Incomplete LU Factorization Research Articles

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.

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.

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.

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

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.

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 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.

Aggregation-Based Algebraic Multilevel Preconditioning

We propose a preconditioning technique that is applicable in a "black box" fashion to linear systems arising from second order scalar elliptic PDEs discretized by finite differences or finite elements with nodal basis functions. This technique is based on an algebraic multilevel scheme with coarsening by aggregation. We introduce a new aggregation method which, for the targeted class of applications, produces semicoarsening effects whenever desirable, while the number of nodes is decreased by a factor of about 4 at each level, regardless of the problem at hand. Moreover, the number of nonzero entries per row in the successive coarse grid matrices remains approximately constant, ensuring small set up cost and modest memory requirements. This aggregation technique can be used in an algebraic multigrid (AMG)-like framework, but better results are obtained with an algebraic multilevel scheme based on a block approximate factorization of the matrix. In this scheme, the block pivot corresponding to fine grid nodes is approximated by a modified incomplete LU (MILU) factorization. To enhance robustness and avoid any potential breakdown, the coarsening process is refined by recasting as "coarse" fine grid nodes for which the corresponding pivot in this MILU factorization would be negative or too small. Numerical results display the efficiency, the scalability, and the robustness of the resulting preconditioner on a wide set of discrete scalar PDE problems, ranging from the two-dimensional Poisson equation to three-dimensional convection-diffusion problems with high Reynolds number and strongly varying convection.

Multilevel Preconditioners Constructed From Inverse-Based ILUs

This paper analyzes dropping strategies in a multilevel incomplete LU decomposition context and presents a few strategies for obtaining related ILUs with enhanced robustness. The analysis shows that the incomplete LU factorization resulting from dropping small entries in Gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large. As a consequence a few strategies are developed whose goal is to achieve this feature. A number of "templates" for enabling implementations of these factorizations are presented. Numerical experiments show that the resulting ILUs offer a good compromise between robustness and efficiency.

Localized Jacobian ILU preconditioners for hydraulic fractures

AbstractWe discuss the properties of a class of sparse localized approximations to the Jacobian operator that arises in modelling the evolution of a hydraulically driven fracture in a multi‐layered elastic medium. The governing equations involve a highly non‐linear coupled system of integro‐partial differential equations along with the fracture front free boundary problem. We demonstrate that an incomplete LU factorization of these localized Jacobians yields an efficient preconditioner for the fully populated, stiff, non‐symmetric system of algebraic equations that need to be solved multiple times for every growth increment of the fracture. The performance characteristics of this class of preconditioners is explored via spectral analysis and numerical experiment. Copyright © 2005 John Wiley & Sons, Ltd.

Effectiveness of GMRES-DR and OSP-ILUC for wave diffraction analysis of a very large floating structure (VLFS)

This paper presents the performance of iterative solvers and preconditioners for the non-Hermitian dense linear systems arising from the boundary value problem related to the diffraction wave field around a very large floating structure (VLFS). These systems can be solved iteratively using the GMRES with deflated restarting (GMRES-DR), which has Krylov subspaces with approximate eigenvectors as starting vectors. The number of iteration needed by GMRES or GMRES-DR can be significantly reduced using preconditioning techniques. Matrix-vector products are approximated by utilizing the fast multipole method (FMM), which need not directly calculate the dense matrix of the far field interactions. The combination of the operator splitting preconditioner (OSP) and the Crout version of the incomplete LU factorization (ILUC) does not require the dense matrix of the far field interactions. Numerical experiments from a hybrid-type VLFS, which is composed of pontoon-part and semi-submersible part, whose length is 3000 m are presented.

A robust preconditioner for fluid–structure interaction problems

Two preconditioners are presented for equation systems of strongly coupled fluid–structure interaction computations where the structure is modeled by shell elements. These preconditioners fall into the general category of incomplete LU factorization. The two differ mainly in whether the coefficient matrix is factorized node by node or variable-by-variable. In the variable-wise preconditioner, a modified Schur complement system for pressure is solved approximately with a few iterations using a special preconditioner. The efficiencies of the two preconditioners are compared for different finite element formulations of the fluid mechanics part, including formulations with SUPG and PSPG stabilizations.

Remark on convergence of algebraic multigrid in the form of matrix decomposition

We introduce the convergence of algebraic multigrid in the form of matrix decomposition. The convergence is proved in block versions of the multi-elimination incomplete LU (BILUM) factorization technique and the approximation of their inverses to preserve sparsity. The convergence theorem can be applied to general interpolation operator. Furthermore, we discuss the error caused by the error matrix.

Distributed block independent set algorithms and parallel multilevel ILU preconditioners

We present a class of parallel preconditioning strategies utilizing multilevel block incomplete LU (ILU) factorization techniques to solve large sparse linear systems. The preconditioners are constructed by exploiting the concept of block independent sets (BISs). Two algorithms for constructing BISs of a sparse matrix in a distributed environment are proposed. We compare a few implementations of the parallel multilevel ILU preconditioners with different BIS construction strategies and different Schur complement preconditioning strategies. We also use some diagonal thresholding and perturbation strategies for the BIS construction and for the last level Schur complement ILU factorization. Numerical experiments indicate that our domain-based parallel multilevel block ILU preconditioners are robust and efficient.

ILUCP: a Crout ILU preconditioner with pivoting

We introduce a new preconditioner, ILUCP, to be used with an iterative method for solving sparse linear systems. It is based on an incomplete LU factorization combining Crout's formulation of Gaussian elimination with pivoting by columns. It is usually faster than ILUTP, which is based on a delayed update version of Gaussian elimination with pivoting, but requires more memory. For applications where memory is not a primary concern, ILUCP can be an attractive alternative to ILUTP. Copyright © 2005 John Wiley & Sons, Ltd.

Multilevel ILU With Reorderings for Diagonal Dominance

This paper presents a preconditioning method based on combining two-sided permutations with a multilevel approach. The nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix. The method can be regarded as a complete pivoting version of the incomplete LU factorization. This leads to an effective incomplete factorization preconditioner for general nonsymmetric, irregularly structured, sparse linear systems.