Incomplete LU Factorization Research Articles

An incompleteLU-factorization algorithm based on block bordering

This paper suggests a new algorithm, called IBBLU, for constructing incomplete LU-factorizations of general non-singular sparse matrices. This algorithm is based on the block bordering idea and uses factorized sparse approximate inverses to approximate the inverses of pivot blocks. The triangular factors L and U are represented in explicit-implicit block form, which enhances the flop performance of the preconditioning. The algorithm suggested is theoretically justified for M- and H-matrices, and its competitiveness with the best available algebraic preconditioning methods in both symmetric and unsymmetric cases is demonstrated numerically. Copyright © 2000 John Wiley & Sons, Ltd.

Preconditioned Krylov subspace methods used in solving two-dimensional transient two-phase flows

We investigate the performance of preconditioned Krylov subspace methods used in a previously presented two-fluid model developed for the simulation of separated and intermittent gas-liquid flows. The two-fluid model has momentum and mass balances for each phase. The equations comprising this model are solved numerically by applying a two-step semi-implicit time integration procedure. A finite difference numerical scheme with a staggered mesh is used. Previously, the resulting linear algebraic equations were solved by a Gaussian band solver. In this study, these algebraic equations are also solved using the generalized minimum residual (GMRES) and the biconjugate gradient stabilized (Bi-CGSTAB) Krylov subspace iterative methods preconditioned with incomplete LU factorization using the ILUT(p, r) algorithm. The decrease in the computational time using the iterative solvers instead of the Gaussian band solver is shown to be considerable

PARALLEL INCOMPLETE LU FACTORIZATION AS A PRECONDITIONER FOR KRYLOV SUBSPACE METHODS

In this paper we describe a new method for the ILU(0) factorization of sparse systems in distributed memory multiprocessor architectures. This method uses a symbolic reordering technique, so the final system can be grouped in blocks where the rows are independent and the factorization of these entries can be carried out in parallel. The parallel ILU(0) factorization has been tested on the Cray T3E multicomputer using the MPI communication library. The performance was analysed using matrices from the Harwell–Boeing collection.

Parallelization of the ILU(0) preconditioner for CFD problems on shared-memory computers

The use of ILU(0) factorization as a preconditioner is quite frequent when solving linear systems of CFD computations. This is because of its efficiency and moderate memory requirements. For a small number of processors, this preconditioner, parallelized through coloring methods, shows little savings when compared with a sequential one using adequate reordering of the unknowns. Level scheduling techniques are applied to obtain the same preconditioning efficiency as in a sequential case, while taking advantage of parallelism through block algorithms. Numerical results obtained from the parallel solution of the compressible Navier–Stokes equations show that this technique gives interesting savings in computational times on a small number of processors of shared-memory computers. In addition, it does this while keeping all the benefits of an ILU(0) factorization with an adequate reordering of the unknowns, and without the loss of efficiency of factorization associated with a more scalable coloring strategy. Copyright © 1999 John Wiley & Sons, Ltd.

Estimation of conflict-free matrix product preconditioner on vector supercomputers

Two classes of preconditioners are considered for nonsymmetric linear systems arising from second order difference discretization of non-self-adjoint elliptic partial differential equations. Experimental results show that the usual incomplete LU and modified ILU factorization preconditioner have bad effect due to bank conflict on current vector supercomputers. That is, the bank conflict is an underlying factor for attaining high performance of the vector supercomputers. To avoid the effect of bank conflict, a matrix product preconditioner is introduced. In experiments with two-dimensional problems with constant and variable coefficients on a uniform mesh, the matrix product preconditioner will be shown to be more efficient than the standard incomplete factorization preconditioners.

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

In previous works [21–23] we proposed the use of $\tau$ [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As $\tau$ and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis.

