椭圆PDE-约束优化问题的一个预条件子

针对由Galerkin有限元离散椭圆PDE-约束优化问题产生的具有特殊结构的3×3块线性鞍点系统,提出了一个预条件子并给出了预处理矩阵特征值及特征向量的具体表达形式.数值结果表明了该预条件子能够有效地加速Krylov子空间方法的收敛速率,同时也验证了理论结果.     

分布式通信响应优化问题及其内点法求解

为了优化移动互联网的分布式通信系统的响应速度,建立分布式通信系统响应速度最优化问题的数学模型,并设计和改进求解该最优化问题的内点法.针对该最优化问题发展一套高效率预条件方法来帮助求解内点法,不但改善计算方法的数值稳定性,而且提高算法的计算效率.通过数值实验验证该预条件对算法稳定性和效率的提高.     

Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs

This paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in, for example, optimal control problems constrained by such equations. As a model problem, we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual Petrov–Galerkin variational formulation of the heat equation in space–time results in a generalized least squares problem. This formulation admits a unique, quasi‐optimal solution in the natural space–time Hilbert space and serves as a basis for the development of space–time compressive solution algorithms. The solution of the heat equation is obtained by applying the conjugate gradient method to the normal equations of the generalized least squares problem. Starting from stable subspace splittings in space and in time, multilevel space–time preconditioners for the normal equations are derived. In order to reduce the complexity of the full space–time problem, all computations are performed in a compressed or sparse format called the hierarchical Tucker format, supposing that the input data are available in this format. In order to maintain sparsity, compression of the iterates within the hierarchical Tucker format is performed in each conjugate gradient iteration. Its application to vectors in the hierarchical Tucker format is detailed. Finally, numerical results in up to five spatial dimensions based on the recently developed htucker toolbox for MATLAB are presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimization of the parameterized Uzawa preconditioners for saddle point matrices

The parameterized Uzawa preconditioners for saddle point problems are studied in this paper. The eigenvalues of the preconditioned matrix are located in (0, 2) by choosing the suitable parameters. Furthermore, we give two strategies to optimize the rate of convergence by finding the suitable values of parameters. Numerical computations show that the parameterized Uzawa preconditioners can lead to practical and effective preconditioned GMRES methods for solving the saddle point problems.     

Fast solvers with block‐diagonal preconditioners for linear FEM–BEM coupling

The purpose of this paper is to present optimal preconditioned iterative methods to solve indefinite linear systems of equations arising from symmetric coupling of finite elements and boundary elements. This is a block‐diagonal preconditioner together with a conjugate residual method and a preconditioned inner–outer iteration. We prove the efficiency of these methods by showing that the number of iterations to preserve a given accuracy is bounded independent of the number of unknowns. Numerical examples underline the efficiency of these methods. Copyright © 2008 John Wiley & Sons, Ltd.     

An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions

The problem of interpolation of scattered data on the unit sphere has many applications in geodesy and Earth science in which the sphere is taken as a model for the Earth. Spherical radial basis functions provide a convenient tool for constructing the interpolant. However, the underlying linear systems tend to be ill-conditioned. In this paper, we present an additive Schwarz preconditioner for accelerating the solution process. An estimate for the condition number of the preconditioned system will be discussed. Numerical experiments using MAGSAT satellite data will be presented.     

A note on spectrum distribution of constraint preconditioned generalized saddle point matrices

In this note, results concerning the eigenvalue distribution and form of the eigenvectors of the constraint preconditioned generalized saddle point matrix and its minimal polynomial are given. These results extend previous ones that appeared in the literature. Copyright © 2009 John Wiley & Sons, Ltd.     

An improved progressive preconditioning method for steady non‐cavitating and sheet‐cavitating flows

An improved progressive preconditioning method for analyzing steady inviscid and laminar flows around fully wetted and sheet‐cavitating hydrofoils is presented. The preconditioning matrix is adapted automatically from the pressure and/or velocity flow‐field by a power‐law relation. The cavitating calculations are based on a single fluid approach. In this approach, the liquid/vapour mixture is treated as a homogeneous fluid whose density is controlled by a barotropic state law. This physical model is integrated with a numerical resolution derived from the cell‐centered Jameson's finite volume algorithm. The stabilization is achieved via the second‐and fourth‐order artificial dissipation scheme. Explicit four‐step Runge–Kutta time integration is applied to achieve the steady‐state condition. Results presented in the paper focus on the pressure distribution on hydrofoils wall, velocity profiles, lift and drag forces, length of sheet cavitation, and effect of the power‐law preconditioning method on convergence speed. The results show satisfactory agreement with numerical and experimental works of others. The scheme has a progressive effect on the convergence speed. The results indicate that using the power‐law preconditioner improves the convergence rate, significantly. Copyright © 2010 John Wiley & Sons, Ltd.     

Schur complement‐based domain decomposition preconditioners with low‐rank corrections

This paper introduces a robust preconditioner for general sparse matrices based on low‐rank approximations of the Schur complement in a Domain Decomposition framework. In this ‘Schur Low Rank’ preconditioning approach, the coefficient matrix is first decoupled by a graph partitioner, and then a low‐rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface unknowns. The method avoids explicit formation of the Schur complement. We show the feasibility of this strategy for a model problem and conduct a detailed spectral analysis for the relation between the low‐rank correction and the quality of the preconditioner. We first introduce the SLR preconditioner for symmetric positive definite matrices and symmetric indefinite matrices if the interface matrices are symmetric positive definite. Extensions to general symmetric indefinite matrices as well as to nonsymmetric matrices are also discussed. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

A comparison of preconditioners for incompressible Navier–Stokes solvers

We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the so‐called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium‐sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright © 2007 John Wiley & Sons, Ltd.