Incomplete LU Factorization Research Articles

Stimulus Protocol Determines the Most Computationally Efficient Preconditioner for the Bidomain Equations

The efficient solution of the bidomain equations is a fundamental tool in the field of cardiac electrophysiology. When choosing a finite element discretization of the coupled system, one has to deal with the solution of a large, highly sparse system of linear equations. The conjugate gradient algorithm, along with suitable preconditioning, is the natural choice in this scenario. In this study, we identify the optimal preconditioners with respect to both stimulus protocol and mesh geometry. The results are supported by a comprehensive study of the mesh-dependence properties of several preconditioning techniques found in the literature. Our results show that when only intracellular stimulus is considered, incomplete LU factorization remains a valid choice for current cardiac geometries. However, when extracellular shocks are delivered to tissue, preconditioners that take into account the structure of the system minimize execution time and ensure mesh-independent convergence.

An Efficient Algebraic Multigrid Solver Strategy for Adaptive Implicit Methods in Oil-Reservoir Simulation

Summary We propose a new, efficient, adaptive algebraic multigrid (AMG) solver strategy for the discrete systems of partial-differential equations (PDEs) arising from structured or unstructured grid models in reservoir simulation. The proposed strategy has been particularly tailored to linear systems of equations arising in adaptive implicit methods (AIMs). The coarsening process of the AMG method designed automatically employs information on the physical structure of the models; as a smoother, an adaptive incomplete LU factorization with thresholding (ILUT) method is employed, taking care of an efficient solution of the hyperbolic parts while providing adequately smooth errors for the elliptic parts. To achieve a good compromise of high efficiency and robustness for a variety of problem classes—ranging from simple, small black-oil to challenging, large compositional models—an automatic, adaptive ILUT parameter and AMG solver switching strategy, α-SAMG. has been developed. Its efficiency is demonstrated for eight industrial benchmark cases by comparison against standard one-level and AMG solvers, including constraint pressure residual (CPR), as well as the pure one-level variant of the proposed new strategy. In addition, very promising results of first parallel runs are shown.

Parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with Lagrange multipliers

A parallel approach to solve three-dimensional viscous incompressible fluid flow problems using discontinuous pressure finite elements and a Lagrange multiplier technique is presented. The strategy is based on non-overlapping domain decomposition methods, and Lagrange multipliers are used to enforce continuity at the boundaries between subdomains. The novelty of the work is the coupled approach for solving the velocity–pressure-Lagrange multiplier algebraic system of the discrete Navier–Stokes equations by a distributed memory parallel ILU (0) preconditioned Krylov method. A penalty function on the interface constraints equations is introduced to avoid the failure of the ILU factorization algorithm. To ensure portability of the code, a message based memory distributed model with MPI is employed. The method has been tested over different benchmark cases such as the lid-driven cavity and pipe flow with unstructured tetrahedral grids. It is found that the partition algorithm and the order of the physical variables are central to parallelization performance. A speed-up in the range of 5–13 is obtained with 16 processors. Finally, the algorithm is tested over an industrial case using up to 128 processors. In considering the literature, the obtained speed-ups on distributed and shared memory computers are found very competitive.

Kaporin-Kon’shin’s method of parallel implementation of block preconditioners for asymmetric matrices in problems of filtration of a multicomponent mixture in a porous medium

The ILU class preconditioners (ILU(0), ILU(1), ILUT) employed for iterative algorithms for non-symmetrical linear sparse matrix systems are considered. Test matrices used in this study originate from discretization of systems of partial differential equations describing multicomponent fluid flows in porous media. A new parallel algorithm for block ILU factorization is suggested. This algorithm demonstrates a good convergence and significant speed-up in comparison with sequential algorithms. New integrated approach was tested on the wide range of matrices resulted from real hydrodynamic simulations of oil fields of Western Siberia and demonstrated significant reduction in computational time.

A Block FSAI-ILU Parallel Preconditioner for Symmetric Positive Definite Linear Systems

A novel parallel preconditioner for symmetric positive definite matrices is developed coupling a generalized factored sparse approximate inverse (FSAI) with an incomplete LU (ILU) factorization. The generalized FSAI, called block FSAI, is derived by requiring the preconditioned matrix to resemble a block-diagonal matrix in the sense of the minimal Frobenius norm. An incomplete block Jacobi algorithm is then effectively used to accelerate the convergence of a Krylov subspace method. The block FSAI-ILU preconditioner proves superior to both FSAI and the incomplete block Jacobi by themselves in a number of realistic finite element test cases and is fully scalable for a given number of blocks.