BILUM: Block Versions of Multielimination and Multilevel ILU Preconditioner for General Sparse Linear Systems

We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typically enjoyed by multigrid methods. Several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed. The advantages of block ILUM over point ILUM include increased robustness and efficiency. We compare several versions of the block ILUM, point ILUM, and the dual-threshold-based ILUT preconditioners. In particular, tests with some convection-diffusion problems show that it may be possible to obtain convergence that is nearly independent of the Reynolds number as well as of the grid size.

Iterative solution of the drift-diffusion equations

The drift-diffusion model can be described by a nonlinear Poisson equation for the electrostatic potential coupled with a system of convection-reaction-diffusion equations for the transport of charge. We use a Gummel-like process [10] to decouple this system. Each of the obtained equations is discretised with the finite element method. We use a local scaling method to avoid breakdown in the numerical algorithm introduced by the use of Slotboom variables. Solution of the discrete nonlinear Poisson equation is accomplished with quasi-Newton methods. The nonsymmetric discrete transport equations are solved using an incomplete LU factorization preconditioner in conjunction with some robust linear solvers, such as (CGS), (BI-CGSTAB) and (GMRES). We investigate the behaviour of these iterative methods to define the effective strategy for this class of problems.

BILUTM: A Domain-Based Multilevel Block ILUT Preconditioner for General Sparse Matrices

This paper describes a domain-based multilevel block ILU preconditioner (BILUTM) for solving general sparse linear systems. This preconditioner combines a high accuracy incomplete LU factorization with an algebraic multilevel recursive reduction. Thus, in the first level the matrix is permuted into a block form using (block) independent set ordering and an ILUT factorization for the reordered matrix is performed. The reduced system is the approximate Schur complement associated with the partitioning, and it is obtained implicitly as a by-product of the partial ILUT factorization with respect to the complement of the independent set. The incomplete factorization process is repeated with the reduced systems recursively. The last reduced system is factored approximately using ILUT again. The successive reduced systems are not stored. This implementation is efficient in controlling the fill-in elements during the multilevel block ILU factorization, especially when large size blocks are used in domain decomposition-type implementations. Numerical experiments are used to show the robustness and efficiency of the proposed technique for solving some difficult problems.

Matrix Renumbering ILU: An Effective Algebraic Multilevel ILU Preconditioner for Sparse Matrices

In this paper a multilevel-like ILU preconditioner is introduced. The ILU factorization generates its own ordering during the elimination process. Both ordering and dropping depend on the size of the entries. The method can handle structured and unstructured problems. Results are presented for some important classes of matrices and for several well-known test examples. The results illustrate the efficiency of the method and show in several cases near grid independent convergence.

Parallel iterative solution for h and p approximations of the shallow water equations

A p finite element scheme and parallel iterative solver are introduced for a modified form of the shallow water equations. The governing equations are the three-dimensional shallow water equations. After a harmonic decomposition in time and rearrangement, the resulting equations are a complex Helmholz problem for surface elevation, and a complex momentum equation for the horizontal velocity. Both equations are nonlinear and the resulting system is solved using the Picard iteration combined with a preconditioned biconjugate gradient (PBCG) method for the linearized subproblems. A subdomain-based parallel preconditioner is developed which uses incomplete LU factorization with thresholding (ILUT) methods within subdomains, overlapping ILUT factorizations for subdomain boundaries and under-relaxed iteration for the resulting block system. The method builds on techniques successfully applied to linear elements by introducing ordering and condensation techniques to handle uniform p refinement. The combined methods show good performance for a range of p (element order), h (element size), and N (number of processors). Performance and scalability results are presented for a field scale problem where up to 512 processors are used.

Non-overlapping preconditioners for a parallel implicit Navier-Stokes solver

Parallel implicit iterative solution techniques are considered for application to a compressible hypersonic Navier-Stokes solver on unstructured meshes. The construction of parallel preconditioners with quasi-optimal convergence properties with respect to their serial counterpart is a key issue in the design of modern parallel implicit schemes. Two types of non-overlapping preconditioners are presented and compared. The first one is an additive Schwarz preconditioner requiring overlapping of the mesh and the second one is based on a Schur complement formulation. Both are using incomplete LU factorisation at the subdomain level but scale differently. Results are presented for computations on the Cray T3D under the message passing interface MPI.

