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Abstract: Abstract In this paper, we present a new iterative method for solving a linear system, whose coefficient matrix is an M-matrix. This method includes four parameters that are obtained by the accelerated overrelaxation (AOR) splitting and using the Taylor approximation. First, under some standard assumptions, we establish the convergence properties of the new method. Then, by minimizing the Frobenius norm of the iteration matrix, we find the optimal parameters. Meanwhile, numerical results on test examples show the efficiency of the new proposed method in contrast with the Hermitian and skew-Hermitian splitting (HSS), AOR methods and a modified version of the AOR (QAOR) iteration. PubDate: 2022-06-01
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Abstract: Abstract We propose a modification of the golden ratio algorithm for solving pseudomonotone equilibrium problems with a Lipschitz-type condition in Hilbert spaces. A new non-monotone stepsize rule is used in the method. Without such an additional condition, the theorem of weak convergence is proved. Furthermore, with strongly pseudomonotone condition, the R-linear convergence rate of the method is established. The results obtained are applied to a variational inequality problem, and the convergence rate of the problem under the condition of error bound is considered. Finally, numerical experiments on several specific problems and comparison with other algorithms show the superiority of the algorithm. PubDate: 2022-06-01
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Abstract: Abstract Recently, Wang (2017) has introduced the K-nonnegative double splitting using the notion of matrices that leave a cone K ⊆ ℝn invariant and studied its convergence theory by generalizing the corresponding results for the nonnegative double splitting by Song and Song (2011). However, the convergence theory for K-weak regular and K-nonnegative double splittings of type II is not yet studied. In this article, we first introduce this class of splittings and then discuss the convergence theory for these sub-classes of matrices. We then obtain the comparison results for two double splittings of a K-monotone matrix. Most of these results are completely new even for \(K = \mathbb{R}_ + ^n\) . The convergence behavior is discussed by performing numerical experiments for different matrices derived from the discretized Poisson equation. PubDate: 2022-06-01
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Abstract: Abstract We propose a feasible primal-dual path-following interior-point algorithm for semidefinite least squares problems (SDLS). At each iteration, the algorithm uses only full Nesterov-Todd steps with the advantage that no line search is required. Under new appropriate choices of the parameter β which defines the size of the neighborhood of the central-path and of the parameter θ which determines the rate of decrease of the barrier parameter, we show that the proposed algorithm is well defined and converges to the optimal solution of SDLS. Moreover, we obtain the currently best known iteration bound for the algorithm with a short-update method, namely, \({\cal O}(\sqrt n \log (n/\varepsilon))\) . Finally, we report some numerical results to illustrate the efficiency of our proposed algorithm. PubDate: 2022-06-01
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Abstract: Abstract We investigate the Cohen-Grosberg differential equations with mixed delays and time-varying coefficient: Several useful results on the functional space of such functions like completeness and composition theorems are established. By using the fixed-point theorem and some properties of the doubly measure pseudo almost automorphic functions, a set of sufficient criteria are established to ensure the existence, uniqueness and global exponential stability of a (μ, ν)-pseudo almost automorphic solution. The theory of this work generalizes the classical results on weighted pseudo almost automorphic functions. Finally, a numerical example is provided to illustrate the validity of the proposed theoretical results. PubDate: 2022-06-01
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Abstract: Abstract A method for the detection of the initial stress tensor is proposed. The method is based on measuring distances between pairs of points located on the wall of underground opening in the excavation process. This methods is based on solving twelve auxiliary problems in the theory of elasticity with force boundary conditions, which is done using the least squares method. The optimal location of the pairs of points on the wall of underground openings is studied. The pairs must be located so that the condition number of the least square matrix has the minimal value, which guarantees a reliable estimation of initial stress tensor. PubDate: 2022-06-01
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Abstract: Abstract Recently, Na Huang and Changfeng Ma in (2016) proposed two kinds of typical practical choices of the PPS method. In this paper, we extrapolate two versions of the PPS iterative method, and we introduce the extrapolated Hermitian and skew-Hermitian positive definite and positive semi-definite splitting (EHPPS) iterative method and extrapolated triangular positive definite and positive semi-definite splitting (ETPPS) iterative method. We also investigate convergence analysis and consistency of the proposed iterative methods. Then, we study upper bounds for the spectral radius of iteration matrices and give upper bounds for the extrapolation parameter of the methods. Moreover, the optimal parameters which minimize upper bounds of the spectral radius are obtained. Finally, several numerical examples are given to show the efficiency of the presented method. PubDate: 2022-06-01
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Abstract: Abstract We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data u0 ∈ Hs−1+ε, w0 ∈ Hs−1 and b0 ∈ Hs for \(s > {3 \over 2}\) and any 0 < ε < 1. The initial regularity of the micro-rotational velocity w is weaker than velocity of the fluid u. PubDate: 2022-04-01
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Abstract: Abstract A new model is proposed to mimic the response of a class of seemingly viscoplastic materials. Using the proposed model, the steady, fully developed flow of the fluid is studied in a cylindrical pipe. The semi-inverse approach is applied to obtain an analytical solution for the velocity profile. The model is used to fit the shear-stress data of several supposedly viscoplastic materials reported in the literature. A numerical procedure is developed to solve the governing ODE and the procedure is validated by comparison with the analytical solution. PubDate: 2022-04-01
