Indefinite Matrices Research Articles

Absolute Value Preconditioning for Symmetric Indefinite Linear Systems

We introduce a novel strategy for constructing symmetric positive definite (SPD) preconditioners for linear systems with symmetric indefinite matrices. The strategy, called absolute value preconditioning, is motivated by the observation that the preconditioned minimal residual method with the inverse of the absolute value of the matrix as a preconditioner converges to the exact solution of the system in at most two steps. Neither the exact absolute value of the matrix nor its exact inverse are computationally feasible to construct in general. However, we provide a practical example of an SPD preconditioner that is based on the suggested approach. In this example we consider a model problem with a shifted discrete negative Laplacian, and suggest a geometric multigrid (MG) preconditioner, where the inverse of the matrix absolute value appears only on the coarse grid, while operations on finer grids are based on the Laplacian. Our numerical tests demonstrate practical effectiveness of the new MG preconditioner, which leads to a robust iterative scheme with minimalist memory requirements.

Recursive approximation of the dominant eigenspace of an indefinite matrix

We consider here the problem of tracking the dominant eigenspace of an indefinite matrix by updating recursively a rank k approximation of the given matrix. The tracking uses a window of the given matrix, which increases at every step of the algorithm. Therefore, the rank of the approximation increases also, and hence a rank reduction of the approximation is needed to retrieve an approximation of rank k. In order to perform the window adaptation and the rank reduction in an efficient manner, we make use of a new anti-triangular decomposition for indefinite matrices. All steps of the algorithm only make use of orthogonal transformations, which guarantees the stability of the intermediate steps. We also show some numerical experiments to illustrate the performance of the tracking algorithm.

A generalization of [formula omitted]-step variants of gradient methods

The s-step methods were proposed by Chronopoulos to gain efficiency in parallel programming of iterative methods for linear systems. They are variants of classical iterative methods based on the construction of a Krylov subspace basis on each iteration. These s-step methods were inferred from algorithms like the Conjugate Gradient, Generalized Conjugate Residual or the Minimal Residual. They converge for all symmetric, nonsymmetric definite and some nonsymmetric indefinite matrices. In this paper, we introduce an s-step variant of a General Orthogonalization Algorithm, that is, a generalization of s-step variants of gradient methods. We prove convergence and obtain error estimates. We also describe an Orthomin variant, together with a convergence theorem. From this we derive the well known s-step methods as particular cases, and some which are newfound to our knowledge. This provides a unified framework to derive and study s-step methods. Some of the methods obtained are convergent for every nonsingular matrix. Finally, we give some numerical results for the new proposed methods, showing that the parallel implementation of these overcomes the original ones.

A general approach to analyse preconditioners for two‐by‐two block matrices

SUMMARYTwo‐by‐two block matrices arise in various applications, such as in domain decomposition methods or when solving boundary value problems discretised by finite elements from the separation of the node set of the mesh into ‘fine’ and ‘coarse’ nodes. Matrices with such a structure, in saddle point form arise also in mixed variable finite element methods and in constrained optimisation problems.A general algebraic approach to construct, analyse and control the accuracy of preconditioners for matrices in two‐by‐two block form is presented. This includes both symmetric and nonsymmetric matrices, as well as indefinite matrices. The action of the preconditioners can involve element‐by‐element approximations and/or geometric or algebraic multigrid/multilevel methods. Copyright © 2011 John Wiley & Sons, Ltd.

Partial factorization of a dense symmetric indefinite matrix

At the heart of a frontal or multifrontal solver for the solution of sparse symmetric sets of linear equations, there is the need to partially factorize dense matrices (the frontal matrices) and to be able to use their factorizations in subsequent forward and backward substitutions. For a large problem, packing (holding only the lower or upper triangular part) is important to save memory. It has long been recognized that blocking is the key to efficiency and this has become particularly relevant on modern hardware. For stability in the indefinite case, the use of interchanges and 2 × 2 pivots as well as 1 × 1 pivots is equally well established. In this article, the challenge of using these three ideas (packing, blocking, and pivoting) together is addressed to achieve stable factorizations of large real-world symmetric indefinite problems with good execution speed. The ideas are not restricted to frontal and multifrontal solvers and are applicable whenever partial or complete factorizations of dense symmetric indefinite matrices are needed.

