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.     

Parallel solvers for flow control based on domain decomposition

We sketch a non-overlapping domain decomposition method (DDM) for a linear quadratic optimal control problem governed by the Oseen equations. The DDM is applied to the system of necessary and sufficient optimality conditions. The approach extends balanced Neumann Neumann DDMs from single partial differential equations (PDEs) to the optimization control context, and it extends previous work on balanced Neumann Neumann DDMs for the optimal control of scalar elliptic PDEs to the optimal control of the Oseen equations. This extension requires a careful handling of the incompressibility constraint and resulting compatibility conditions, as well as a careful handling of the advection term. The DDM is used to parallelize the matrix-vector operations for the optimality system, as well as to parallelize the preconditioner. We present two approaches. One tackles the global optimality system directly, the other forms the Schur complement corresponding to variables on the subdomain interfaces. We present numerical experiments which clearly show the potential of the approaches. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bilus: A block version of ilus factorization

ILUS factorization has many desirable properties such as its amenability to the skyline format, the ease with which stability may be monitored, and the possibility of constructing a preconditioner with symmetric structure. In this paper we introduce a new preconditioning technique for general sparse linear systems based on the ILUS factorization strategy. The resulting preconditioner has the same properties as the ILUS preconditioner. Some theoretical properties of the new preconditioner are discussed and numerical experiments on test matrices from the Harwell-Boeing collection are tested. Our results indicate that the new preconditioner is cheaper to construct than the ILUS preconditioner.     

An iterative solver for the Oseen problem and numerical solution of incompressible Navier–Stokes equations

Incompressible unsteady Navier–Stokes equations in pressure–velocity variables are considered. By use of the implicit and semi‐implicit schemes presented the resulting system of linear equations can be solved by a robust and efficient iterative method. This iterative solver is constructed for the system of linearized Navier–Stokes equations. The Schur complement technique is used. We present a new approach of building a non‐symmetric preconditioner to solve a non‐symmetric problem of convection–diffusion and saddle‐point type. It is shown that handling the differential equations properly results in constructing efficient solvers for the corresponding finite linear algebra systems. The method has good performance for various ranges of viscosity and can be used both for 2D and 3D problems. The analysis of the method is still partly heuristic, however, the mathematically rigorous results are proved for certain cases. The proof is based on energy estimates and basic properties of the underlying partial differential equations. Numerical results are provided. Additionally, a multigrid method for the auxiliary convection–diffusion problem is briefly discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Influence of matrix reordering on the performance of iterative methods for solving linear systems arising from interior point methods for linear programming

This study analyzes the influence of sparse matrix reordering on the solution of linear systems arising from interior point methods for linear programming. In particular, such linear systems are solved by the conjugate gradient method with a two-phase hybrid preconditioner that uses the controlled Cholesky factorization during the initial iterations and later adopts the splitting preconditioner. This approach yields satisfactory computational results for the solution of linear systems with symmetric positive-definite matrices. Three reordering heuristics are analyzed in this study: the reverse Cuthill–McKee heuristic, the Sloan algorithm, and the minimum degree heuristic. Through numerical experiments, it was observed that these heuristics can be advantageous in terms of accelerating the convergence of the conjugate gradient method and reducing the processing time.     

Hermite collocation solution of partial differential equations via preconditioned Krylov methods

We are concerned with the numerical solution of partial differential equations (PDEs) in two spatial dimensions discretized via Hermite collocation. To efficiently solve the resulting systems of linear algebraic equations, we choose a Krylov subspace method. We implement two such methods: Bi‐CGSTAB [1] and GMRES [2]. In addition, we utilize two different preconditioners: one based on the Gauss–Seidel method with a block red‐black ordering (RBGS); the other based upon a block incomplete LU factorization (ILU). Our results suggest that, at least in the context of Hermite collocation, the RBGS preconditioner is superior to the ILU preconditioner and that the Bi‐CGSTAB method is superior to GMRES. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:120–136, 2001     

