Incomplete LU Factorization Research Articles

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.

Independent columns: A new parallel ILU preconditioner for the PCG method

This paper considers the parallel solution of finite element equations using the preconditioned conjugate gradient method on shared memory multiprocessors. The preconditioner used is an incomplete LU factorization of the stiffness matrix. We have designed a new method for implementing parallel forward and backward substitutions which requires the L and U factors to have an independent column structure, which we refer to as I 2LU . The algorithm has been implemented on an Encore Multimax and parallel efficiencies up to 80% of the maximum possible theoretical value have been obtained using around 6–8 processors. These figures are shown to be comparable with the efficiencies of an implementation of a similar row-oriented approach, based on level scheduling, tested with the same problems.

A NON-LINEAR ADAPTIVE FULL TRI-TREE MULTIGRID METHOD FOR THE MIXED FINITE ELEMENT FORMULATION OF THE NAVIER-STOKES EQUATIONS

The full adaptive multigrid method is based on the tri-tree grid generator. The solution of the Navier–Stokes equations is first found for a low Reynolds number. The velocity boundary conditions are then increased and the grid is adapted to the scaled solution. The scaled solution is then used as a start vector for the multigrid iterations. During the multigrid iterations the grid is first recoarsed a specified number of grid levels. The solution of the Navier–Stokes equations with the multigrid residual as right-hand side is smoothed in a fixed number of Newton iterations. The linear equation system in the Newton algorithm is solved iteratively by CGSTAB preconditioned by ILU factorization with coupled node fill-in. The full adaptive multigrid algorithm is demonstr ated for cavity flow. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 1037ndash;1047, 1997.

HIGH-ORDER SCHEME IMPLEMENTATION USING NEWTON-KRYLOV SOLUTION METHODS

Implementation of high-order discretization for the connective transport terms in the inexact Newton method for a benchmark fluid flow and heat transfer problem using various solution configurations at two Reynolds numbers has been investigated. These configurations include fully consistent discretization of the Jacobian, preconditioner and residual of the Newton method, low-order preconditioning using a matrix-free method to approximate the action of the Jacobian, and defect correction or low-order Jacobian and preconditioning. The residual in each case employs high-order discretization to preserve the high-order solution. Two preconditioners, point incomplete lower-upper factorization ILU( k) and block incomplete lower-upper factorization BILUO(K) for k = 0,1,2 were applied. Also, “one-way” muitigrid-ing and capping the inner iterations was applied to determine the behavior of the solution performance. It was determined that, overall, the configuration using low-order preconditioning with ILU(1), BILU(1), or BILU(2), mesh sequencing, and inner linear solve iterations capped at the same value of the dimension n, used with the GMRES(n) iterative solver( i.e., no restarts), performed best for time, memory, and robustness considerations.

Parallel solution of a traffic flow simulation problem

Computational Fluid Dynamics (CFD) methods for solving traffic flow continuum models have been studied and efficiently implemented in traffic simulation codes in the past. This is the first time that such methods are studied from the point of view of parallel computing. We studied and implemented an implicit numerical method for solving the high-order flow conservation traffic model on parallel computers. Implicit methods allow much larger time-step than explicit methods, for the same accuracy. However, at each time-step a nonlinear system must be solved. We used the Newton method coupled with a linear iterative method ( Orthomin). We accelerated the convergence of Orthomin with parallel incomplete LU factorization preconditionings. We ran simulation tests with real traffic data from an 12-mile freeway section (in Minnesota) on the nCUBE2 parallel computer. These tests gave the same accuracy as past tests, which were performed on one-processor computers, and the overall execution time was significantly reduced.

