Incomplete LU Research Articles

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.

Analysis of wire antennas mounted on large perfectly conducting platforms using MLFMA

The electric field integral equation (EFIE) combined with the multilevel fast multipole algorithm (MLFMA) is applied to analyze the radiation and impedance properties of wire antennas mounted on complex conducting platforms to realize fast, accurate solutions. Wire, surface and junction basis functions are used to model the current distribution on the object. Application of MLFMA reduces memory requirement and computing time compared to conventional methods, such as method of moment (MOM), especially for the antenna on a large-sized platform. Generalized minimal residual (GMRES) solver with incomplete LU factorization preconditioner using a dual dropping strategy (ILUT) is applied to reduce the iterative number. Several typical numerical examples are presented to validate this algorithm and show the accuracy and computational efficiency.

A multilevel Crout ILU preconditioner with pivoting and row permutation

AbstractIn this paper, we present a new incomplete LU factorization using pivoting by columns and row permutation. Pivoting by columns helps to avoid small pivots and row permutation is used to promote sparsity. This factorization is used in a multilevel framework as a preconditioner for iterative methods for solving sparse linear systems. In most multilevel incomplete ILU factorization preconditioners, preprocessing (scaling and permutation of rows and columns of the coefficient matrix) results in further improvements. Numerical results illustrate that these preconditioners are suitable for a wide variety of applications. Copyright © 2007 John Wiley & Sons, Ltd.

JADAMILU: a software code for computing selected eigenvalues of large sparse symmetric matrices

A new software code for computing selected eigenvalues and associated eigenvectors of a real symmetric matrix is described. The eigenvalues are either the smallest or those closest to some specified target, which may be in the interior of the spectrum. The underlying algorithm combines the Jacobi–Davidson method with efficient multilevel incomplete LU (ILU) preconditioning. Key features are modest memory requirements and robust convergence to accurate solutions. Parameters needed for incomplete LU preconditioning are automatically computed and may be updated at run time depending on the convergence pattern. The software is easy to use by non-experts and its top level routines are written in FORTRAN 77. Its potentialities are demonstrated on a few applications taken from computational physics. Program summary Program title: JADAMILU Catalogue identifier: ADZT_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZT_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 101 359 No. of bytes in distributed program, including test data, etc.: 7 493 144 Distribution format: tar.gz Programming language: Fortran 77 Computer: Intel or AMD with g77 and pgf; Intel EM64T or Itanium with ifort; AMD Opteron with g77, pgf and ifort; Power (IBM) with xlf90. Operating system: Linux, AIX RAM: problem dependent Word size: real:8; integer: 4 or 8, according to user's choice Classification: 4.8 Nature of problem: Any physical problem requiring the computation of a few eigenvalues of a symmetric matrix. Solution method: Jacobi–Davidson combined with multilevel ILU preconditioning. Additional comments: We supply binaries rather than source code because JADAMILU uses the following external packages: • MC64. This software is copyrighted software and not freely available. COPYRIGHT (c) 1999 Council for the Central Laboratory of the Research Councils. • AMD. Copyright (c) 2004–2006 by Timothy A. Davis, Patrick R. Amestoy, and Iain S. Duff. All Rights Reserved. Source code is distributed by the authors under the GNU LGPL licence. • BLAS. The reference BLAS is a freely-available software package. It is available from netlib via anonymous ftp and the World Wide Web. • LAPACK. The complete LAPACK package or individual routines from LAPACK are freely available on netlib and can be obtained via the World Wide Web or anonymous ftp. • For maximal benefit to the community, we added the sources we are proprietary of to the tar.gz file submitted for inclusion in the CPC library. However, as explained in the README file, users willing to compile the code instead of using binaries should first obtain the sources for the external packages mentioned above (email and/or web addresses are provided). Running time: Problem dependent; the test examples provided with the code only take a few seconds to run; timing results for large scale problems are given in Section 5.

