Block ILU Preconditioners for a Nonsymmetric Block-Tridiagonal M-Matrix

We propose block ILU (incomplete LU) factorization preconditioners for a nonsymmetric block-tridiagonal M-matrix whose computation can be done in parallel based on matrix blocks. Some theoretical properties for these block ILU factorization preconditioners are studied and then we describe how to construct them effectively for a special type of matrix. We also discuss a parallelization of the preconditioner solver step used in nonstationary iterative methods with the block ILU preconditioners. Numerical results of the right preconditioned BiCGSTAB method using the block ILU preconditioners are compared with those of the right preconditioned BiCGSTAB using a standard ILU factorization preconditioner to see how effective the block ILU preconditioners are.     

Kronecker product approximation preconditioners for convection–diffusion model problems

We consider the iterative solution of linear systems arising from four convection–diffusion model problems: scalar convection–diffusion problem, Stokes problem, Oseen problem and Navier–Stokes problem. We design preconditioners for these model problems that are based on Kronecker product approximations (KPAs). For this we first identify explicit Kronecker product structure of the coefficient matrices, in particular for the convection term. For the latter three model cases, the coefficient matrices have a 2 × 2 block structure, where each block is a Kronecker product or a summation of several Kronecker products. We then use this structure to design a block diagonal preconditioner, a block triangular preconditioner and a constraint preconditioner. Numerical experiments show the efficiency of the three KPA preconditioners, and in particular of the constraint preconditioner that usually outperforms the other two. This can be explained by the relationship that exists between these three preconditioners: the constraint preconditioner can be regarded as a modification of the block triangular preconditioner, which at its turn is a modification of the block diagonal preconditioner based on the cell Reynolds number. Copyright © 2009 John Wiley & Sons, Ltd.     

Modified Block SSOR Preconditioners for Symmetric Positive Definite Linear Systems

A class of modified block SSOR preconditioners is presented for the symmetric positive definite systems of linear equations, whose coefficient matrices come from the hierarchical-basis finite-element discretizations of the second-order self-adjoint elliptic boundary value problems. These preconditioners include a block SSOR iteration preconditioner, and two inexact block SSOR iteration preconditioners whose diagonal matrices except for the (1,1)-block are approximated by either point symmetric Gauss–Seidel iterations or incomplete Cholesky factorizations, respectively. The optimal relaxation factors involved in these preconditioners and the corresponding optimal condition numbers are estimated in details through two different approaches used by Bank, Dupont and Yserentant (Numer. Math. 52 (1988) 427–458) and Axelsson (Iterative Solution Methods (Cambridge University Press, 1994)). Theoretical analyses show that these modified block SSOR preconditioners are very robust, have nearly optimal convergence rates, and especially, are well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.     

FSAI-based parallel Mixed Constraint Preconditioners for saddle point problems arising in geomechanics

In this paper we propose and describe a parallel implementation of a block preconditioner for the solution of saddle point linear systems arising from Finite Element (FE) discretization of 3D coupled consolidation problems. The Mixed Constraint Preconditioner developed in [L. Bergamaschi, M. Ferronato, G. Gambolati, Mixed constraint preconditioners for the solution to FE coupled consolidation equations, J. Comput. Phys., 227(23) (2008), 9885-9897] is combined with the parallel FSAI preconditioner which is used here to approximate the inverses of both the structural (1, 1) block and an appropriate Schur complement matrix. The resulting preconditioner proves effective in the acceleration of the BiCGSTAB iterative solver. Numerical results on a number of test cases of size up to 2×10⁶ unknowns and 1.2×10⁸ nonzeros show the perfect scalability of the overall code up to 256 processors.     

FSAI-based parallel Mixed Constraint Preconditioners for saddle point problems arising in geomechanics

In this paper we propose and describe a parallel implementation of a block preconditioner for the solution of saddle point linear systems arising from Finite Element (FE) discretization of 3D coupled consolidation problems. The Mixed Constraint Preconditioner developed in [L. Bergamaschi, M. Ferronato, G. Gambolati, Mixed constraint preconditioners for the solution to FE coupled consolidation equations, J. Comput. Phys., 227(23) (2008), 9885–9897] is combined with the parallel FSAI preconditioner which is used here to approximate the inverses of both the structural (1, 1) block and an appropriate Schur complement matrix. The resulting preconditioner proves effective in the acceleration of the BiCGSTAB iterative solver. Numerical results on a number of test cases of size up to 2×10⁶ unknowns and 1.2×10⁸ nonzeros show the perfect scalability of the overall code up to 256 processors.     