圧縮性流れの陰的非構造解法における前処理付反復法の比較

Preconditioned iterative methods in an implicit unstructured grid solver of compressible flows are compared. The three-dimensional Euler and Navier-Stokes equations are discretized by the edge-based finite volume method. The artificial dissipation scheme for the inviscid numerical fluxes is developed for the unstructured grid from Jameson's scheme for the structured grid. The following methods are compared to the supersonic and transonic inviscid flows and subsonic viscous laminar and turbulent flows : the central difference methods with the scalar and matrix dissipation scheme and Roe's upwind method with the MUSCL sheme for the inviscid fluxes ; the scalar and matrix dissipation method for the left-hand side operator in the implicit scheme ; the ILU, DILU and SGS factorizations for the preconditioner ; and GMRES and Bi-CGSTAB methods for the linear system solver. The results show that a combination of the matrix dissipation in the inviscid fluxes and the scalar dissipation in the left-hand side works better than the other shemes. In the preconditioned iterative methods, a combination of DILU or SGS and the Bi-CGSTAB is recommended for the aspect of required memory.

A high-accuracy discretization scheme for numerical device simulations

To avoid non-physical oscillations resulting from discretization schemes, the finite difference scheme developed by Scharfetter and Gummel (1969) (the SG scheme) is widely used in numerical device simulations. However, this scheme is equivalent to the optimal first-order upwind scheme, so the crosswind diffusion is inherent in the SG scheme for multidimensional problems. To reduce its influence on the results of numerical device simulations, a new discretization approach with higher accuracy than the SG scheme for the convection term included in semiconductor device models is presented. To compare their results, the present scheme is interpolated into MINIMOS 4.0, which is a popular software in numerical MOSFET simulations. The discretization equations are solved by the conjugate gradient scheme for non-symmetrical linear equations with the preconditioning technique of the incomplete LU factorization (ILUCGS scheme). The numerical results show that the convection term must be discretized by those schemes with high accuracy, or the results may not represent the realistic solutions of the physical model.

Finite Difference Preconditioning for Solving Orthogonal Collocation Equations for Boundary Value Problems

A technique to construct a low-order finite difference preconditioner for solving orthogonal collocation equations for boundary value problems is presented. It is shown numerically and theoretically that the spectral condition numbers of the preconditioned collocation matrices are bounded by constants independent of the number of mesh nodes when certain exact low-order finite difference preconditionings are used. Preconditioners based on incomplete LU factorization are also discussed. Numerical experiments show the efficiency and robustness of the preconditioning.

ILUM: A Multi-Elimination ILU Preconditioner for General Sparse Matrices

Standard preconditioning techniques based on incomplete LU (ILU) factorizations offer a limited degree of parallelism, in general. A few of the alternatives advocated so far consist of either using some form of polynomial preconditioning or applying the usual ILU factorization to a matrix obtained from a multicolor ordering. In this paper we present an incomplete factorization technique based on independent set orderings and multicoloring. We note that in order to improve robustness, it is necessary to allow the preconditioner to have an arbitrarily high accuracy, as is done with ILUs based on threshold techniques. The ILUM factorization described in this paper is in this category. It can be viewed as a multifrontal version of a Gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism. The, emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on unstructured meshes.

A NON-LINEAR ADAPTED TRI-TREE MULTIGRID SOLVER FOR FINITE ELEMENT FORMULATION OF THE NAVIER-STOKES EQUATIONS

An iterative adaptive equation multigrid solver for solving the implicit Navier–Stokes equations simultaneously with tri-tree grid generation is developed. The tri-tree grid generator builds a hierarchical grid structur e which is mapped to a finite element grid at each hierarchical level. For each hierarchical finite element multigrid the Navier–Stokes equations are solved approximately. The solution at each level is projected onto the next finer grid and used as a start vector for the iterative equation solver at the finer level. When the finest grid is reached, the equation solver is iterated until a tolerated solution is reached. The iterative multigrid equation solver is preconditioned by incomplete LU factorization with coupled node fill-in. The non-linear Navier–Stokes equations are linearized by both the Newton method and grid adaption. The efficiency and behaviour of the present adaptive method are compared with those of the previously developed iterative equation solver which is preconditioned by incomplete LU factorization with coupled node fill-in.

