New gauss–seidel like block iterative methods to solve discrete boundary value problems

The concept of new Gauss–Seidel like iterative methods, which was introduced in [3], will be extended so as to obtain a class of convergent Gauss–Seidel like block iterative methods to solve linear matrix equations Ax=b with an M-Matrix A. New block iterative methods will be applied to finite difference approximations of the Laplace's equation on a square (“model problem” [8]) which surpass even the block successive overrelaxation iterative method with optimum relaxation factor in this example.     

A Gauss–Seidel type solver for the fast computation of input-constrained control systems

An important class of nonlinear control systems can be represented as the feedback interconnection of two parts: a linear time-invariant system and a block of decentralized nonlinearities. When the linear time-invariant part has a nontrivial feedthrough term or the nonlinearity has a feedback gain, control computation involves the online implementation of a multivariable algebraic loop which must be resolved at each sampling instant. The requirements for such online computation may result in several implementation issues, especially in real-time and embedded control applications. This paper considers the implementation of such algebraic loops arising from several input-constrained systems. The proposed solution algorithm is globally convergent for a very large class of feedthrough or feedback gains and shows promise for real-time and embedded control applications where a fast but approximate solution is of the essence.     

基于迭代方法的补偿器系统设计

针对F数为4的高次非球面,设计了一套基于平行光束照射的高精度补偿器。系统工作波长为632.8 nm,设计残差为0.0018λ。提出了求解两片检测透镜组成补偿器初始结构的新方法。该方法合理假设补偿器部分结构参数,通过判断补偿器与非球面的Seidel系数之和最小获得未知参数。比较非球面在补偿器第一面位置处的高斯像高与假设的入射平行光高度,通过MATLAB迭代计算得到高度差小于预设误差,进而确定平行光入射高度。重新复算得到最终的初始结构。最后将迭代计算得到的各个面Seidel系数和ZEMAX给出的结果进行比较,证实了该方法的可靠性。     

高速移动环境下OTSM迭代检测算法研究

正交时序复用( Orthogonal Time Sequency Multiplexing, OTSM)通过级联时分和沃尔什-哈达玛(WHT)复用将信息符号在时延和序列域进行复用。由于WHT在调制解调过程不需要进行复杂的乘法运算,相比于正交时频空(OTFS)调制有更低的调制复杂度。该文针对高速移动环境下的OTSM系统提出了一种二级均衡器：首先利用信道矩阵的稀疏性和带状结构在时域逐块进行低复杂度MMSE检测;随后采用高斯-赛德尔(GS)迭代检测进一步消除残余符号干扰。仿真结果表明,所提算法与基于单抽头频域均衡的GS迭代检测算法相比,采用16QAM调制且误码率为10^–4时有1.8 dB性能增益。     

Linear and nonlinear solvers for variational phase‐field models of brittle fracture

The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns but is computationally demanding. The model is a strongly nonlinear non‐convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill‐conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss–Seidel iteration and employ over‐relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton's method, to further reduce the time to solution, and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton's method. We investigate the improvements in efficiency on several examples from the literature; the new solver is five to six times faster on a majority of the test cases considered. © 2016 The Authors International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

Exact performance analytical model for spectrum allocation in flexible grid optical networks

Dynamic flexible grid optical networks have gained much attention because of the advantages of high spectrum efficiency and flexibility, while the performance analysis will be more complex compared with fixed grid optical networks. An analytical Markov model is first presented in the paper, which can exactly describe the stochastic characteristics of the spectrum allocation in flexible grid optical networks considering both random-fit and first-fit resource assignment policies. We focus on the effect of spectrum contiguous constraint which has not been systematically studied in respect of mathematical modeling, and three major properties of the model are presented and analyzed. The model can expose key performance features and act as the foundation of modeling the Routing and Spectrum Assignment (RSA) problem with diverse topologies. Two heuristic algorithms are also proposed to make it more tractable. Finally, several key parameters, such as blocking probability, resource utilization rate and fragmentation rate are presented and computed, and the corresponding Monte Carlo simulation results match closely with analytical results, which prove the correctness of this mathematical model.     

鱼眼镜头后光组初值设计

在鱼眼镜头的设计过程中,设计者往往依赖设计经验选择专利作为初始结构,缺乏理论指导。在鱼眼镜头前光组结构和后光组的光焦度以及光学间隔的基础上,提出一种求解鱼眼镜头后光组曲率半径初始参数的方法。通过前、后光组球差和彗差的波像差平衡条件建立方程组,求解该方程组后得到后光组的各光学面曲率半径初值,并以曲率半径初值作为参考值,应用Zemax软件进一步优化,得到初始结构。研究表明,应用这种方法求解鱼眼镜头后光组的初始光学参数,能使优化算法在较短时间内搜索到最优的参数解。     

Iterative solution of a hybrid method for Maxwell's equations in the frequency domain

A hybrid method for solution of Maxwell's equations of electromagnetics in the frequency domain is developed as a combination between the method of moments and the approximation in physical optics. The equations are discretized by a Galerkin method and solved by an iterative block Gauss–Seidel method. The convergence of the iterations is studied theoretically and in numerical experiments. The accuracy of the hybrid method is compared to the method of moments for a cylinder with an incident field for different wavenumbers. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite-element Discretization of Static Hamilton-Jacobi Equations based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton–Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss–Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.Dedicated to Peter Deuflhard on the occasion of his 60th birthday     

Error bounds on block Gauss–Seidel solutions of coupled multiphysics problems

Mathematical models in many fields often consist of coupled sub‐models, each of which describes a different physical process. For many applications, the quantity of interest from these models may be written as a linear functional of the solution to the governing equations. Mature numerical solution techniques for the individual sub‐models often exist. Rather than derive a numerical solution technique for the full coupled model, it is therefore natural to investigate whether these techniques may be used by coupling in a block Gauss–Seidel fashion. In this study, we derive two a posteriori bounds for such linear functionals. These bounds may be used on each Gauss–Seidel iteration to estimate the error in the linear functional computed using the single physics solvers, without actually solving the full, coupled problem. We demonstrate the use of the bound first by using a model problem from linear algebra, and then a linear ordinary differential equation example. We then investigate the effectiveness of the bound using a non‐linear coupled fluid‐temperature problem. One of the bounds derived is very sharp for most linear functionals considered, allowing us to predict very accurately when to terminate our block Gauss–Seidel iteration. Copyright © 2011 John Wiley & Sons, Ltd.