Approximate sparsity patterns for the inverse of a matrix and preconditioning

doi:10.1016/S0168-9274(98)00117-2

Applied Numerical Mathematics

Volume 30, Issues 2–3, June 1999, Pages 291-303

https://doi.org/10.1016/S0168-9274(98)00117-2 Get rights and content

Abstract

We consider a general sparse matrix A. Computing a sparse approximate inverse matrix M by minimizing ∥AM−E∥ in the Frobenius norm is very useful for deriving preconditioners in iterative solvers, especially in a parallel environment. The problems, that appear in this connection in a distributed memory setting, are the distribution of the data—mainly submatrices of A—on the different processors. An a-priori knowledge of the data that has to be sent to a processor would be very helpful in this connection in order to reduce the communication time.

In this paper, we compare different strategies for choosing a-priori an approximate sparsity structure of A⁻¹. Such a sparsity pattern can be used as a maximum pattern of allowed entries for the sparse approximate inverse M. Then, with this maximum pattern we are able to distribute the claimed data in one step to each processor. Therefore, communication is necessary only at the beginning and at the end of a subprocess.

Using the characteristic polynomials and the Neumann series associated with A and A^TA, we develop heuristic methods to find good and sparse approximate patterns for A⁻¹. Furthermore, we exactly determine the submatrices that are used in the SPAI algorithm to compute one new column of the sparse approximate inverse. Hence, it is possible to predict in advance an upper bound for the data that each processor will need.

Based on numerical examples we compare the different methods with regard to the quality of the resulting approximation M.

References (0)

Cited by (97)