A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems

In this paper, a class of generalized shift-splitting preconditioners with two shift parameters are implemented for nonsymmetric saddle point problems with nonsymmetric positive definite (1, 1) block. The generalized shift-splitting (GSS) preconditioner is induced by a generalized shift-splitting of the nonsymmetric saddle point matrix, resulting in an unconditional convergent fixed-point iteration. By removing the shift parameter in the (1, 1) block of the GSS preconditioner, a deteriorated shift-splitting (DSS) preconditioner is presented. Some useful properties of the DSS preconditioned saddle point matrix are studied. Finally, numerical experiments of a model Navier–Stokes problem are presented to show the effectiveness of the proposed preconditioners.     

An Algebraic Multigrid-Based Physical Factorization Preconditioner for the Multi-Group Radiation Diffusion Equations in Three Dimensions

The paper investigates the robustness and parallel scaling properties of a novel physical factorization preconditioner with algebraic multigrid subsolves in the iterative solution of a cell-centered finite volume discretization of the three-dimensional multi-group radiation diffusion equations. The key idea is to take advantage of a particular kind of block factorization of the resulting system matrix and approximate the left-hand block matrix selectively spurred by parallel processing considerations. The spectral property of the preconditioned matrix is then analyzed. The practical strategy is considered sequentially and in parallel. Finally, numerical results illustrate the numerical robustness, computational efficiency and parallel strong and weak scalabilities over the real-world structured and unstructured coupled problems, showing its competitiveness with many existing block preconditioners.     

Local preconditioners for two‐level non‐overlapping domain decomposition methods

We consider additive two‐level preconditioners, with a local and a global component, for the Schur complement system arising in non‐overlapping domain decomposition methods. We propose two new parallelizable local preconditioners. The first one is a computationally cheap but numerically relevant alternative to the classical block Jacobi preconditioner. The second one exploits all the information from the local Schur complement matrices and demonstrates an attractive numerical behaviour on heterogeneous and anisotropic problems. We also propose two implementations based on approximate Schur complement matrices that are cheaper alternatives to construct the given preconditioners but that preserve their good numerical behaviour. Through extensive computational experiments we study the numerical scalability and the robustness of the proposed preconditioners and compare their numerical performance with well‐known robust preconditioners such as BPS and the balancing Neumann–Neumann method. Finally, we describe a parallel implementation on distributed memory computers of some of the proposed techniques and report parallel performances. Copyright © 2001 John Wiley & Sons, Ltd.     

Modified block preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

ABSTRACT

In this paper, based on the preconditioners presented by Zhang [A new preconditioner for generalized saddle matrices with highly singular(1,1) blocks. Int J Comput Maths. 2014;91(9):2091-2101], we consider a modified block preconditioner for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and the minimal polynomial. Finally, numerical examples show the eigenvalue distribution with the presented preconditioner and confirm our analysis.     

A hybrid domain decomposition method based on aggregation

A new two‐level black‐box preconditioner based on the hybrid domain decomposition technique is proposed and studied. The preconditioner is a combination of an additive Schwarz preconditioner and a special smoother. The smoother removes dependence of the condition number on the number of subdomains and variations of the diffusion coefficient and leaves minor sensitivity to the problem size. The algorithm is parallel and pure algebraic which makes it a convenient framework for the construction parallel black‐box preconditioners on unstructured meshes. Copyright © 2004 John Wiley & Sons, Ltd.     

SINE TRANSFORM MATRIX FOR SOLVING TOEPLITZ MATRIX PROBLEMS

1. IntroductionStrang[1] first studied the use of circulallt matrices C for solving systems of linear eqllationsTi x = b witha symmetric positive definite Toeplitz matrix.Numerous authors such as T.Chan[2],R.Chan,etc.[3],[4],[5], Tyrtyshnikov[6], Huckle[7] and T.Ku and C.Kuo[8] proposed differentfamilies of circulallt / skew- circulant precondit ioners.Appling the preconditioned conjugate gradient algorithm(PCGA) to solve the systems Ti x -b, we must find a preconditioner P such that P…     

On parallel solution of linear elasticity problems. Part II: Methods and some computer experiments

This is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coefficients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method. In the first part of the trilogy block‐diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg‐method, i.e. inner iterations were performed. As preconditioner, we used modified incomplete factorization MIC(0), where possibly the element matrices were modified in order to give M‐matrices, i.e. in order to guarantee the existence of the MIC(0) factorization. In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner/outer iterations we also study a variant of the block‐diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)‐factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system. In the third part of the trilogy, we will focus on the use of higher‐order finite elements. Copyright © 2002 John Wiley & Sons, Ltd.     

