Multilevel Incomplete LU Research Articles

HIFIR: Hybrid Incomplete Factorization with Iterative Refinement for Preconditioning Ill-Conditioned and Singular Systems

We introduce a software package called Hybrid Incomplete Factorization with Iterative Refinement (HIFIR) for preconditioning sparse, unsymmetric, ill-conditioned, and potentially singular systems. HIFIR computes a hybrid incomplete factorization (HIF) , which combines multilevel incomplete LU factorization with a truncated, rank-revealing QR (RRQR) factorization on the final Schur complement. This novel hybridization is based on the new theory of ϵ-accurate approximate generalized inverse (AGI) . It enables near-optimal preconditioners for consistent systems and enables flexible GMRES to solve inconsistent systems when coupled with iterative refinement. In this article, we focus on some practical algorithmic and software issues of HIFIR. In particular, we introduce a new inverse-based rook pivoting (IBRP) into ILU, which improves the robustness and the overall efficiency for some ill-conditioned systems by significantly reducing the size of the final Schur complement for some systems. We also describe the software design of HIFIR in terms of its efficient data structures for supporting rook pivoting in a multilevel setting, its template-based generic programming interfaces for mixed-precision real and complex values in C++, and its user-friendly high-level interfaces in MATLAB and Python. We demonstrate the effectiveness of HIFIR for ill-conditioned or singular systems arising from several applications, including the Helmholtz equation, linear elasticity, stationary incompressible Navier–Stokes (INS) equations, and time-dependent advection-diffusion equation.

Accelerating advanced preconditioning methods on hybrid architectures

Many problems, in diverse areas of science and engineering, involve the solution of largescale sparse systems of linear equations. In most of these scenarios, they are also a computational bottleneck, and therefore their efficient solution on parallel architectureshas motivated a tremendous volume of research.This dissertation targets the use of GPUs to enhance the performance of the solution of sparse linear systems using iterative methods complemented with state-of-the-art preconditioned techniques. In particular, we study ILUPACK, a package for the solution of sparse linear systems via Krylov subspace methods that relies on a modern inverse-based multilevel ILU (incomplete LU) preconditioning technique.We present new data-parallel versions of the preconditioner and the most important solvers contained in the package that significantly improve its performance without affecting its accuracy. Additionally we enhance existing task-parallel versions of ILUPACK for shared- and distributed-memory systems with the inclusion of GPU acceleration. The results obtained show a sensible reduction in the runtime of the methods, as well as the possibility of addressing large-scale problems efficiently.

VBARMS: A variable block algebraic recursive multilevel solver for sparse linear systems

Sparse matrices arising from the solution of systems of partial differential equations often exhibit a perfect block structure, meaning that the nonzero blocks in the sparsity pattern are fully dense (and typically small), e.g., when several unknown quantities are associated with the same grid point. Similar block orderings can be sometimes unravelled also on general unstructured matrices, by ordering consecutively rows and columns with a similar sparsity pattern, and treating some zero entries of the reordered matrix as nonzero elements, with a little sacrifice of memory. We show how we can take advantage of these frequently occurring structures in the design of the multilevel incomplete LU factorization preconditioner ARMS (Saad and Suchomel, 2002 [14]) and maximize computational efficiency.

SYMMETRIC INVERSE-BASED MULTILEVEL ILU PRECONDITIONING FOR SOLVING DENSE COMPLEX NON-HERMITIAN SYSTEMS IN ELECTROMAGNETICS

Boundary element discretizations of exterior Maxwell problems lead to dense complex non-Hermitian systems of linear equations that are di-cult to solve from a linear algebra point of view. We show that the recently developed class of inverse-based multilevel incomplete LU factorization has very good potential to precondition these systems efiectively. This family of algorithms can produce numerically stable factorizations and exploits e-ciently the possible symmetry of the underlying integral formulation. The results are highlighted by calculating the radar-cross-section of a full aircraft, and by a numerical comparison against other standard preconditioners.

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.

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.

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.

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.

A grid-based multilevel incomplete LU factorization preconditioning technique for general sparse matrices

We design a grid-based multilevel incomplete LU preconditioner (GILUM) for solving general sparse matrices. This preconditioner combines a high accuracy ILU factorization with an algebraic multilevel recursive reduction. The GILUM preconditioner is a compliment to the domain-based multilevel block ILUT preconditioner. A major difference between these two preconditioners is the way that the coarse level nodes are chosen. The approach of GILUM is analogous to that of algebraic multigrid method. The GILUM approach avoids some controversial issues in algebraic multigrid method such as how to construct the interlevel transfer operators and how to compute the coarse level operator. Numerical experiments are conducted to compare GILUM with other ILU preconditioners.

A Multilevel Dual Reordering Strategy for Robust Incomplete LU Factorization of Indefinite Matrices

A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. The first part with large diagonal dominance is reordered using a graph-based strategy, followed by an ILU factorization. A partial ILU factorization is applied to the second part to yield an approximate Schur complement matrix. The whole process is repeated on the Schur complement matrix and continues for a few times to yield a multilevel ILU factorization. Analyses are conducted to show how the Schur complement approach removes small diagonal elements of indefinite matrices and how the stability of the LU factor affects the quality of the preconditioner. Numerical results are used to compare the new preconditioning strategy with two popular ILU preconditioning techniques and a multilevel block ILU threshold preconditioner.