Rapid calculation of large-scale acoustic scattering from complex targets by a dual-level fast direct solver
2023, Computers and Mathematics with Applications
Citation Excerpt :
More details about the sparse approximate inverse matrix technique can be found in [39,44,45]. It is noted that the sparse approximate inverse matrix is only used as the preconditioner of the GMRES in [39,44,45]. The GMRES still has to solve the fully dense linear equations.
Rapid calculation of large-scale acoustic scattering from complex targets is the basis of many acoustic technologies. The key issue is to efficiently solve the dense and ill-conditioned linear equations caused by semi-analytical boundary collocation method. This article aims to establish a dual-level fast direct solver to replace the GMRES for solving such a large-scale dense linear system. The core idea of the dual-level fast direct solver is to construct a sparse approximate inverse matrix to indirectly approximate the solution of linear equations. The solution process consists of the smoothing process and the correction process. Therefore, the dual-level fast direct solver only needs to solve a small-scale coarse-mesh matrix and a series of small-scale least squares problems defined on fine mesh. The drawback that the GMRES cannot efficiently solve large-scale dense linear system is overcome. Several practical engineering problems are analyzed by the dual-level fast direct solver, such as the acoustic scattering from an A320 aircraft and acoustic scattering from a human head.
Benchmarking of finite-difference time-domain method and fast multipole boundary element method for room acoustics
2022, Applied Acoustics
Citation Excerpt :
The construction of efficient preconditioner to the system is an active research field. In this discussion, the sparse approximate inverse preconditioner based on Frobenius norm minimization [43] is implemented to improve the convergence. The following list describes the material input data for each simulation scenario:
Compared to geometrical acoustics, wave-based methods which solve the wave equation either in the time domain or in the frequency domain are known for their high accuracy. However, their systematic use as professional room acoustic simulation tools is less popular due to the modelling effort and computational time requirements, especially in the case of complex scenarios. This paper aims at providing guidelines for the use of two wave-based methods in complex room acoustics simulations, namely the finite-difference time-domain (FDTD) method and the fast multipole boundary element method (FMBEM). Numerical experiments are conducted to address the convergence issues of the two solvers, more specifically, the selection of the convergence tolerance of the iterative solver in FMBEM and the temporal sampling frequency in FDTD. To evaluate the capability of the solvers in simulating complex scenarios, five cases with increasing complexity of material input data are presented. The results show that both solvers give close predictions for various room acoustics parameters. In addition, an uncertainty sensitivity study is performed in a case where experimental data is available. Large deviations between measured and simulated reverberation time reveal that typical material data-sets poorly represent the behaviour of real materials in a room acoustics context. Lastly, the efficiency of the two solvers is discussed. With parallelization implemented, both solvers can simulate sizeable room acoustic problems with good accuracy within a reasonable time.
A transformation approach that makes SPAI, PSAI and RSAI procedures efficient for large double irregular nonsymmetric sparse linear systems
2019, Journal of Computational and Applied Mathematics
A sparse matrix is called double irregular sparse if it has at least one relatively dense column and row, and it is double regular sparse if all the columns and rows of it are sparse. The sparse approximate inverse preconditioning procedures SPAI, PSAI( $t o l$ ) and RSAI( $t o l$ ) are costly and even impractical to construct preconditioners for a large sparse nonsymmetric linear system with the coefficient matrix being double irregular sparse, but they are efficient for double regular sparse problems. Double irregular sparse linear systems have a wide range of applications, and 24.4% of the nonsymmetric matrices in the Florida University collection are double irregular sparse. For this class of problems, we propose a transformation approach, which consists of four steps: (i) transform a given double irregular sparse problem into a small number of double regular sparse ones with the same coefficient matrix $\hat{A}$ , (ii) use SPAI, PSAI( $t o l$ ) and RSAI( $t o l$ ) to construct sparse approximate inverses $M$ of $\hat{A}$ , (iii) solve the preconditioned double regular sparse linear systems by Krylov solvers, and (iv) recover an approximate solution of the original problem with a prescribed accuracy from those of the double regular sparse ones. A number of theoretical and practical issues are considered on the transformation approach. Numerical experiments on a number of real-world problems confirm the very sharp superiority of the transformation approach to the standard approach that preconditions the original double irregular sparse problem by SPAI, PSAI( $t o l$ ) or RSAI( $t o l$ ) and solves the resulting preconditioned system by Krylov solvers.
Block preconditioning for fault/fracture mechanics saddle-point problems
2019, Computer Methods in Applied Mechanics and Engineering
The efficient simulation of fault and fracture mechanics is a key issue in several applications and is attracting a growing interest by the scientific community. Using a formulation based on Lagrange multipliers, the Jacobian matrix resulting from the Finite Element discretization of the governing equations has a non-symmetric generalized saddle-point structure. In this work, we propose a family of block preconditioners to accelerate the convergence of Krylov methods for such problems. We critically review possible advantages and difficulties of using various Schur complement approximations, based on both physical and algebraic considerations. The proposed approaches are tested in a number of real-world applications, showing their robustness and efficiency also in large-size and ill-conditioned problems.
Sparsity preserving optimal control of discretized PDE systems
2018, Computer Methods in Applied Mechanics and Engineering
Citation Excerpt :
Furthermore, we ignore possible zero entries created by coincidence in the numerical values of the matrix entries. This is a standard assumption that is equivalent to the assumption that all the entries of the matrices used for the a priori pattern analysis are positive, see for example [51] and references therein (see also Remark 3.2). In this paper we have presented two computationally efficient methods for computing (sparse) banded approximate solutions of the generalized Lyapunov equations with banded matrices.
We focus on the problem of optimal control of large-scale systems whose models are obtained by discretization of partial differential equations using the Finite Element (FE) or Finite Difference (FD) methods. The motivation for studying this pressing problem originates from the fact that the classical numerical tools used to solve low-dimensional optimal control problems are computationally infeasible for large-scale systems. Furthermore, although the matrices of large-scale FE or FD models are usually sparse banded or highly structured, the optimal control solution computed using the classical methods is dense and unstructured. Consequently, it is not suitable for efficient centralized and distributed real-time implementations. We show that the a priori (sparsity) patterns of the exact solutions of the generalized Lyapunov equations for FE and FD models are banded matrices. The a priori pattern predicts the dominant non-zero entries of the exact solution. We furthermore show that for well-conditioned problems, the a priori patterns are not only banded but also sparse matrices. On the basis of these results, we develop two computationally efficient methods for computing sparse approximate solutions of generalized Lyapunov equations. Using these two methods and the inexact Newton method, we show that the solution of the generalized Riccati equation can be approximated by a banded matrix. This enables us to develop a novel computationally efficient optimal control approach that is able to preserve the sparsity of the control law. We perform extensive numerical experiments that demonstrate the effectiveness of our approach.
Sparse solution of the Lyapunov equation for large-scale interconnected systems
2016, Automatica
We consider the problem of computing an approximate banded solution of the continuous-time Lyapunov equation $\underline{A} \underline{X} + \underline{X} {\underline{A}}^{T} = \underline{P}$ , where the coefficient matrices $\underline{A}$ and $\underline{P}$ are large, symmetric banded matrices. The (sparsity) pattern of $\underline{A}$ describes the interconnection structure of a large-scale interconnected system. Recently, it has been shown that the entries of the solution $\underline{X}$ are spatially localized or decaying away from a banded pattern. We show that the decay of the entries of $\underline{X}$ is faster if the condition number of $\underline{A}$ is smaller. By exploiting the decay of entries of $\underline{X}$ , we develop two computationally efficient methods for approximating $\underline{X}$ by a banded matrix. For a well-conditioned and sparse banded $\underline{A}$ , the computational and memory complexities of the methods scale linearly with the state dimension. We perform extensive numerical experiments that confirm this, and that demonstrate the effectiveness of the developed methods. The methods proposed in this paper can be generalized to (sparsity) patterns of $\underline{A}$ and $\underline{P}$ that are more general than banded matrices. The results of this paper open the possibility for developing computationally efficient methods for approximating the solution of the large-scale Riccati equation by a sparse matrix.

View all citing articles on Scopus

View full text