Calcolo最新文献

The nonlinear Kaczmarz method was recently proposed to solve the system of nonlinear equations. In this paper, we first discuss two greedy selection rules, i.e., the maximum residual and maximum distance rules, for the nonlinear Kaczmarz iteration. Then, based on them, two kinds of greedy randomized sampling methods are presented. Furthermore, we also devise four corresponding greedy randomized block methods, i.e., the multiple samples-based methods. The linear convergence in expectation of all the proposed methods is proved. Numerical results show that, in some applications, including brown almost linear function and generalized linear model, the greedy selection rules give faster convergence rates than the existing ones, and the block methods outperform the single sample-based ones.

Abstract

A constructive numerical approximation of the two-dimensional unsteady stochastic Navier–Stokes equations of an incompressible fluid is proposed via a pseudo-compressibility technique involving a penalty parameter (varepsilon ) . Space and time are discretized through a finite element approximation and an Euler method. The convergence analysis of the suggested numerical scheme is investigated throughout this paper. It is based on a local monotonicity property permitting the convergence toward the unique strong solution of the stochastic Navier–Stokes equations to occur within the originally introduced probability space.

Abstract

Recent developments in the field of the radial basis function-finite difference (RBF-FD) framework have been focused on conditionally positive definite polyharmonic splines (PHS). Within this context, our research focuses on deriving analytical weights for the RBF-FD+polynomials method within the framework of PHS. We provide convergence analyses for various stencils. To validate the accuracy of our derived formulations, we conduct a series of computational experiments across a range of test problems.

The proposed work discusses discrete collocation and discrete Galerkin methods for second kind Fredholm–Hammerstein integral equations on half line ([0,infty )) using Kumar and Sloan technique. In addition, the finite section approximation method is applied to transform the domain of integration from ([0, infty )) to ([0,alpha ],~ alpha >0). In contrast to previous studies in which the optimal order of convergence is achieved for projection methods, we attained superconvergence rates in uniform norm using piecewise polynomial basis function. Moreover, these superconvergence rates are further enhanced by using discrete multi-projection (collocation and Galerkin) methods. In order to support the provided theoretical framework, numerical examples are included as well.

Abstract

In this article, band structure calculations of two dimensional (2D) anisotropic photonic-crystal fibers (PhCFs) are considered. In 2D PhCFs, Maxwell’s equations for the transversal electric and magnetic mode become decoupled, but the difficulty, arising from the anisotropic permittivity ({{varvec{varepsilon }}}) and/or permeability ({{varvec{mu }}},) plaguing the frequency-domain finite difference method, especially the original Yee’s scheme, is our top concern. To resolve this difficulty, we re-establish the connection between the lowest order finite element method with the quasi-periodic condition and Yee’s scheme using 2D non-orthogonal mesh, whereby the decoupled Maxwell’s equations in 2D anisotropic PhCFs are readily discretized into a generalized eigenvalue problem (GEP). Moreover, we spell out the nullspace of the resulting GEP, if it exists, and explicitly construct the Moore–Penrose pseudoinverse of the singular coefficient matrix, whose smallest positive eigenvalues can be solved by the inverse Lanczos method. Extensive band structures of 2D PhCFs are calculated and benchmarked against reliable results to demonstrate the accuracy and efficiency of our method.