Normalized explicit finite element approximate inverse preconditioning

Normalized explicit approximate inverse matrix techniques for computing explicitly various families of normalized approximate inverses based on normalized approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference and finite element discretization of partial differential equations are presented. Normalized explicit preconditioned conjugate gradient-type schemes in conjunction with normalized approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear systems. Theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.     

Explicit approximate inverse preconditioning techniques

Summary  The numerical treatment and the production of related software for solving large sparse linear systems of algebraic equations, derived mainly from the discretization of partial differential equation, by preconditioning techniques has attracted the attention of many researchers. In this paper we give an overview of explicit approximate inverse matrix techniques for computing explicitly various families of approximate inverses based on Choleski and LU—type approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference, finite element and the domain decomposition discretization of elliptic and parabolic partial differential equations. Composite iterative schemes, using inner-outer schemes in conjunction with Picard and Newton method, based on approximate inverse matrix techniques for solving non-linear boundary value problems, are presented. Additionally, isomorphic iterative methods are introduced for the efficient solution of non-linear systems. Explicit preconditioned conjugate gradient—type schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear system of algebraic equations. Theoretical estimates on the rate of convergence and computational complexity of the explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.     

Explicit isomorphic iterative methods for solving arrow-type linear systems

A new class of approximate inverse arrow-type matrix techniques based on the concept of sparse approximate LU-type factorization procedures is introduced for computing explicitly approximate inverses without inverting the decomposition factors. Isomorphic methods in conjunction with explicit preconditioned schemes based on approximate inverse matrix techniques are presented for the efficient solution of arrow-type linear systems. Applications of the proposed method on linear systems is discussed and numerical results are given     

Design and implementation of parallel approximate inverse classes using OpenMP

A new parallel normalized optimized approximate inverse algorithm, based on the concept of antidiagonal wave pattern, for computing classes of explicitly approximate inverses, is introduced for symmetric multiprocessor systems. The parallel normalized explicit approximate inverses are used in conjunction with parallel normalized explicit preconditioned conjugate gradient schemes for the efficient solution of finite element sparse linear systems. The parallel design and implementation issues of the new algorithm are discussed and the parallel performance is presented using OpenMP. Copyright © 2008 John Wiley & Sons, Ltd.     

Parallel and Systolic Solution of Normalized Explicit Approximate Inverse Preconditioning

A new class of normalized approximate inverse matrix techniques, based on the concept of sparse normalized approximate factorization procedures are introduced for solving sparse linear systems derived from the finite difference discretization of partial differential equations. Normalized explicit preconditioned conjugate gradient type methods in conjunction with normalized approximate inverse matrix techniques are presented for the efficient solution of sparse linear systems. Theoretical results on the rate of convergence of the normalized explicit preconditioned conjugate gradient scheme and estimates of the required computational work are presented. Application of the new proposed methods on two dimensional initial/boundary value problems is discussed and numerical results are given. The parallel and systolic implementation of the dominant computational part is also investigated.     

On the Solution of Boundary Value Problems by Using Fast Generalized Approximate Inverse Banded Matrix Techniques

A class of finite difference schemes in conjunction with approximate inverse banded matrix techniques based on the concept of LU-type factorization procedures is introduced for computing fast explicit approximate inverses. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of banded linear systems. A theorem on the rate of convergence and estimates of the computational complexity required to reduce the L-norm of the error is presented. Applications of the method on linear and non-linear systems are discussed and numerical results are given.     

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.     

A parallel Self Mesh-Adaptive N-body method based on approximate inverses

A new parallel Self Mesh-Adaptive N-body method based on approximate inverses is proposed. The scheme is a three-dimensional Cartesian-based method that solves the Poisson equation directly in physical space, using modified multipole expansion formulas for the boundary conditions. Moreover, adaptive-mesh techniques are utilized to form a class of separate smaller n-body problems that can be solved in parallel and increase the total resolution of the system. The solution method is based on multigrid method in conjunction with the symmetric factored approximate sparse inverse matrix as smoother. The design of the parallel Self Mesh-Adaptive method along with discussion on implementation issues for shared memory computer systems is presented. The new parallel method is evaluated through a series of benchmark simulations using N-body models of isolated galaxies or galaxies interacting with dwarf companions. Furthermore, numerical results on the performance and the speedups of the scheme are presented.     