Reduced-Order Preconditioning for Bidomain Simulations

Simulations of the bidomain equations involve solving large, sparse, linear systems of the form Ax = b. Being an initial value problems, it is solved at every time step. Therefore, efficient solvers are essential to keep simulations tractable. Iterative solvers, especially the preconditioned conjugate gradient (PCG) method, are attractive since memory demands are minimized compared to direct methods, albeit at the cost of solution speed. However, a proper preconditioner can drastically speed up the solution process by reducing the number of iterations. In this paper, a novel preconditioner for the PCG method based on system order reduction using the Arnoldi method (A-PCG) is proposed. Large order systems, generated during cardiac bidomain simulations employing a finite element method formulation, are solved with the A-PCG method. Its performance is compared with incomplete LU (ILU) preconditioning. Results indicate that the A-PCG estimates an approximate solution considerably faster than the ILU, often within a single iteration. To reduce the computational demands in terms of memory and run time, the use of a cascaded preconditioner was suggested. The A-PCG was applied to quickly obtain an approximate solution, and subsequently a cheap iterative method such as successive overrelaxation (SOR) is applied to further refine the solution to arrive at a desired accuracy. The memory requirements are less than those of direct LU but more than ILU method. The proposed scheme is shown to yield significant speedups when solving time evolving systems.

Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model

The bidomain equations are considered to be one of the most complete descriptions of the electrical activity in cardiac tissue, but large scale simulations, as resulting from discretization of an entire heart, remain a computational challenge due to the elliptic portion of the problem, the part associated with solving the extracellular potential. In such cases, the use of iterative solvers and parallel computing environments are mandatory to make parameter studies feasible. The preconditioned conjugate gradient (PCG) method is a standard choice for this problem. Although robust, its efficiency greatly depends on the choice of preconditioner. On structured grids, it has been demonstrated that a geometric multigrid preconditioner performs significantly better than an incomplete LU (ILU) preconditioner. However, unstructured grids are often preferred to better represent organ boundaries and allow for coarser discretization in the bath far from cardiac surfaces. Under these circumstances, algebraic multigrid (AMG) methods are advantageous since they compute coarser levels directly from the system matrix itself, thus avoiding the complexity of explicitly generating coarser, geometric grids. In this paper, the performance of an AMG preconditioner (BoomerAMG) is compared with that of the standard ILU preconditioner and a direct solver. BoomerAMG is used in two different ways, as a preconditioner and as a standalone solver. Two 3-D simulation examples modeling the induction of arrhythmias in rabbit ventricles were used to measure performance in both sequential and parallel simulations. It is shown that the AMG preconditioner is very well suited for the solution of the bidomain equation, being clearly superior to ILU preconditioning in all regards, with speedups by factors in the range 5.9-7.7.

A Study of preconditioned Krylov subspace methods with reordering for linear systems from a biphasic v–p finite element formulation

A study was conducted on combinations of preconditioned iterative methods with matrix reordering to solve the linear systems arising from a biphasic velocity–pressure (v–p) finite element formulation used to simulate soft hydrated tissues in the human musculoskeletal system. Krylov subspace methods were tested due to the symmetric indefiniteness of our systems, specifically the generalized minimal residual (GMRES), transpose-free quasi-minimal residual (TFQMR), and biconjugate gradient stabilized (BiCGSTAB) methods. Standard graph reordering techniques were used with incomplete LU (ILU) preconditioning. Performance of the methods was compared on the basis of convergence rate, computing time, and memory requirements. Our results indicate that performance is affected more significantly by the choice of reordering scheme than by the choice of Krylov method. Overall, BiCGSTAB with one-way dissection (OWD) reordering performed best for a test problem representative of a physiological tissue layer. The preferred methods were then used to simulate the contact of the humeral head and glenoid tissue layers in glenohumeral joint of the shoulder, using a penetration-based method to approximate contact. The distribution of pressure and stress fields within the tissues shows significant through-thickness effects and demonstrates the importance of simulating soft hydrated tissues with a biphasic model.

On a [formula omitted] preconditioning of non-conforming mixed FEM elliptic problems

In this paper we analyze a preconditioner for mixed finite element systems arising in the approximation of a second order elliptic problem with Neumann boundary conditions by triangular non-conforming elements. This problem stems from the so called projection methods for the unsteady Navier–Stokes equations and is one of the most computationally intensive parts of the method. The present study is focused on the efficient implementation of the modified incomplete LU factorization MIC ( 0 ) as a preconditioner in the PCG iterative method for the reduced Schur complement system following from the mixed formulation of the pressure Poisson problem. The presented model analysis and the related set of numerical tests illustrate well the potential of the proposed approach.

Scaling, reordering, and diagonal pivoting in ILU preconditionings

In this paper we consider several techniques for construction of general-purpose ILU pre- conditionings. We also provide some new theoretical results supporting the proposed preconditioning strategy. For the preprocession stage we consider row/column norm-equalizing two-side scaling, as well as diagonal dominance improvement by unsymmetric permutations. Within the incomplete LU- factorization, we use a variant of the inverse norm-reducing Complete Diagonal Pivoting. Numerical results are given for 67 test problems chosen from publicly available matrix sets.