Block-oriented J-Jacobi methods for Hermitian matrices

The paper describes a way how one-sided Jacobi-type algorithm of Veselić for computing the hyperbolic singular value decomposition of rectangular matrices can be modified to work with blocks. The proposed modification preserves the relative accuracy property of the original algorithm and essentially improves its performance. Special attention is devoted to proving the global convergence of the method under some important classes of block-oriented pivot strategies. As numerical tests indicate, the block-oriented J-Jacobi methods combined with the Hermitian indefinite factorization, become efficient and accurate eigensolvers for Hermitian indefinite matrices.

Stability analysis of block LDLT factorization for symmetric indefinite matrices

Stability analysis of block LDLT factorization for symmetric indefinite matrices

A two-phase preconditioning strategy of sparse approximate inverse for indefinite matrices

A two-phase preconditioning strategy based on a factored sparse approximate inverse is proposed for solving sparse indefinite matrices. In each phase, the strategy first makes the original matrix diagonally dominant to enhance the stability by a shifting method, and constructs an inverse approximation of the shifted matrix by utilizing a factored sparse approximate inverse preconditioner. The two inverse approximation matrices produced from each phase are then combined to be used as a preconditioner. Experimental results show that the presented strategy improves the accuracy and the stability of the preconditioner on solving indefinite sparse matrices. Furthermore, the strategy ensures that convergence rate of the preconditioned iterations of the two-phase preconditioning strategy is much better than that of the standard sparse approximate inverse ones for solving indefinite matrices.

Constraint Preconditioners for Symmetric Indefinite Matrices

We study the eigenvalue bounds of block two-by-two nonsingular and symmetric indefinite matrices whose $(1,1)$ block is symmetric positive definite and Schur complement with respect to its $(2,2)$ block is symmetric indefinite. A constraint preconditioner for this matrix is constructed by simply replacing the $(1,1)$ block by a symmetric and positive definite approximation, and the spectral properties of the preconditioned matrix are discussed. Numerical results show that, for a suitably chosen $(1,1)$ block-matrix, this constraint preconditioner outperforms the block-diagonal and the block-tridiagonal ones in iteration step and computing time when they are used to accelerate the GMRES method for solving these block two-by-two symmetric positive indefinite linear systems. The new results extend the existing ones about block two-by-two matrices of symmetric negative semidefinite $(2,2)$ blocks to those of general symmetric $(2,2)$ blocks.

Algebraic Multilevel Preconditioner for the Helmholtz Equation in Heterogeneous Media

An algebraic multilevel (ML) preconditioner is presented for the Helmholtz equation in heterogeneous media. It is based on a multilevel incomplete $LDL^T$ factorization and preserves the inherent (complex) symmetry of the Helmholtz equation. The ML preconditioner incorporates two key components for efficiency and numerical stability: symmetric maximum weight matchings and an inverse-based pivoting strategy. The former increases the block-diagonal dominance of the system, whereas the latter controls $\|L^{-1}\|$ for numerical stability. When applied recursively, their combined effect yields an algebraic coarsening strategy, similar to algebraic multigrid methods, even for highly indefinite matrices. The ML preconditioner is combined with a Krylov subspace method and applied as a “black-box” solver to a series of challenging two- and three-dimensional test problems, mainly from geophysical seismic imaging. The numerical results demonstrate the robustness and efficiency of the ML preconditioner, even at higher frequency regimes.