Multigrid preconditioning for time-periodic Navier–Stokes problems

We propose to solve time-periodic Navier–Stokes problems by a discrete Fourier transform in time. Truncating the Fourier series yields a nonlinear system of equations for the unknown Fourier coefficients. Its solution by Picard iteration requires to solve a sequence of linear systems of equations. The focus of this work is on an efficient method to solve these linear systems. We employ GMRES, complemented by an optimal block triangular preconditioner. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Using groups in the splitting preconditioner computation for interior point methods

Interior point methods usually rely on iterative methods to solve the linear systems of large scale problems. The paper proposes a hybrid strategy using groups for the preconditioning of these iterative methods. The objective is to solve large scale linear programming problems more efficiently by a faster and robust computation of the preconditioner. In these problems, the coefficient matrix of the linear system becomes ill conditioned during the interior point iterations, causing numerical difficulties to find a solution, mainly with iterative methods. Therefore, the use of preconditioners is a mandatory requirement to achieve successful results. The paper proposes the use of a new columns ordering for the splitting preconditioner computation, exploring the sparsity of the original matrix and the concepts of groups. This new preconditioner is designed specially for the final interior point iterations; a hybrid approach with the controlled Cholesky factorization preconditioner is adopted. Case studies show that the proposed methodology reduces the computational times with the same quality of solutions when compared to previous reference approaches. Furthermore, the benefits are obtained while preserving the sparse structure of the systems. These results highlight the suitability of the proposed approach for large scale problems.     

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two‐by‐two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean‐value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix–vector multiplications for the off‐diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen–Loève expansion and a good preconditioned for the mean‐value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

稀疏近似逆与多层块ILU预条件技术

设计了一种求解一般稀疏线性方程组的健壮且有效的可并行化预条件子,这种预条件子涉及在多层块ILU预条件子(BILUM)中使用稀疏近似逆(AINV)技术．所得的预条件子保持了BILUM的健壮性,它比标准的BILUM预条件子有两点优势:控制稀疏性的能力和增强了并行性．数值例子显示了新预条件子的有效性和效率．     

Circulant-block preconditioners for solving ordinary differential equations

Boundary value methods for solving 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 circulant-block preconditioner is proposed to speed up the convergence rate of the GMRES method. Theoretical and practical arguments are given to show that this preconditioner is more efficient than some other circulant-type preconditioners in some cases. A class of waveform relaxation methods is also proposed to solve the linear systems.     

A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations

In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier-Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixed-point iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included.     

Convergence Analysis of Modified Iterative Methods to Solve Linear Systems

In the past years, a growing interest to solve linear systems with modified iterative methods has been shown by researchers. Recently, Dehghan and Hajarian (J Vib Control, doi:10.1177/10775463124, 2012)based on preconditioned methods, introduced modified accelerated overrelaxation methods for solving linear systems. These authors stated that the best property of the mentioned methods is that they can be used under mild conditions than the Milaszewicz’s (Linear Algebra Appl 93:161–170, 1987) preconditioner. In this paper, we show that the Milaszewicz’s preconditioner is applicable under mild conditions and also, under these conditions, Milaszewicz’s preconditioner is superior to the Dehghan and Hajarian’s preconditioner. Numerical examples are also given to illustrate our results.     

A domain decomposition preconditioner for hermite collocation problems

We propose a preconditioning method for linear systems of equations arising from piecewise Hermite bicubic collocation applied to two‐dimensional elliptic PDEs with mixed boundary conditions. We construct an efficient, parallel preconditioner for the GMRES method. The main contribution of the article is a novel interface preconditioner derived in the framework of substructuring and employing a local Hermite collocation discretization for the interface subproblems based on a hybrid fine‐coarse mesh. Interface equations based on this mesh depend only weakly on unknowns associated with subdomains. The effectiveness of the proposed method is highlighted by numerical experiments that cover a variety of problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 135–151, 2003     

Parallel preconditioners and multigrid solvers for stochastic polynomial chaos discretizations of the diffusion equation at the large scale