Implicit multigrid schemes for challenging aerodynamic simulations on block-structured grids

In this paper, the use of implicit multigrid smoothers for challenging aerodynamic simulations is explored. The block lower–upper symmetric Gauss–Seidel (LU-SGS) and hybrid Runge–Kutta/LU-SGS schemes are implemented in Bombardier’s multiblock Navier–Stokes solver, FANSC. The schemes are compared to the existing Runge–Kutta and point-Jacobi preconditioned explicit multistage smoothers. Through tests ranging from 2D airfoils to 3D wing-body-engine cases, the computational speed-up and robustness of the implicit schemes are evaluated. It is shown that the implicit smoothers present a computational speed-up of at least two, and are significantly more robust, especially for flow problems involving the “power-on” engine boundary condition.     

Multigrid Methods for the Stokes Equations using Distributive Gauss–Seidel Relaxations based on the Least Squares Commutator

A distributive Gauss–Seidel relaxation based on the least squares commutator is devised for the saddle-point systems arising from the discretized Stokes equations. Based on that, an efficient multigrid method is developed for finite element discretizations of the Stokes equations on both structured grids and unstructured grids. On rectangular grids, an auxiliary space multigrid method using one multigrid cycle for the Marker and Cell scheme as auxiliary space correction and least squares commutator distributive Gauss–Seidel relaxation as a smoother is shown to be very efficient and outperforms the popular block preconditioned Krylov subspace methods.     

Explicit preconditioned domain decomposition schemes for solving nonlinear boundary value problems

A new class of inner-outer iterative procedures in conjunction with Picard-Newton methods based on explicit preconditioning iterative methods for solving nonlinear systems is presented. Explicit preconditioned iterative schemes, based on the explicit computation of a class of domain decomposition generalized approximate inverse matrix techniques are presented for the efficient solution of nonlinear boundary value problems on multiprocessor systems. Applications of the new composite scheme on characteristic nonlinear boundary value problems are discussed and numerical results are given.     

High Performance Inverse Preconditioning

The derivation of parallel numerical algorithms for solving sparse linear systems on modern computer systems and software platforms has attracted the attention of many researchers over the years. In this paper we present an overview on the design issues of parallel approximate inverse matrix algorithms, based on an anti-diagonal “wave pattern” approach and a “fish-bone” computational procedure, for computing explicitly various families of exact and approximate inverses for solving sparse linear systems. Parallel preconditioned conjugate gradient-type schemes in conjunction with parallel approximate inverses are presented for the efficient solution of sparse linear systems. Applications of the proposed parallel methods by solving characteristic sparse linear systems on symmetric multiprocessor systems and distributed systems are discussed and the parallel performance of the proposed schemes is given, using MPI, OpenMP and Java multithreading.     

A parallel preconditioned conjugate gradient solution method for finite element problems with coarse–fine mesh formulation

The object of this paper is a parallel preconditioned conjugate gradient iterative solver for finite element problems with coarse-mesh/fine-mesh formulation. An efficient preconditioner is easily derived from the multigrid stiffness matrix. The method has been implemented, for the sake of comparison, both on a IBM-RISC590 and on a Quadrics-QH1, a massive parallel SIMD machine with 128 processors. Examples of solutions of simple linear elastic problems on rectangular grids are presented and convergence and parallel performance are discussed.     

一种针对大波数Helmholtz方程的高性能并行预条件迭代求解算法

针对传统串行迭代法求解大波数Helmholtz方程存在效率低下且受限于单机内存的问题,提出了一种基于消息传递接口(Message Passing Interface,MPI) 的并行预条件迭代法。该算法利用复移位拉普拉斯算子对Helmholtz方程进行预条件处理,联合稳定双共轭梯度法和基于矩阵的多重网格法来求解预条件方程离散后的大规模线性系统,在Linux集群系统上基于 MPI环境实现了求解算法的并行计算,重点解决了多重网格的并行划分、信息传递和多重网格组件的构建问题。数值实验表明,对于大波数问题,提出的算法具有良好的并行加速比,相较于串行算法极大地提高了计算效率。     