Alternating projection method applied to indefinite correlation matrices for generation of synthetic MIMO channels

Complex correlation multiple-input multiple-output (MIMO) matrices estimated from measurements under realistic propagation conditions occasionally have positive, zero and negative eigenvalues, converting them to indefinite matrices. In this framework, this paper presents a novel correction procedure (CP) based on the alternating projection (AP) method to find the Hermitian and positive definite matrix closest to an estimated indefinite full spatial correlation (FSC) MIMO matrix. This corrected matrix allows the Cholesky factorisation usually used for generation of synthetic MIMO channel samples. The applicability of the CP has been analysed using real data from non-line-of-sight (NLOS) indoor measurements. The results reported in this paper show that a proper selection of the CP parameters has not a sensitive impact on the cumulative distribution function (CDF) of the eigenvalues of the estimated FSC MIMO matrices. This characteristic makes the CP useful for channel characterisation, allowing to assess the performance of a MIMO system. As an example of the CP application, a study of ergodic and outage MIMO capacities for different array configurations is presented .

On the optimal complex extrapolation of the complex Cayley transform

The Cayley transform, F ≔ F ( A ) = ( I + A ) - 1 ( I - A ) , with A ∈ C n , n and - 1 ∉ σ ( A ) , where σ ( · ) denotes spectrum, and its extrapolated counterpart F ( ω A ) , ω ∈ C ⧹ { 0 } and - 1 ∉ σ ( ω A ) , are of significant theoretical and practical importance (see, e.g. [A. Hadjidimos, M. Tzoumas, On the principle of extrapolation and the Cayley transform, Linear Algebra Appl., in press]). In this work, we extend the theory in [8] to cover the complex case. Specifically, we determine the optimal extrapolation parameter ω ∈ C ⧹ { 0 } for which the spectral radius of the extrapolated Cayley transform ρ ( F ( ω A ) ) is minimized assuming that σ ( A ) ⊂ H , where H is the smallest closed convex polygon, and satisfies O ( 0 ) ∉ H . As an application, we show how a complex linear system, with coefficient a certain class of indefinite matrices, which the ADI-type method of Hermitian/Skew-Hermitian splitting fails to solve, can be solved in a “best” way by the aforementioned method.

The design and use of a sparse direct solver for skew symmetric matrices

We consider the L D L T factorization of sparse skew symmetric matrices. We see that the pivoting strategies are similar, but simpler, to those used in the factorization of sparse symmetric indefinite matrices, and we briefly describe the algorithms used in a forthcoming direct code based on multifrontal techniques for the factorization of real skew symmetric matrices. We show how this factorization can be very efficient for preconditioning matrices that have a large skew component.

Numerically stable LDLT-factorization of F-type saddle point matrices

We present a new algorithm that constructs a fill-reducing ordering for a special class of saddle point matrices: the ℱ-matrices. This class contains the matrix occurring after discretization of the Stokes equation on a C-grid. The commonly used approach is to construct a fill-reducing ordering for the whole matrix followed by an adaptation of the ordering such that it becomes feasible. We propose to compute first a fill-reducing ordering for an extension of the definite submatrix. This ordering can be easily extended to an ordering for the whole matrix. In this manner, the construction of the ordering is straightforward and it can be computed efficiently. We show that much of the structure of the matrix is preserved during Gaussian elimination. For an ℱ-matrix, the preserved structure allows us to prove that any feasible ordering obtained in this way is numerically stable. The growth factor of this factorization is much smaller than the one for general indefinite matrices and is bounded by a number that depends linearly on the number of indefinite nodes. The algorithm allows for generalization to saddle point problems that are not of ℱ-type and are nonsymmetric, e.g. the incompressible Navier–Stokes equations (with Coriolis force) on a C-grid. Numerical results for ℱ-matrices show that the algorithm is able to produce a factorization with low fill.

On Large-Scale Diagonalization Techniques for the Anderson Model of Localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi-Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete $LDL^T$ factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.