Perturbed incomplete ILU preconditioner for efficient solution of electric field integral equations

To efficiently solve large, dense, complex linear systems that arise in the electric field integral equation (EFIE) formulation of electromagnetic scattering problems, a new modified incomplete LU (ILU) preconditioner is developed and used in the context of the generalised minimal residual iterative method accelerated with the multilevel fast multipole method. The key idea is to perturb the near-field impedance matrix of EFIE with the principle value term of the magnetic field integral equation operator before constructing ILU preconditioners. Numerical experiments indicate that this new perturbation technique is very effective with the ILU preconditioner and the resulted ILU preconditioner can reduce both the iteration number and the computational time substantially.

Incomplete LU Preconditioning with the Multilevel Fast Multipole Algorithm for Electromagnetic Scattering

Iterative solution of large-scale scattering problems in computational electromagnetics with the multilevel fast multipole algorithm (MLFMA) requires strong preconditioners, especially for the electric-field integral equation (EFIE) formulation. Incomplete LU (ILU) preconditioners are widely used and available in several solver packages. However, they lack robustness due to potential instability problems. In this study, we consider various ILU-class preconditioners and investigate the parameters that render them safely applicable to common surface integral formulations without increasing the ${\cal O}(n\log n)$ complexity of MLFMA. We conclude that the no-fill ILU(0) preconditioner is an optimal choice for the combined-field integral equation (CFIE). For EFIE, we establish the need to resort to methods depending on drop tolerance and apply pivoting for problems with high condition estimate. We propose a strategy for the selection of the parameters so that the preconditioner can be used as a black-box method. Robustness and efficiency of the employed preconditioners are demonstrated over several test problems.

Stability and Fast Algorithms of Incomplete LU Factorization with Zero‐Fill for Nine‐Diagonal Matrices

Stability and fast algorithms of zero‐fill incomplete LU factorization for nonsymmetric nine‐diagonal matrices are considered. Based on a convergence analysis of the diagonal sequences in the LU factors of a monotone nine‐diagonal matrix, we give necessary and sufficient conditions for stably computing the ILU factorization in terms of an extreme solution to a system of nonlinear equations. Furthermore, the extreme solution is used to construct a fast ILU factorization with much less cost in flops and storage. Basically we use the standard ILU recursions up to a certain point and then replace the remaining terms of the recursive sequences by their limit values that can be determined by the extreme solution. Two strategies, preconditioning the nonlinear system to suit for Newton iterations and estimating the solution to get a good initial for the iterations, are discussed for computing the extreme solution efficiently. We also generalize the stability analysis and propose fast algorithms for nine‐diagonal matrices with periodically monotone diagonals or other nonconstant diagonals. Numerical examples show the efficiency of the proposed fast algorithms.

A control volume finite‐element method for numerical simulating incompressible fluid flows without pressure correction

AbstractThis paper presents a numerical model to study the laminar flows induced in confined spaces by natural convection. A control volume finite‐element method (CVFEM) with equal‐order meshing is employed to discretize the governing equations in the pressure–velocity formulation. In the proposed model, unknown variables are calculated in the same grid system using different specific interpolation functions without pressure correction. To manage memory storage requirements, a data storage format is developed for generated sparse banded matrices. The performance of various Krylov techniques, including Bi‐CGSTAB (Bi‐Conjugate Gradient STABilized) with an incomplete LU (ILU) factorization preconditioner is verified by applying it to three well‐known test problems. The results are compared to those of independent numerical or theoretical solutions in literature. The iterative computer procedure is improved by using a coupled strategy, which consists of solving simultaneously the momentum and the continuity equation transformed in a pressure equation. Results show that the strategy provides useful benefits with respect to both reduction of storage requirements and central processing unit runtime. Copyright © 2006 John Wiley & Sons, Ltd.

A preconditioner for solving large-scale variational inequality problems by a semismooth inexact approach

The numerical solution of a large-scale variational inequality problem can be obtained using the generalization of an inexact Newton method applied to a semismooth nonlinear system. This approach requires a sparse and large linear system to be solved at each step. In this work we obtain an approximate solution of this system by the LSQR algorithm of Paige and Saunders combined with a convenient preconditioner that is a variant of the incomplete LU–factorization. Since the computation of the factorization of the preconditioning matrix can be very expensive and memory consuming, we propose a preconditioner that admits block-factorization. Thus the direct factorization is only applied to submatrices of smaller sizes. Numerical experiments on a set of test-problems arising from the literature show the effectiveness of this approach.

