nodes, the new estimate improves previous results requiring some theoretical restrictions on (gamma ) which are not needed as shown by the numerical tests carried out to validate the theory.

Based on DDP method proposed by Mohammad and Abubakar, in this paper we use the inertia index and relaxation factor to establish an inertia projection method for solving nonlinear pseudo-monotone equations with convex constraints. This method can generate a sufficient descent direction at each iteration, which is independent of any line search condition. Moreover, we prove the global convergence of the proposed method without assuming that the objective function satisfies the Lipschitz continuity. Numerical results demonstrate the effectiveness of the proposed method by comparing with some existing methods.

This study presents a new branch and bound algorithm designed for the global optimization of the fractional squared least squares model for GPS localization. The algorithm incorporates a novel underestimation approach that provides theoretically superior lower bounds while requiring a comparable computational effort to the current approach. Numerical results demonstrate the substantial efficiency enhancements of the proposed algorithm over the existing algorithm.

We study multivariate polynomial interpolation based on Radon projections corresponding to the intersection of hyperplanes and the coordinate axes of (mathbb {R}^n). We give a characterization of these hyperplanes which determine an interpolation polynomial uniquely. We also establish conditions such that the interpolation projectors based on Radon projections converge to the Taylor projector.

We consider the problem of maximizing the ratio of two generalized quadratic matrix form functions over the Stiefel manifold, i.e., (max limits _{X^{T}X=I} frac{text {tr}(GX^{T}AX)}{text {tr}(GX^{T}BX)}) (RQMP). We utilize the Dinkelbach algorithm to globally solve RQMP, where each subproblem is evaluated by the closed-form solution. For a special case of RQMP with (AB=BA), we propose an equivalent linear programming problem. Numerical experiments demonstrate that it is more efficient than the recent SDP-based algorithm.