SIGEST

In discussions of the tremendous advances in scientific computation over the past few decades, it has often been written and demonstrated that as large as the increases in computer speed have been, they have been matched by improvements from more efficient algorithms. For many problem classes, over the last two to four decades, each has led to speedups of many, many orders of magnitude. The SIGEST paper in this issue, from the SIAM Journal on Scientific Computing, is an example of this tremendous type of algorithmic improvement. It also is an excellent example of how increasingly significant algorithmic advances are obtained by drawing upon a broad range of sophisticated techniques. In “On Large-Scale Diagonalization Techniques for the Anderson Model of Localization” by O. Schenk, M. Bollhöfer, and R. Römer, the authors demonstrate a greatly improved computational approach for an important, common problem: computing a few interior eigenvalues and eigenvectors of large, sparse, real symmetric, and highly indefinite matrices. By “large” they mean matrices with millions to hundreds of millions of rows and columns. The computational improvements that are demonstrated extend to a factor of a thousand or more over previous approaches, while keeping memory requirements modest. The application class is the Anderson model of localization, an important problem in quantum physics with a wide range of applications, and it appears likely that the approach can be applied successfully to large symmetric indefinite eigenvalue problems more generally. An interesting aspect of the paper is that is draws heavily upon recent advances in solving large sparse symmetric indefinite systems of linear equations. Solution of such systems is central to each of the eigenvalue solvers that the paper considers, two Lanczos-based methods and the Jacobi–Davidson method. In considering how to solve the linear systems in a manner that exploits sparsity as well as symmetry, the paper combines a diverse range of topics that includes preprocessing the matrix with symmetric weighted matchings, solving the linear system with Krylov subspace methods, and accelerating the linear system solution with multilevel preconditioners based upon incomplete factorizations. This makes for a very fitting paper for SIAM Review, highlighting a variety of important topics in numerical computation. We expect that a wide range of SIAM readers will find this to be a very readable paper that provides an interesting glimpse into a range of modern research approaches.

On large‐scale diagonalization techniques for the Anderson model of localization

AbstractWe propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large‐scale sparse real and symmetric indefinite matrices of the Anderson model of localization. Our preconditioning approaches for the shift‐and‐invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multi‐level incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill‐conditioned symmetric indefinite Anderson matrices. Our numerical examples reveal that recent algebraic multi‐level preconditioning solvers can accelerate the computation of a large‐scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Towards Stable Mixed Pivoting Strategies for the Sequential and Parallel Solution of Sparse Symmetric Indefinite Systems

We consider the direct solution of sparse symmetric indefinite matrices. We develop new pivoting strategies that combine numerical pivoting and perturbation techniques. Then an iterative refinement process uses our approximate factorization to compute a solution. We show that our pivoting strategies are numerically robust, that few steps of iterative refinement are required, and that the factorization is significantly faster than with previous methods. Furthermore, we propose original approaches that are designed for parallel distributed factorization. A key point of our parallel implementation is the cheap and reliable estimation of the growth factor. This estimation is based on an approximation of the off-diagonal entries and does not require any supplementary messages.

Hybrid reordering strategies for ILU preconditioning of indefinite sparse matrices

Incomplete LU factorization preconditioning techniques often have difficulty on indefinite sparse matrices. We present hybrid reordering strategies to deal with such matrices, which include new diagonal reorderings that are in conjunction with a symmetric nondecreasing degree algorithm. We first use the diagonal reorderings to efficiently search for entries of single element rows and columns and/or the maximum absolute value to be placed on the diagonal for computing a nonsymmetric permutation. To augment the effectiveness of the diagonal reorderings, a nondecreasing degree algorithm is applied to reduce the amount of fill-in during the ILU factorization. With the reordered matrices, we achieve a noticeable improvement in enhancing the stability of incomplete LU factorizations. Consequently, we reduce the convergence cost of the preconditioned Krylov subspace methods on solving the reordered indefinite matrices.

Equivalent Conditions for Convergence of Splittings of Non-Hermitian Indefinite Matrices

We present some sufficient and necessary conditions for convergent splitting of a non-Hermitian indefinite matrix. Some sufficient conditions to determinate a matrix with a (strongly) dominant symmetric part for a class of boundary value problem are also obtained. These results are applicable to identify the convergence of iterative methods for solving large sparse systems of linear equations.