A Fully—Implicit 2‐D Navier-Stokes Algorithm for Unstructured Grids

A fully-implicit algorithm is developed for the two-dimensional, compressible, Favre-averaged Navier-Stokes equations. It incorporates the standard k-ε turbulence model of Launder and Spalding and the low Reynolds number correction of Chien. The equations are solved using an unstructured grid of triangles with the flow variables stored at the centroids of the cells. A generalization of wall functions including pressure gradient effects is implemented to solve the near-wall region for turbulent flows using a separate algorithm and a hybrid grid. The inviscid fluxes are obtained from Roe's flux difference split method. Linear reconstruction of the flow variables to the cell faces provides second-order spatial accuracy. Turbulent and viscous stresses as well as heat transfer are obtained from a discrete representation of Gauss's theorem. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, Trapezoidal or 3-Point Backward differencing. An incomplete LU factorization of the Jacobian matrix is implemented as a preconditioning method. The accuracy of the code and the efficiency of the solution strategy are presented for three test cases: a supersonic turbulent mixing layer, a supersonic laminar compression corner and a supersonic turbulent compression corner.

Comparison of preconditioned generalized conjugate gradient methods to two-dimensional neutron and photon transport equation

We apply and compare the preconditioned generalized conjugate gradient methods to solve the linear system equation that arises in the two-dimensional neutron and photon transport equation in this paper. Several subroutines are developed on the basis of preconditioned generalized conjugate gradient methods for time-independent, two-dimensional neutron and photon transport equation in the transport theory. These generalized conjugate gradient methods are used: TFQMR (transpose free quasi-minimal residual algorithm) CGS (conjugate gradient square algorithm), Bi-CGSTAB (bi-conjugate gradient stabilized algorithm) and QMRCGSTAB (quasi-minimal residual variant of bi-conjugate gradient stabilized algorithm). These subroutines are connected to computer program DORT. Several problems are tested on a personal computer with Intel Pentium CPU. The reasons to choose the generalized conjugate gradient methods are that the methods have better residual (equivalent to error) control procedures in the computation and have better convergent rate. The pointwise incomplete LU factorization ILU, modified pointwise incomplete LU factorization MILU, block incomplete factorization BILU and modified blockwise incomplete LU factorization MBILU are the preconditioning techniques used in the several testing problems. In Bi-CGSTAB, CGS, TFQMR and QMRCGSTAB method, we find that either CGS or Bi-CGSTAB method combined with preconditioner MBILU is the most efficient algorithm in these methods in the several testing problems. The numerical solution of flux by preconditioned CGS and Bi-CGSTAB methods has the same result as those from Cray computer, obtained by either the point successive relaxation method or the line successive relaxation method combined with Gaussian elimination.

Preconditioned Krylov subspace methods for transport equations

Transport equations have many important applications. Because these equations are based on highly non-normal operators, they present difficulties in numerical computations. Iterative methods have been shown to be efficient to solve transport equations. However, because of the nature of transport problems, convergence of these methods tends to slow for many important problems. In this paper, we focus on acceleration techniques for iterative methods. Particularly, we investigate the applicability and performance of some Krylov subspace methods with preconditioners, such as the incomplete LU (ILU) factorization (with no fill-in) and multigrid algorithms (spatial and angular multigrid). Three cases are considered: isotropic equations without absorption, isotropic equations with absorption, and anisotropic equations. Our numerical experiments show that the use of an appropriate multilevel preconditioner can significantly improve Krylov subspace methods, such as GMRES and CGS.