Adaptive Techniques for Improving the Performance of Incomplete Factorization Preconditioning

Three techniques for improving the robustness and performance of incomplete factorization preconditioners for sparse systems with symmetric positive definite or mildly indefinite coefficient matrices are introduced. The primary contribution is two new block algorithms for incomplete factorization that result in an improvement in the performance of both the preconditioner generation and the iterative solution phases. One of the algorithms applies to matrices that have a natural block structure in their original form, and the other applies to matrices without natural dense blocks. Additionally, two relatively simple but highly effective heuristic strategies are introduced. These include selecting the solver based on the definiteness properties of the preconditioner and automatic selection and tuning of incomplete factorization parameters. All three techniques have adaptive components; i.e., the preconditioner-solver combination chooses parameters or algorithmic components based on the properties of the coefficient matrix and its incomplete factors. Two of the three techniques are applicable to incomplete LU factorization for unsymmetric systems as well.

Parallel nonlinear preconditioners on multicore architectures

Parallel nonlinear preconditioners, for solving mildly nonlinear systems, are proposed. These algorithms are based on both the Fletcher---Reeves version of the nonlinear conjugate gradient method and a polynomial preconditioner type based on block two-stage methods. The behavior of these algorithms is analyzed when incomplete LU factorizations are used in order to obtain the inner splittings of the block two-stage method. As our illustrative example we have considered a nonlinear elliptic partial differential equation, known as the Bratu problem. The reported experiments show the performance of the algorithms designed in this work on two multicore architectures.

On the implementation of 3D Galerkin boundary integral equations

In this article, a reverse contribution technique is proposed to accelerate the construction of the dense influence matrices associated with a Galerkin approximation of hypersingular boundary integral equations of mixed-type in potential theory. In addition, a general-purpose sparse preconditioner for boundary element methods has also been developed to successfully deal with ill-conditioned linear systems arising from the discretization of mixed boundary-value problems on non-smooth surfaces. The proposed preconditioner, which originates from the precorrected-FFT method, is sparse, easy to generate and apply in a Krylov subspace iterative solution of discretized boundary integral equations. Moreover, an approximate inverse of the preconditioner is implicitly built by employing an incomplete LU factorization. Numerical experiments involving mixed boundary-value problems for the Laplace equation are included to illustrate the performance and validity of the proposed techniques.

Parallel algorithms for solving linear systems with sparse triangular matrices

In this article, we present two new algorithms for solving given triangular systems in parallel on a shared memory architecture. Multilevel incomplete LU factorization based preconditioners, which have been very successful for solving linear systems iteratively, require these triangular solves. Hence, the algorithms presented here can be seen as parallelizing the application of these preconditioners. The first algorithm solves the triangular matrix by block anti-diagonals. The drawback of this approach is that it can be difficult to choose an appropriate block structure. On the other hand, if a good block partition can be found, this algorithm can be quite effective. The second algorithm takes a hybrid approach by solving the triangular system by block columns and anti-diagonals. It is usually as effective as the first algorithm, but the block structure can be chosen in a nearly optimal manner. Although numerical results indicate that the speed-up can be fairly good, systems with matrices having a strong diagonal structure or narrow bandwidth cannot be solved effectively in parallel. Hence, for these matrices, the results are disappointing. On the other hand, the results are better for matrices having a more uniform distribution of non-zero elements. Although not discussed in this article, these algorithms can possibly be adapted for distributed memory architectures.

Breakdown-free version of ILU factorization for nonsymmetric positive definite matrices

In this paper a new ILU factorization preconditioner for solving large sparse linear systems by iterative methods is presented. The factorization which is based on A -biorthogonalization process is well defined for a general positive definite matrix. Numerical experiments illustrating the performance of the preconditioner are presented. A comparison with the well known preconditioner RIF p of Benzi and Tůma is also included.

