Preconditioners For Matrices Research Articles

Partitioning Sparse Matrices for Parallel Preconditioned Iterative Methods

This paper addresses the parallelization of the preconditioned iterative methods that use explicit preconditioners such as approximate inverses. Parallelizing a full step of these methods requires the coefficient and preconditioner matrices to be well partitioned. We first show that different methods impose different partitioning requirements for the matrices. Then we develop hypergraph models to meet those requirements. In particular, we develop models that enable us to obtain partitionings on the coefficient and preconditioner matrices simultaneously. Experiments on a set of unsymmetric sparse matrices show that the proposed models yield effective partitioning results. A parallel implementation of the right preconditioned BiCGStab method on a PC cluster verifies that the theoretical gains obtained by the models hold in practice.

Two‐level preconditioners for serendipity finite element matrices

AbstractIn this paper an approach to construct algebraic multilevel preconditioners for serendipity finite element matrices is presented. Two‐level preconditioners constructed in the paper allow to obtain multilevel preconditioners in serendipity case using multilevel preconditioners for linear finite element matrices. Copyright © 2006 John Wiley & Sons, Ltd.

Parallel implementation of efficient preconditioned linear solver for grid-based applications in chemical physics. I: Block Jacobi diagonalization

Linear systems in chemical physics often involve matrices with a certain sparse block structure. These can often be solved very effectively using iterative methods (sequence of matrix–vector products) in conjunction with a block Jacobi preconditioner [Numer. Linear Algebra Appl. 7 (2000) 715]. In a two-part series, we present an efficient parallel implementation, incorporating several additional refinements. The present study (paper I) emphasizes construction of the block Jacobi preconditioner matrices. This is achieved in a preprocessing step, performed prior to the subsequent iterative linear solve step, considered in a companion paper (paper II). Results indicate that the block Jacobi routines scale remarkably well on parallel computing platforms, and should remain effective over tens of thousands of nodes.

Parallel iterative methods using factorized preconditioning matrices for solving elliptic equations on triangular grids

Parallel analogs of the variants of the incomplete Cholesky-conjugate gradient method and the modified incomplete Cholesky-conjugate gradient method for solving elliptic equations on uniform triangular and unstructured triangular grids on parallel computer systems with the MIMD architecture are considered. The construction of parallel methods is based on the use of various variants of ordering the grid points depending on the decomposition of the computation domain. Results of the theoretic and experimental studies of the convergence rate of these methods are presented. The solution of model problems on a moderate number processors is used to examine the efficiency of the proposed parallel methods.

Preconditioned Galerkin and minimal residual methods for solving Sylvester equations

This paper presents preconditioned Galerkin and minimal residual algorithms for the solution of Sylvester equations AX − XB = C. Given two good preconditioner matrices M and N for matrices A and B, respectively, we solve the Sylvester equations MAXN − MXBN = MCN. The algorithms use the Arnoldi process to generate orthonormal bases of certain Krylov subspaces and simultaneously reduce the order of Sylvester equations. Numerical experiments show that the solution of Sylvester equations can be obtained with high accuracy by using the preconditioned versions of Galerkin and minimal residual algorithms and this versions are more robust and more efficient than those without preconditioning.

A class of constraint preconditioners for nonsymmetric saddle point matrices

We consider the use of a class of constraint preconditioners for the application of the Krylov subspace iterative method to the solution of large nonsymmetric, indefinite linear systems. The eigensolution distribution of the preconditioned matrix is determined and the convergence behavior of a Krylov subspace method such as GMRES is described. The choices of the parameter matrices and the implementation of the preconditioning step are discussed. Numerical experiments are presented.

Approximate Inverse Preconditioning of Finite Element Discretizations of Elliptic Operators with Nonsmooth Coefficients

A new class of approximate inverse preconditioners for large finite element stiffness matrices arising from second order elliptic partial differential operators is introduced which guarantees a bounded, problem-independent condition number. Since the proposed preconditioner is based on hierarchical matrices, it can be generated, stored, and multiplied by a vector with almost linear complexity. This preconditioner may be used as a black-box method, because it is robust with respect to varying coefficients and because it can be applied to arbitrary quasi-uniform grids.

Approximate Factorization Constraint Preconditioners for Saddle-Point Matrices

We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions.