A domain embedding preconditioner for the Lagrange multiplier system

Finite element approximations for the Dirichlet problem associated to a second-order elliptic differential equation are studied. The purpose of this paper is to discuss domain embedding preconditioners for discrete systems. The essential boundary condition on the interior interface is removed by introducing Lagrange multipliers. The associated discrete system, with a saddle point structure, is preconditioned by a block diagonal preconditioner. The main contribution of this paper is to propose a new operator, constructed from the -inner product, for the block of the preconditioner corresponding to the multipliers.

     

Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations

We consider an abstract parameter dependent saddle-point problem and present a general framework for analyzing robust Schur complement preconditioners. The abstract analysis is applied to a generalized Stokes problem, which yields robustness of the Cahouet-Chabard preconditioner. Motivated by models for two-phase incompressible flows we consider a generalized Stokes interface problem. Application of the general theory results in a new Schur complement preconditioner for this class of problems. The robustness of this preconditioner with respect to several parameters is treated. Results of numerical experiments are given that illustrate robustness properties of the preconditioner.     

Approximate Inverse Preconditioners for Some Large Dense Random Electrostatic Interaction Matrices

A sparse mesh-neighbour based approximate inverse preconditioner is proposed for a type of dense matrices whose entries come from the evaluation of a slowly decaying free space Green’s function at randomly placed points in a unit cell. By approximating distant potential fields originating at closely spaced sources in a certain way, the preconditioner is given properties similar to, or better than, those of a standard least squares approximate inverse preconditioner while its setup cost is only that of a diagonal block approximate inverse preconditioner. Numerical experiments on iterative solutions of linear systems with up to four million unknowns illustrate how the new preconditioner drastically outperforms standard approximate inverse preconditioners of otherwise similar construction, and especially so when the preconditioners are very sparse. AMS subject classification (2000) 65F10, 65R20, 65F35, 78A30     

区域分解界面预条件子构造的一般框架

1．引言考虑模型问题：其中ΩR2是多边形区域,常数n≥0．将Ω作非重叠区域分解：Ω＝假定：（i）当i≠j时,（ii）当Ωi与Ωj相邻时,是Ωi和Ωj的一条公共边记称为界面）;（iii）每个闪的尺寸为d,即存在常数co和q,使出包含（包含在）一个直径为C（）（Cod）的圆（国内）．非重叠区域分解方法的实质是,引进两个变量：内部变量。h和界面变量～．先在几上并行未解子问题,将。。消去（即用～表示）,得到～的方程（称为界面方程）;再求解界面方程,得到～的值;最后将～回代,得到。人的值（即原问题的解）．这类区域分解方法是否比重…     

Evaluation of ST preconditioners for saddle point problems

The general block ST decomposition of the saddle point problem is used as a preconditioner to transform the saddle point problem into an equivalent symmetric and positive definite system. Such a decomposition is called a block ST preconditioner. Two general block ST preconditioners are proposed for saddle point problems with symmetric and positive definite (1,1)-block. Some estimations of the condition number of the preconditioned system are given. The same study is done for singular (1,1)-block.     

BCCB preconditioners for systems of BVM‐based numerical integrators

Boundary value methods (BVMs) for ordinary differential equations require the solution of non‐symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A block‐circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is A_k1,k2‐stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block‐circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.     

Preconditioners for block Toeplitz systems based on circulant preconditioners

We study the numerical solution of a block system T _m,n x=b by preconditioned conjugate gradient methods where T _m,n is an m×m block Toeplitz matrix with n×n Toeplitz blocks. These systems occur in a variety of applications, such as two-dimensional image processing and the discretization of two-dimensional partial differential equations. In this paper, we propose new preconditioners for block systems based on circulant preconditioners. From level-1 circulant preconditioner we construct our first preconditioner q ₁(T _m,n ) which is the sum of a block Toeplitz matrix with Toeplitz blocks and a sparse matrix with Toeplitz blocks. By setting selected entries of the inverse of level-2 circulant preconditioner to zero, we get our preconditioner q ₂(T _m,n ) which is a (band) block Toeplitz matrix with (band) Toeplitz blocks. Numerical results show that our preconditioners are more efficient than circulant preconditioners.     

A family of constrained pressure residual preconditioners for parallel reservoir simulations

Large‐scale reservoir simulations are extremely time‐consuming because of the solution of large‐scale linear systems arising from the Newton or Newton–Raphson iterations. The problem becomes even worse when highly heterogeneous geological models are employed. This paper introduces a family of multi‐stage preconditioners for parallel black oil simulations, which are based on the famous constrained pressure residual preconditioner. Numerical experiments demonstrate that our preconditioners are robust, efficient, and scalable. Copyright © 2015 John Wiley & Sons, Ltd.