Solution of finite arrays using stationary block iterative techniques

The finite element-boundary integral method is used in the context of stationary iteration techniques to solve for the radiation patterns of linear and planar finite arrays of identical elements. The radiating element is a cavity-backed slot/patch (CBS/P) antenna which is decomposed into an interior region and an aperture region. The array elements are coupled to each other through the aperture unknowns and the free-space Green's function. The resulting block matrix system is solved using one of three stationary block iterative techniques namely the Jacobi method, the Gauss–Seidel method and the successive over-relaxation method. An incomplete LU factorisation of the single-element matrix is used within the block iteration scheme. The rate of convergence and computational statistics of these block iterative techniques are examined and compared with each other. Conclusions are drawn on the effectiveness of each of these iterative techniques to tackle moderate to large finite arrays of CBS/P antennas.

Symmetric Permutations for I‐Matrices to Avoid Small Pivots During Incomplete Factorization

AbstractIncomplete LU–factorizations have been very successful as preconditioners for solving sparse linear systems iteratively. However, for unsymmetric, indefinite systems small pivots (or even zero pivots) are often very detrimental to the quality of the preconditioner. A fairly recent strategy to deal with this problem has been to permute the rows of the matrix and to scale rows and columns to produce an I–matrix, a matrix having elements of modulus one on the diagonal and elements of at most modulus one elsewhere. These matrices are generally more suited for incomplete LU–factorization. I–matrices are preserved by symmetric permutation, i.e. by applying the same permutation to rows and columns of a matrix. We discuss different approaches for constructing such permutations which aim at improving the sparsity and diagonal dominance of an initial block. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Convergence behaviors of multisplitting methods with [formula omitted] relaxed parameters

In this paper, two multisplitting methods with K + 1 relaxed parameters are established for solving a linear system whose coefficient matrix is a large sparse M -matrix or H -matrix and the corresponding convergence behaviors are studied. Then the implementation of these two methods with ILU factorizations as inner splittings is investigated. Finally, some numerical experiments are presented to illustrate the effectiveness of the preconditioners obtained from our methods when combined with BiCGSTAB.

Analysis of incomplete factorizations for a nine-point approximation to a convection–diffusion model problem

We study the stability of zero-fill incomplete LU factorizations of a nine-point coefficient matrix arising from a high-order compact discretisation of a two-dimensional constant-coefficient convection–diffusion problem. Nonlinear recurrences for computing entries of the lower and upper triangular matrices are derived and we show that the sequence of diagonal entries of the lower triangular factor is unconditionally convergent. A theoretical estimate of the limiting value is derived and we show that this estimate is a good predictor of the computed value. The unconditional convergence of the diagonal sequence of the lower triangular factor to a positive limit implies that the incomplete factorization process never encounters a zero pivot and that the other diagonal sequences are also convergent. The characteristic polynomials associated with the lower and upper triangular solves that occur during the preconditioning step are studied and conditions for the stability of the triangular solves are derived in terms of the entries of the tridiagonal matrices appearing in the lower and upper subdiagonals of the block triangular system matrix and a triplet of parameters which completely determines the solution of the nonlinear recursions. Results of ILU-preconditioned GMRES iterations and the effects of orderings on their convergence are also described.

Fast transform based preconditioners for 2D finite-difference frequency-domain – Waveguides and periodic structures

The fields scattered by dielectric objects placed inside parallel-plate waveguides and periodic structures in two dimensions may efficiently be computed via a finite-difference frequency-domain (FDFD) method. This involves large, sparse linear systems of equations that may be solved using preconditioned Krylov subspace methods. Our preconditioners involve fast discrete trigonometric transforms and are based on a physical approximation. Simulations show significant gain in terms of computation time and iteration count in comparison with results obtained with preconditioners based on incomplete LU (ILU) factorization. Moreover, with the new preconditioners, the required number of iterations is independent of the grid size.

THE EFFICIENT ANALYSIS OF ELECTROMAGNETIC SCATTERING FROM COMPOSITE STRUCTURES USING HYBRID CFIE-IEFIE

The efficient algorithm is presented for the analysis of electromagnetic scattering from composite structures with coexisting open and closed conductors. A hybrid combined-field integral equation-the improved electric-field integral equation (CFIE-IEFIE) formulation with the incomplete LU factorization (ILU) preconditioner is proposed. Numerical results are given to demonstrate that the efficiency of our algorithm can be significantly improved when compared with the conventional EFIE formulation and the hybrid CFIE-EFIE formulation.