Fast 3-D Capacitance Extraction by Inexact Factorization and Reduction

Capacitance-extraction algorithms based on the boundary element method (BEM) have to solve large linear systems. The number of unknowns equals the number of discretization panels n, which is much greater than the number of conductors m. The authors present a capacitance-extraction algorithm RedCap that first reduces the BEM system of size n into a small system of size O(m) and then solves the small system to compute the capacitances. RedCap uses a number of techniques, including the hierarchical-refinement technique of HiCap [Shi, 2002], the dense-to-sparse transformation of PHiCap [Yan, 2005], a reordering of the sparse linear system, and an incomplete LU factorization, to obtain the reduced system. RedCap achieves a significant speed improvement over previous methods. On benchmark problems with conductors in uniform and multilayer dielectrics, RedCap is up to 100 times faster than FastCap [Nabors and White, 1991] and up to four times faster than PHiCap [Yan, 2005], while restricting error to within 2% of FastCap

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.

Evaluation and enhancement of COBRA-TF efficiency for LWR calculations

Detailed representations of the reactor core generate computational meshes with a high number of cells where the fluid dynamics equations must be solved. An exhaustive analysis of the CPU times needed by the thermal-hydraulic subchannel code COBRA-TF for different stages in the solution process has revealed that the solution of the linear system of pressure equations is the most time consuming process. To improve code efficiency two optimized matrix solvers, Super LU library and Krylov non-stationary iterative methods have been implemented in the code and their performance has been tested using a suite of five test cases. The results of performed comparative analyses have demonstrated that for large cases, the implementation of the Bi-Conjugate Gradient Stabilized (Bi-CGSTAB) Krylov method combined with the incomplete LU factorization with dual truncation strategy (ILUT) pre-conditioner reduced the time used by the code for the solution of the pressure matrix by a factor of 20. Both new solvers converge smoothly regardless of the nature of simulated cases and the mesh structures and improve the stability and accuracy of results compared to the classic Gauss–Seidel iterative method. The obtained results indicate that the direct inversion method is the best option for small cases.

Parallel Newton two‐stage methods based on ILU factorizations for nonlinear systems

AbstractParallel iterative algorithms based on the Newton method and on two of its variants, the Shamanskii method and the Chord method, for solving nonlinear systems are proposed. These algorithms are based on two‐stage multisplitting methods where incomplete LU factorizations are considered as a mean of constructing the inner splittings. Convergence properties of these parallel methods are studied for H‐matrices. Computational results of these methods on two parallel computing systems are discussed. The reported experiments show the effectiveness of these methods. Copyright © 2006 John Wiley & Sons, Ltd.

Iterative‐solvers for numerical analysis of photonic devices

AbstractAn assessment of four linear‐system iterative solvers is carried out through the simulation of a key 2D photonic crystal structure. Among these solvers, the biconjugate gradient stabilized (BCGS) in conjunction with the incomplete LU (ILU) decomposition, shows the best performance and can be regarded as a promising candidate for large‐scale computation. In addition, the iterative solver with the best performance is applied to analysis of the directional optical coupler and transverse anisotropic waveguide. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1182–1186, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21571

Iterative solution of high-order boundary element method for acoustic impedance boundary value problems

A high-order boundary element method for time-harmonic acoustic impedance boundary value problems is presented. The method is based on the Galerkin-type formulation of the Burton–Miller integral equation (BMIE) with high-order polynomial basis and testing functions. This formulation has several important advantages: It is free of the interior resonance problem, the hypersingular integral operator of the traditional BMIE formulation can be avoided, it leads to faster convergence in terms of the number of unknowns than the low-order methods and it shows a good performance with iterative solvers. To avoid the numerical difficulties associated to the implementation of the singular surface integral equations, the singularity extraction technique is applied to evaluate the integrals in the singular and near-singular cases. The resulting matrix equation is solved iteratively with the generalized minimal residual method (GMRES) and a simple preconditioner based on the incomplete LU factorization is applied to expedite the convergence. Numerical results indicate that the BMIE formulation with Galerkin method and high-order basis functions has good convergence properties for various geometries and boundary conditions on a wide frequency range when the GMRES method is applied to solve the matrix equation iteratively. This, in turn, indicates that the formulation is well suited for an efficient application of fast solution procedures, such as the fast multipole method.