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Abstract: Abstract We propose a new and efficient nonmonotone adaptive trust region algorithm to solve unconstrained optimization problems. This algorithm incorporates two novelties: it benefits from a radius dependent shrinkage parameter for adjusting the trust region radius that avoids undesirable directions and exploits a new strategy to prevent sudden increments of objective function values in nonmonotone trust region techniques. Global convergence of this algorithm is investigated under some mild conditions. Numerical experiments demonstrate the efficiency and robustness of the proposed algorithm in solving a collection of unconstrained optimization problems from the CUTEst package. PubDate: 2022-04-01
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Abstract: Abstract The aim of this paper is to propose two modified forward-backward splitting algorithms for zeros of the sum of a maximal monotone operator and a Bregman inverse strongly monotone operator in reflexive Banach spaces. We prove weak and strong convergence theorems of the generated sequences by the proposed methods under some suitable conditions. We apply our results to study the variational inequality problem and the equilibrium problem. Finally, a numerical example is given to illustrate the proposed methods. The results presented in this paper improve and generalize many known results in recent literature. PubDate: 2022-04-01
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Abstract: Abstract There has been much interest in studying symmetric cone complementarity problems. In this paper, we study the circular cone complementarity problem (denoted by CCCP) which is a type of nonsymmetric cone complementarity problem. We first construct two smoothing functions for the CCCP and show that they are all coercive and strong semismooth. Then we propose a smoothing algorithm to solve the CCCP. The proposed algorithm generates an infinite sequence such that the value of the merit function converges to zero. Moreover, we show that the iteration sequence must be bounded if the solution set of the CCCP is nonempty and bounded. At last, we prove that the proposed algorithm has local superlinear or quadratical convergence under some assumptions which are much weaker than Jacobian nonsingularity assumption. Some numerical results are reported which demonstrate that our algorithm is very effective for solving CCCPs. PubDate: 2022-04-01
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Abstract: Abstract We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain D ⊂ ℝd and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by \({\cal P}\) . We associate to Problem \({\cal P}\) an optimal control problem, denoted by \({\cal Q}\) . Then, using appropriate Tykhonov triples, governed by a nonlinear operator G and a convex \(\tilde K\) , we provide results concerning the well-posedness of problems \({\cal P}\) and \({\cal Q}\) . Our main results are Theorems 4.2 and 5.2, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem \({\cal P}\) , constructed with particular choices of G and \(\tilde K\) . We prove that Theorems 4.2 and 5.2 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results. PubDate: 2022-04-01
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Abstract: Abstract We study linear stability of solutions to the Navier-Stokes equations with stochastic viscosity. Specifically, we assume that the viscosity is given in the form of a stochastic expansion. Stability analysis requires a solution of the steady-state Navier-Stokes equation and then leads to a generalized eigenvalue problem, from which we wish to characterize the real part of the rightmost eigenvalue. While this can be achieved by Monte Carlo simulation, due to its computational cost we study three surrogates based on generalized polynomial chaos, Gaussian process regression and a shallow neural network. The results of linear stability analysis assessment obtained by the surrogates are compared to that of Monte Carlo simulation using a set of numerical experiments. PubDate: 2022-03-01
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Abstract: Abstract In this paper, we deal with the optimal choice of the parameter γ for augmented Lagrangian preconditioning of GMRES method for efficient solution of linear systems obtained from discretization of the incompressible Navier-Stokes equations. We consider discretization of the equations using the B-spline based isogeometric analysis approach. We are interested in the dependence of the convergence on the parameter γ for various problem parameters (Reynolds number, mesh refinement) and especially for various isogeometric discretizations (degree and interelement continuity of the B-spline discretization bases). The idea is to be able to determine the optimal value of γ for a problem that is relatively cheap to compute and, based on this value, predict suitable values for other problems, e.g., with finer mesh, different discretization, etc. The influence of inner solvers (direct or iterative based on multigrid method) is also discussed. PubDate: 2022-02-17
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Abstract: Abstract This article deals with the low Mach number limit of the compressible Euler-Korteweg equations. It is justified rigorously that solutions of the compressible Euler-Korteweg equations converge to those of the incompressible Euler equations as the Mach number tends to zero. Furthermore, the desired convergence rates are also obtained. PubDate: 2022-02-16
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Abstract: Abstract We study existence, uniqueness, continuous dependence, general decay of solutions of an initial boundary value problem for a viscoelastic wave equation with strong damping and nonlinear memory term. At first, we state and prove a theorem involving local existence and uniqueness of a weak solution. Next, we establish a sufficient condition to get an estimate of the continuous dependence of the solution with respect to the kernel function and the nonlinear terms. Finally, under suitable conditions to obtain the global solution, we prove the general decay property with positive initial energy for this global solution. PubDate: 2022-02-08
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Abstract: Abstract We prove a uniqueness result of weak solutions to the nD (n ⩾ 3) Cauchy problem of a Keller-Segel-Navier-Stokes system with a logistic term. PubDate: 2022-02-01
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Abstract: Abstract In this paper, we propose a smoothing Levenberg-Marquardt method for the symmetric cone complementarity problem. Based on a smoothing function, we turn this problem into a system of nonlinear equations and then solve the equations by the method proposed. Under the condition of Lipschitz continuity of the Jacobian matrix and local error bound, the new method is proved to be globally convergent and locally superlinearly/quadratically convergent. Numerical experiments are also employed to show that the method is stable and efficient. PubDate: 2022-02-01
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