An Implementation of A Parallel Iterative Algorithm for the Solution of Large Banded System on A Cluster of Workstations

In this paper, we present a parallel iterative solution for large banded systems of linear equations based on incomplete LU-factorization (ILU). A master—workers parallel computing scheme is used. The proposed algorithm incurs reduced storage and communication overhead as compared to previous methods. The reduction in communication overhead has been achieved by pipelining the inter-workers message passing and limiting the communication between master and workers to initial work distribution and final results collection.The performance of the proposed algorithm is evaluated analytically and experimentally using Parallel Virtual Machine (PVM) as a parallel programming environment on a cluster of Linux work-stations. The obtained performance results show that the proposed algorithm exhibits higher efficiency for coarse grain computations corresponding to large matrix sizes and reduced number of processors.

Symmetric Permutations for I-matrices to Delay and Avoid Small Pivots During Factorization

In this article, we present several new permutations for I-matrices making these more suitable for incomplete LU-factorization preconditioners used in solving linear systems by iterative methods. A general matrix can be transformed by row permutation as well as row and columns scaling into an I-matrix, i.e., a matrix having elements of modulus 1 on the diagonal and elements of modulus of no more than 1 elsewhere. Reordering rows and columns by the same permutation clearly preserves I-matrices. In this article, we consider such reordering techniques which make the permuted matrix more suitable for an incomplete LU-factorization preconditioner than the original I-matrix. We use a multilevel ILUC, an incomplete LU-factorization preconditioner using Crout's implementation of Gaussian elimination without pivoting to test these reorderings. The combination of I-matrix preprocessing with the various algorithms presented here and the multilevel incomplete LU-factorizations forms a powerful preconditioning method for unsymmetric, highly indefinite problems. The C++ code has been made available in the software package ILU++.

Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier–Stokes Equations

We study preconditioners for the iterative solution of the linear systems arising in the implicit time integration of the compressible Navier-Stokes equations. The spatial discretization is carried out using a discontinuous Galerkin method with fourth order polynomial interpolations on triangular elements. The time integration is based on backward difference formulas resulting in a nonlinear system of equations which is solved at each timestep. This is accomplished using Newton's method. The resulting linear systems are solved using a preconditioned GMRES iterative algorithm. We consider several existing preconditioners such as block Jacobi and Gauss-Seidel combined with multilevel schemes which have been developed and tested for specific applications. While our results are consistent with the claims reported, we find that these preconditioners lack robustness when used in more challenging situations involving low Mach numbers, stretched grids, or high Reynolds number turbulent flows. We propose a preconditioner based on a coarse scale correction with postsmoothing based on a block incomplete LU factorization with zero fill-in (ILU0) of the Jacobian matrix. The performance of the ILU0 smoother is found to depend critically on the element numbering. We propose a numbering strategy based on minimizing the discarded fill-in in a greedy fashion. The coarse scale correction scheme is found to be important for diffusion dominated problems, whereas the ILU0 preconditioner with the proposed ordering is effective at handling the convection dominated case. While little can be said in the way of theoretical results, the proposed preconditioner is shown to perform remarkably well for a broad range of representative test problems. These include compressible flows ranging from very low Reynolds numbers to fully turbulent flows using the Reynolds averaged Navier-Stokes equations discretized on highly stretched grids. For low Mach number flows, the proposed preconditioner is more than one order of magnitude more efficient than the other preconditioners considered.

An Efficient ILU Preconditioning for Highly Sparse Matrices Constructed using the FDFD Method

Finding a good preconditioner to solve a given sparse linear system is often considered a difficult but important task. One of the simplest ways of defining a preconditioner is to perform an incomplete LU decomposition (ILU) of the original matrix. The ILU factorization is considered to be an easy and inexpensive preconditioner to use. However, it fails to provide a solution of a sparse linear system generated from general three-dimensional problems with three unknowns. In this paper, a modified ILU preconditioner is proposed to provide an efficient preconditioner for highly sparse matrices, especially for matrices constructed using the finite-difference frequency-domain (FDFD) method for three-dimensional applications. It has been proven that the proposed ILU preconditioner provides a valid solution when the classical ILU preconditioner fails. The efficiency of the proposed ILU preconditioner is also demonstrated by the small memory requirements relative to traditional ILU preconditioners.