Approximate inverse preconditioning for the fast multipole BEM in acoustics

Abstact Acoustic radiation from vibrating structures is simulated by a Galerkin boundary element method based on the Burton–Miller approach. The boundary element operators are evaluated by the fast multipole method that allows large-scale computations in the medium frequency range. Two iterative solvers are considered: the generalized minimal residual method and a multigrid solver. Both approaches can be accelerated greatly by the presented approximate inverse preconditioner. Communicated by: U. Langer Research of the author is supported by the Deutsche Forschungsgemeinschaft in the framework of the collaborative research centre SFB 404 “Multifield Problems in Solid and Fluid Mechanics”     

An efficient multigrid preconditioner for Maxwell’s equations in micromagnetism

We consider a system of Maxwell’s and Landau-Lifshitz-Gilbert equations describing magnetization dynamics in micromagnetism. The problem is discretized by a convergent, unconditionally stable finite element method. A multigrid preconditioned Uzawa type method for the solution of the algebraic system resulting from the discretized Maxwell’s equations is constructed. The efficiency of the method is demonstrated on numerical experiments and the results are compared to those obtained by simplified models.     

A rapid method of approximating the asymptote to an iterative sequence

We discuss a procedure for the adaptive construction of sparse approximate inverse preconditionings for general sparse linear systems. The approximate inverses are based on minimizing a consistent norm of the difference between the identity and the preconditioned matrix. The analysis provides positive definiteness and condition number estimates for the preconditioned system under certain circumstances. We show that for the 1-norm, restricting the size of the difference matrix below 1 may require dense approximate inverses. However, this requirement does not hold for the 2-norm, and similarly reducing the Frobenius norm below 1 does not generally require that much fill-in. Moreover, for the Frobenius norm, the calculation of the approximate inverses yields naturally column-oriented parallelism. General sparsity can be exploited in a straightforward fashion. Numerical criteria are considered for determining which columns of the sparse approximate inverse require additional fill-in. Spare algorithms are discussed for the location of potential fill-in within each column. Results using a minimum-residual-type iterative method are presented to illustrate the potential of the method.     

Explicit preconditioned iterative methods for solving large unsymmetric finite element systems

A class of Generalized Approximate Inverse Matrix (GAIM) techniques, based on the concept of LU-sparse factorization procedures, is introduced for computing explicitly approximate inverses of large sparse unsymmetric matrices of irregular structure, without inverting the decomposition factors. Explicit preconditioned iterative methods, in conjunction with modified forms of the GAIM techniques, are presented for solving numerically initial/boundary value problems on multiprocessor systems. Application of the new methods on linear boundary-value problems is discussed and numerical results are given.     

科恩-沈方程与含时科恩-沈方程的一个自适应有限元数值方法

在材料分析、纳米光学等研究中,高质量数值模拟多体系统电子密度的随时间演化是一类重要研究内容.演化中产生的时间依赖偶极子等物理量,是更进一步研究的基础.此类数值模拟分为两个步骤.即多体系统的基态求解、及以基态为初值的系统的动态演化模拟.这两个步骤可以分别通过数值求解科恩-沈(Kohn-Sham)方程及含时科恩-沈(time-dependent Kohn-Sham)方程实现.本文中,我们提出一类基于有限元方法的数值求解框架,为这两个步骤提供一个统一的模拟实现.在基态求解中,我们利用一类自洽场迭代对方程进行线性化,采用局部最优块预处理共轭梯度法求解导出的广义特征值问题,并设计了一个基于多重网格方法的预优对求解进行有效加速.在动态演化模拟中,针对方程的结构,我们提出了一个基于隐式中点公式的数值方法,利用预估-校正方法对方程进行线性化处理,并设计了一个针对复值线性系统的代数多重网格求解器用于加速时间推进.特别地,我们基于提出的数值方法,分别针对科恩-沈及含时科恩-沈方程导出了残量型后验误差估计子,并实现了基于局部加密的网格自适应方法,用于进一步改善数值模拟效率.数值解展示了方法的有效性.     

Parallel algorithms for the solution of certain large sparse linear systems

A couple of approximate inversion techniques are presented which provide a parallel enhancement to several iterative methods for solving linear systems arising from the discretization of boundary value problems. In particular, the Jacobi, Gauss‐Seidel, and successive overrelaxation methods can be improved substantially in a parallel environment by the extensions considered. A special case convergence proof is presented. The use of our approximate inverses with the preconditioned conjugate gradient method is examined and comparisons are made with some recently proposed algorithms in this area that also employ approximate inverses. The methods considered are compared under sequential and parallel hardware assumptions.