AN ADAPTIVE UNSTRUCTURED TRI-TREE ITERATIVE SOLVER FOR MIXED FINITE ELEMENT FORMULATION OF THE STOKES EQUATIONS

An iterative adaptive equation solver for solving the implicit Stokes equations simultaneously with tri-tree grid generation is developed. The tri-tree grid generator builds a hierarchical grid structure which is mapped to a finite element grid at each hierarchical level. For each hierarchical finite element grid the Stokes equations are solved. The approximate solution at each level is projected onto the next finer grid and used as a start vector for the iterative equation solver at the finer level. When the finest grid is reached, the equation solver is iterated until a tolerated solution is reached. In order to reduce the overall work, the element matrices are integrated analytically beforehand. The efficiency and behaviour of the present adaptive method are compared with those of the previously developed iterative equation solver which is preconditioned by incomplete LU factorization with coupled node fill-in. The efficiency of the incomplete coupled node fill-in preconditioner is shown to be largely dependent on the global node numbering. The preconditioner is therefore tested for the natural node ordering of the tri-tree grid generator and for different ways of sorting the nodes.

A simple ILU preconditioning coupled to Orthomin method

Conjugate gradient type methods with preconditionings based on incompleteLU factorization (ILU) are known to be very useful for solving nonsymmetric linear systems of equations. In particular, the effectiveness of such iterative methods has been demonstrated on sparse linear systems which arise from discretizing non-self adjoint elliptic problems. In this paper, we present a simplifiedILU preconditioning coupled to a variational iterative method (Orthomin) on a model problem. It is shown that this simplified version ofILU is easy to implement and when coupled to Orthomin method, is as competitive or better than the standardILU with Orthomin method.

Efficient Iterative Methods Applied to the Solution of Transonic Flows

We investigate the use of an inexact Newton's method to solve the potential equations in the transonic regime. As a test case, we solve the two-dimensional steady transonic small disturbance equation. Approximate factorization/ADI techniques have traditionally been employed for implicit solutions of this nonlinear equation. Instead, we apply Newton's method using an exact analytical determination of the Jacobian with preconditioned conjugate gradient-like iterative solvers for solution of the linear systems in each Newton iteration. Two iterative solvers are tested; a block s-step version of the classical Orthomin(k) algorithm called orthogonal s-step Orthomin (OSOmin) and the well-known GMRES method. The preconditioner is a vectorizable and parallelizable version of incomplete LU (ILU) factorization. Efficiency of the Newton–Iterative method on vector and parallel computer architectures is the main issue addressed. In vectorized tests on a single processor of the Cray C-90, the performance of Newton–OSOmin is superior to Newton–GMRES and a more traditional monotone AF/ADI method (MAF) for a variety of transonic Mach numbers and mesh sizes. Newton–GMRES is superior to MAF for some cases. The parallel performance of the Newton method is also found to be very good on multiple processors of the Cray C-90 and on the massively parallel thinking machine CM-5, where very fast execution rates (up to 9 Gflops) are found for large problems.

ITERATIVE METHODS IN SOLVING NAVIER-STOKES EQUATIONS BY THE BOUNDARY ELEMENT METHOD

The solution of Navier–Stokes equations of time-dependent incompressible viscous fluid flow in planar geometry by the Boundary Domain Integral Method (BDIM) is discussed. The introduction of a subdomain technique to fluid flow problems is considered and improved in order to maintain the stability of BDIM. To avoid problems with flow kinematics computation in the sudomain mesh, a segmentation technique is proposed which combines the original BDIM with its subdomain variant and preserves its numerical stability. In order to reduce the computational cost of BDIM, which greatly depends on the solution of systems of linear equations, iterative methods are used. Conjugate gradient methods, conjugate gradients squared and an improved version of the biconjugate gradient method BiCGSTAB, together with the generalized minimal residual method, are used as iterative solvers. Different types of preconditioning, from simple Jacobi to incomplete LU factorization, are carried out and the performance of chosen iterative methods and preconditioners are reported. Test examples include backward facing step flow and flow through tubular heat exchangers. Test computation results show that BDIM is an accurate approximation technique which, together with the subdomain technique and powerful iterative solvers, can exhibit some significant savings in storage and CPU time requirements.