This paper presents parallel preconditioners and multigrid solvers for solving linear systems of equations arising from stochastic polynomial chaos formulations of the diffusion equation with random coefficients. These preconditioners and solvers are extensions of the preconditioner developed in an earlier paper for strongly coupled systems of elliptic partial differential equations that are norm equivalent to systems that can be factored into an algebraic coupling component and a diagonal differential component. The first preconditioner, which is applied to the norm equivalent system, is obtained by sparsifying the inverse of the algebraic coupling component. This sparsification leads to an efficient method for solving these systems at the large scale, even for problems with large statistical variations in the random coefficients. An extension of this preconditioner leads to stand‐alone multigrid methods that can be applied directly to the actual system rather than to the norm equivalent system. These multigrid methods exploit the algebraic/differential factorization of the norm equivalent systems to produce variable‐decoupled systems on the coarse levels. Moreover, the structure of these methods allows easy software implementation through re‐use of robust high‐performance software such as the Hypre library package. Two‐grid matrix bounds will be established, and numerical results will be given. Copyright © 2015 John Wiley & Sons, Ltd.     

Iterative Solution of SPARK Methods Applied to DAEs

In this article a broad class of systems of implicit differential–algebraic equations (DAEs) is considered, including the equations of mechanical systems with holonomic and nonholonomic constraints. Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive Runge–Kutta (SPARK) methods. Several properties of the SPARK coefficients, satisfied by the family of Lobatto IIIA-B-C-C^*-D coefficients, are crucial to deal properly with the presence of constraints and algebraic variables. A main difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations. Inexact modified Newton iterations can be used to solve these systems. Linear systems of the modified Newton method can be solved approximately with a preconditioned linear iterative method. Preconditioners can be obtained after certain transformations to the systems of nonlinear and linear equations. These transformations rely heavily on specific properties of the SPARK coefficients. A new truly parallelizable preconditioner is presented.     

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.     

广义鞍点问题基于PSS的约束预条件子

对于(1,1)块为非Hermitian阵的广义鞍点问题,本文给出了一种基于正定和反对称分裂(Positive definite andskew-Hermitian splitting, PSS)的约束预条件子.该预条件子的(1,1)块由求解非Hermitian正定线性方程组时的PSS迭代法所构造得到.文中分析了PSS约束预条件子的一些性质并证明了预处理迭代法的收敛性.最后用数值算例验证了该预条件子的有效性.     

A preconditioned GMRES for complex dense linear systems from electromagnetic wave scattering problems

We consider the numerical solution of linear systems arising from the discretization of the electric field integral equation (EFIE). For some geometries the associated matrix can be poorly conditioned making the use of a preconditioner mandatory to obtain convergence. The electromagnetic scattering problem is here solved by means of a preconditioned GMRES in the context of the multilevel fast multipole method (MLFMM). The novelty of this work is the construction of an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups in the MLFMM, as preconditioner for the GMRES iterations. As experience shows, the efficiency of an ILU preconditioning for such systems essentially depends on a sufficient fill-in, which apparently sacrifices the sparsity of the near-field matrix. In the light of this experience we propose a multilevel near-field matrix and its corresponding HSS representation as a hierarchical preconditioner in order to substantially reduce the number of iterations in the solution of the resulting system of equations.     

A relaxed positive semi-definite and skew-Hermitian splitting preconditioner for non-Hermitian generalized saddle point problems

For non-Hermitian saddle point linear systems, Pan, Ng and Bai presented a positive semi-definite and skew-Hermitian splitting (PSS) preconditioner (Pan et al. Appl. Math. Comput. 172, 762–771 2006), to accelerate the convergence rate of the Krylov subspace iteration methods like the GMRES method. In this paper, a relaxed positive semi-definite and skew-Hermitian (RPSS) splitting preconditioner based on the PSS preconditioner for the non-Hermitian generalized saddle point problems is considered. The distribution of eigenvalues and the form of the eigenvectors of the preconditioned matrix are analyzed. Moreover, an upper bound on the degree of the minimal polynomial is also studied. Finally, numerical experiments of a model Navier-Stokes equation are presented to illustrate the efficiency of the RPSS preconditioner compared to the PSS preconditioner, the block diagonal preconditioner (BD), and the block triangular preconditioner (BT) in terms of the number of iteration and computational time.