Fast Newton-Krylov method for unstructured grids

Three algorithms are presented and compared for the solution of the steady Euler equations on unstructured triangulargrids.AllarevariationsonNewton' smethod— onequasi-andtwofull-Newtonschemes— andemploythe BILU(n)-preconditioned generalized minimum residualmethod (GMRES)algorithm to solvetheJacobian matrix problem that arises at each iteration. The quasi-Newton scheme uses a e rst-order approximation to the Jacobian matrixwiththestandardGMRESimplementation,inwhichmatrix-vectorproductsareformedintheusualexplicit manner. The full-Newton schemes are distinguished by the implementation ofGMRES: One employs thestandard GMRES algorithm, and theother is matrix free using Frechet derivatives. Thematrix-free, full-Newton algorithm is shown to be the fastest of the three algorithms. Optimal preconditioning, reordering, and storage strategies for thematrix-free, full-Newton algorithm are presented. Registerand cache performanceissues arebriee y discussed. thetradeoffbetweenspeedandmemory.Manyauthorshaveusedthe block ILU factorization with no additional e ll, BILU (0), using the natural four-by-four block size for the two-dimensional Euler equa- tions.OtherILUvariantsincludeILU (n)andBILU(n),usingscalar- and block-based storage schemes, respectively, which allow e ll en- tries based on a level-of-e ll parameter n, and ILUT(p, t), which employs a threshold drop tolerance t to introduce a maximum of p e ll entries in each factor. 19 Possible reordering algorithms include reverse Cuthill- McKee, one-way dissection, nested dissection, and quotient minimum degree. 20 The goal of the present paper is to examine the tradeoffs inherent in the various Newton- GMRES implementations and to develop a fast, robust algorithm. In particular, we compare two full-Newton algorithms,usingmatrix-freeandstandardimplementationsofGM- RES, and a quasi-Newton algorithm. The primary criterion is CPU usage. Results from an extensive parameter study of each solver are presented. Optimalstrategies for the choiceof GMRESKrylov sub- space dimension and exit tolerance, BILU (n) e ll level, reordering technique, and storage model are presented.

Parallelization of the GESIMA mesoscale atmospheric model

The implementation of the GESIMA mesoscale atmospheric model on message passing, distributed memory parallel computers is presented. Particular emphasis is given to the parallelization of the conjugate gradient solver using pre-conditioning by an incomplete LU factorization. Performance results are presented for the Cray T3D and Cray T3E systems, which show good scalability over a range of problem sizes and numbers of processors.

Experimental study of ILU preconditioners for indefinite matrices

Incomplete LU factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices. However, their failure rate is still too high for them to be useful as black-box library software for general matrices. Besides fatal breakdowns due to zero pivots, the major causes of failure are inaccuracy, and instability of the triangular solves. When there are small pivots, both these problems can occur, but these problems can also occur without small pivots. Through examples from actual problems, this paper shows how these problems evince themselves, how these problems can be detected, and how these problems can sometimes be circumvented through pivoting, reordering, scaling, perturbing diagonal elements, and preserving symmetric structure. The goal of this paper is to gain a better practical understanding of ILU preconditioners and help improve their reliability.

Approximate Inverse Techniques for Block-Partitioned Matrices

This paper proposes some preconditioning options when the system matrix is in block-partitioned form. This form may arise naturally, for example, from the incompressible Navier--Stokes equations, or may be imposed after a domain decomposition reordering. Approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner. The storage requirements forthese preconditioners may be much less than for incomplete LU factorization (ILU) preconditioners for tough, large-scale computational fluid dynamics (CFD) problems. The numerical experiments show that these preconditioners can help solve difficult linear systems whose coefficient matrices are highly indefinite.

ILUS: An incomplete LU preconditioner in sparse skyline format

Incomplete LU factorizations are among the most effective preconditioners for solving general large, sparse linear systems arising from practical engineering problems. This paper shows how an ILU factorization may be easily computed in sparse skyline storage format, as opposed to traditional row-by-row schemes. This organization of the factorization has many advantages, including its amenability when the original matrix is in skyline format, the ability to dynamically monitor the stability of the factorization and the fact that factorizations may be produced with symmetric structure. Numerical results are presented for Galerkin finite element matrices arising from the standard square lid-driven cavity problem. © 1997 John Wiley & Sons, Ltd.