Robust linear and nonlinear strategies for solution of the transonic Euler equations

Full Newton nonlinear iteration using a Jacobian based on a high order discretization is compared with a widely used technique which uses a first order Jacobian. The linearized equations are solved using a level based block incomplete LU factorization with CGSTAB acceleration. Numerical tests are carried out using various standard airfoil shapes, at transonic and supersonic conditions. The full Newton, high order Jacobian technique is more robust and efficient than the usual approach. The Jacobian matrices can be readily solved using at least a level (2) block incomplete factorization preconditioner.

Levelized incomplete LU factorization and its application to large-scale circuit simulation

In the simulation of large circuits, the CPU time for solving the resulting linear equations may exceed the time required for evaluating the circuit elements. The circuit size above which this occurs depends on the applied transistor model and is roughly 10/sup 4/ devices for a vectorizing table model. To further speed up large-scale circuit simulation, one therefore has to focus on the solution algorithm. In this paper the excessive propagation of fill-in elements during sparse matrix factorization is identified as the major source of the superlinear increase of solution time. The idea of truncating the fill-in propagation in a variable manner forms the basis for the construction of a hierarchical solver with the same robustness as Newton's method but much less effort for large circuits. The method was applied to MOS circuits with up to 63000 transistors and in all cases the predominance of the solution part was broken. The new algorithm can be used efficiently both on sequential and vector architectures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

ILU factorization well suited to the vector processor using a variant of the 5-point difference scheme

The implementation on the vector processor of Incomplete LU factorization in the Preconditioned Conjugate Gradient method and other Preconditioned CG-like methods for the numerical solutions of the partial differential equations is discussed. For sufficient vectorization, a variant of the standard 5-point difference scheme for 2-dimensional problems is introduced. The convergence property of ILU factorization for this variant is examined in comparison with that vectorized with the usual hyperplane ordering method for the standard 5-point difference scheme on the vector processor. The efficiency and effectiveness of this factorization for the variant of the difference scheme is demonstrated through several computational experiments.

On the stability of the incomplete LU-factorizations and characterizations of H-matrices

Meijerink and van der Vorst [8] have shown that the incomplete LU-factorizations are numerically stable for M-matrices. Varga, Saff and Mehrmann [16] gave some characterizations of the H-matrices by using the incomplete LU-factorizations of them. The purpose of this paper is to show that the incomplete LU-factorizations of an H-matrix are at least as stable as the complete LU-factorizations of its comparison matrix. We give also some new characterizations of the H-matrices in connection with their incomplete LU-factorizations.

A framework for block ILU factorizations using block-size reduction

We propose a block ILU factorization technique for block tridiagonal matrices that need not necessarily be M-matrices. The technique explores reduction by a coarse-vector restriction of the block size of the approximate Schur complements computed throughout the factorization process. Then on the basis of the Sherman-Morrison-Woodbury formula these are easily inverted. We prove the existence of the proposed factorization techniques in the case of (nonsymmetric, in general) M-matrices. For block tridiagonal matrices with positive definite symmetric part we show the existence of a limit version of the factorization (exact inverses of the reduced matrices are needed). The theory is illustrated with numerical tests.

A Framework for Block ILU Factorizations Using Block-Size Reduction

We propose a block ILU factorization technique for block tridiagonal matrices that need not necessarily be M-matrices. The technique explores reduction by a coarse-vector restriction of the block size of the approximate Schur complements computed throughout the factorization process. Then on the basis of the Sherman-Morrison-Woodbury formula these are easily inverted. We prove the existence of the proposed factorization techniques in the case of (nonsymmetric, in general) M-matrices. For block tridiagonal matrices with positive definite symmetric part we show the existence of a limit version of the factorization (exact inverses of the reduced matrices are needed). The theory is illustrated with numerical tests.