Structured preconditioners for nonsingular matrices of block two-by-two structures

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

On Inexact Preconditioners for Nonsymmetric Matrices

Inexact versions of the block-triangular preconditioners for nonsymmetric matrices of block two-by-two structures in [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput., 21 (2000), pp. 1969--1972] and [I. C. F. Ipsen, SIAM J. Sci. Comput., 23 (2001), pp. 1050--1051] are presented, and the two preconditioners for symmetric block two-by-two matrices in [C. Durazzi and V. Ruggiero, Numer. Linear Algebra Appl., 10 (2003), pp. 673--688] are extended to general nonsymmetric matrices. Moreover, we precisely describe the spectral properties of the preconditioned matrices and the finite-step termination properties of the preconditioned Krylov subspace iteration methods with an optimal or Galerkin property, with respect to these preconditioners. Several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners.

Factorized Banded Inverse Preconditioners for Matrices with Toeplitz Structure

In this paper, we study factorized banded inverse preconditioners for matrices with Toeplitz structure. We show that if a Toeplitz matrix T has certain off-diagonal decay property, then the factorized banded inverse preconditioner approximates T-1 accurately, and the spectra of these preconditioned matrices are clustered around 1. In nonlinear image restoration applications, Toeplitz-related systems of the form I + T*DT are required to solve, where D is a positive nonconstant diagonal matrix. We construct inverse preconditionersfor such matrices. Numerical results show that the performance of our proposed preconditioners are superior to that of circulant preconditioners. A two-dimensional nonlinear image restoration example is also presented to demonstrate the effectiveness of the proposed preconditioner.

Accelerating the transient simulation of semiconductor devices using filter-bank transforms

Simulation of high frequency semiconductor devices, where non-local and hot carrier transport cannot be ignored, requires solution of Poisson's equation and at least the first three moments of the Boltzmann transport equation (hydrodynamic transport model). These equations form non-linear, coupled and time-dependent partial differential equations. One of the most efficient solvers of such system of equations is decoupled solver. In conventional decoupled methods, the fully implicit, semi-implicit and explicit methods are used to solve the equations. In fully or semi-implicit schemes, the method is unconditionally stable for any Δt or for very large Δt compared to explicit scheme. Thus, these schemes are very suitable and efficient for transient simulations. But, using these techniques leads to a large system of linear equations. Here for the first time, a filter bank-based preconditioning method is used to facilitate the iterative solution of this system. This method provides efficient preconditioners for matrices arising from discretizing of the PDEs, using finite difference techniques. Numerical results show that the condition number and iteration number are significantly reduced. The most important advantage of this preconditioner is its low computational complexity which can be reduced to O(N). Copyright © 2006 John Wiley & Sons, Ltd.

Construction and analysis of structured preconditioners for block two-by-two matrices

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.

Data-sparse algebraic multigrid methods for large scale boundary element equations

This paper presents new algebraic multigrid (AMG) preconditioners for data-sparse boundary element matrices arising from the adaptive cross approximation (ACA) to dense boundary element matrices. Here we mainly consider the single layer potential integral equation, resulting from the direct boundary integral formulation of the interior, or exterior Dirichlet boundary value problems for the Laplace equation in two, or three spatial dimensions, as the most interesting boundary integral equation. The standard collocation, or Galerkin boundary element discretizations lead to fully populated system matrices which require O ( N h 2 ) of storage units and cause the same complexity for a single matrix-by-vector multiplication, where N h denotes the number of boundary unknowns. Data-sparse matrix approximation schemes such as ACA reduce this complexity to an almost linear behavior in N h . Since the single layer potential operator is a pseudo-differential operator of the order −1, the resulting boundary element matrices have large condition numbers on fine meshes. Iterative solvers dramatically suffer from this property for growing N h . Our AMG-preconditioners avoid the increase of the iteration numbers and result in almost optimal solvers with respect to the total complexity. This behavior is confirmed by our numerical experiments. Let us mention that our AMG-preconditioners use only single grid information provided by the ACA system matrix and by the usual mesh data on the finest level anyway. Similar results are valid for the hypersingular integral operator that is easier to handle since it is a pseudo-differential operator of the order +1.

Efficient preconditioners for boundary element matrices based on grey‐box algebraic multigrid methods

AbstractThis paper is concerned with the iterative solution of the boundary element equations arising from standard Galerkin boundary element discretizations of first‐kind boundary integral operators of positive and negative order. We construct efficient preconditioners on the basis of so‐called grey‐box algebraic multigrid methods that are well adapted to the treatment of boundary element matrices. In particular, the coarsening is based on an auxiliary matrix that represents the underlying topology in a certain sense. This auxiliary matrix is additionally used for the construction of the smoothers and the transfer operators. Finally, we present the results of some numerical studies that show the efficiency of the proposed algebraic multigrid preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.

How to prove that a preconditioner cannot be superlinear

In the general case of multilevel Toeplitz matrices, we recently proved that any multilevel circulant preconditioner is not superlinear (a cluster it may provide cannot be proper). The proof was based on the concept of quasi-equimodular matrices, although this concept does not apply, for example, to the sine-transform matrices. In this paper, with a new concept of partially equimodular matrices, we cover all trigonometric matrix algebras widely used in the literature. We propose a technique for proving the nonsuperlinearity of certain frequently used preconditioners for some representative sample multilevel matrices. At the same time, we show that these preconditioners are, in a certain sense, the best among the sublinear preconditioners (with only a general cluster) for multilevel Toeplitz matrices.

Simple preconditioners for the conjugate gradient method: experience with test day models

Preconditioned conjugate gradient method can be used to solve large mixed model equations quickly. Convergence of the method depends on the quality of the preconditioner. Here, the effect of simple preconditioners on the number of iterations until convergence was studied by solving breeding values for several test day models. The test day records were from a field data set, and several simulated data sets with low and high correlations among regression coefficients. The preconditioner matrices had diagonal or block diagonal parts. Transformation of the mixed model equations by diagonalization of the genetic covariance matrix was studied as well. Preconditioner having the whole block of the fixed effects was found to be advantageous. A block diagonal preconditioner for the animal effects reduced the number of iterations the higher the correlations among animal effects, but increased memory usage of the preconditioner. Diagonalization of the animal genetic covariance matrix often reduced the number of iterations considerably without increased memory usage.

Matrix algebra preconditioners for multilevel Toeplitz matrices are not superlinear

Let f be a d-variate 2 π periodic continuous function and let { T n ( f)} n , n=( n 1,…, n d ), be the multiindexed sequence of multilevel N× N Toeplitz matrices ( N= N( n)=∏ i n i ) generated by f. Let A={ A N} N be a sequence of matrix algebras simultaneously diagonalized by unitary transforms. We show that there exist infinitely many linearly independent trigonometric polynomials (and continuous nonpolynomial functions) f such that rank ϵ ( T n ( f)− P N )≠o( N( n)∑ i=1 d n i −1) for any matrix sequence P={P N}∈ A . This implies that no superlinear matrix algebra preconditioner exists in the multilevel Toeplitz case. The above mentioned result improves the analysis of the author and E. Tyrtyshnikov [SIAM J. Matrix Anal. Appl. 21 (2) (1999) 431] where the same was proved under the assumption that the involved algebras are of circulant type.

A note on the fast algorithm for block Toeplitz systems with tensor structure

We study the solutions of block Toeplitz systems T mn x= b by using the preconditioned conjugate gradient (PCG) method. Here T mn = T m ⊗ T n and T i, i=m,n are Toeplitz matrices. In [X. Jin, Appl. Math. Comput. 73 (1995) 115–124], Jin introduced a fast algorithm for these systems by applying the PCG method. This fast algorithm allows a tensor problem to be reduced to a one-dimensional problem. It was proved that if the mn-by- mn system is well conditioned, then the PCG method converges superlinearly and only O( mnlog mn) operations are required in solving the preconditioned system. However, only well-conditioned systems were considered in Jin, 1995. In this paper, we apply this fast algorithm with the { ω}-circulant preconditioners proposed in [D. Potts, G. Steidl, Preconditioners for Ill-Conditioned toeplitz matrices, BIT, to be appeared] to solve the ill-conditioned systems. Numerical results are included to illustrate the effectiveness of the algorithm for solving the preconditioned systems by using the PCG method. An application in image restoration is also given.

A robust AINV-type method for constructing sparse approximate inverse preconditioners in factored form

This paper suggests a new method, called AINV-A, for constructing sparse approximate inverse preconditioners for positive-definite matrices, which can be regarded as a modification of the AINV method proposed by Benzi and Túma. Numerical results on SPD test matrices coming from different applications demonstrate the robustness of the AINV-A method and its superiority to the original AINV approach. Copyright © 2001 John Wiley & Sons, Ltd.