Incomplete LU Factorization Preconditioner Research Articles

An Asynchronous Incomplete Block LU Preconditioner for Computational Fluid Dynamics on Unstructured Grids

We present a study of the effectiveness of asynchronous incomplete LU factorization preconditioners for the time-implicit solution of compressible flow problems while exploiting thread-parallelism within a compute node. A block variant of the asynchronous fine-grained parallel preconditioner adapted to a finite volume discretization of the compressible Navier--Stokes equations on unstructured grids is presented, and convergence theory is extended to the new variant. Experimental (numerical) results on the performance of these preconditioners on inviscid and viscous laminar two-dimensional steady-state test cases are reported. It is found, for these compressible flow problems, that the block variant performs much better in terms of convergence, parallel scalability, and reliability than the original scalar asynchronous ILU preconditioner. For viscous flow, it is found that the ordering of unknowns may determine the success or failure of asynchronous block-ILU preconditioning, and an ordering of grid cells suitable for solving viscous problems is presented.

Revisiting the BE SBS algorithm and applying it to solve torsion problems in composite bars: Robustness and efficiency study

The construction of truly competitive boundary-element (BE) codes, capable of analyzing real-life engineering problems, unavoidably requires the devising of coupling strategies to make the modeling of complex heterogeneous domains user-friendly. Thereby, high-performance algorithms for solving the highly sparse resulting system of equations are essential. Moreover, as discontinuous boundary elements are necessary to alleviate the modeling process of coupled domains, efficient (low-order) quadratures for integrating singular and nearly-singular fundamental kernels over the boundary elements must be implemented. This paper newly discusses the construction of the boundary-element subregion-by-subregion (BE SBS) technique based on a BEM formulation for torsion problems in general composite bars. One also presents details of the formulation of the Krylov solvers BiCG and BiCGSTAB-(l), embedded in the coupling algorithm. In addition, the BE SBS matrix structure itself is used to form an efficient sparse incomplete LU factorization (SILU) preconditioner to accelerate the iterative solution process. Torsion problems in bars with complex composite patterns (e.g. with many different materials) are analyzed to attest the efficiency and robustness of the whole boundary-element technique.

Spectral properties of discrete models of multi-dimensional elliptic problems with mixed derivatives

The influence of the spectrum of original and preconditioned matrices on a convergence rate of iterative methods for solving systems of finite-difference equations applicable to two-dimensional elliptic equations with mixed derivatives is investigated. It is shown that the efficiency of the bi-conjugate gradient iterative methods for systems with asymmetric matrices significantly depends not only on the matrix spectrum boundaries, but also on the heterogeneity of the distribution of the spectrum components, as well as on the magnitude of the imaginary part of complex eigenvalues. For test matrices with a fixed condition number, three variants of the spectral distribution were studied and the dependences of the number of iterations on the dimension of matrices were estimated. It is shown that the non-uniformity in the eigenvalue distribution within the fixed spectrum boundaries leads to a significant increase in the number of iterations with increasing dimension of the matrices. The increasing imaginary part of the eigenvalues has a similar effect on the convergence rate. Using as an example the model potential distribution problem in a square domain, including anisotropic ring inhomogeneity, a comparative analysis of the matrix structure and the convergence rate of the bi-conjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners is performed. It is shown that the advantages of the Fourier – Jacobi preconditioner are associated with a more uniform distribution of the spectrum of the preconditioned matrix along the real axis and a better suppression of the imaginary part of the spectrum compared to the preconditioner based on the incomplete LU factorization.

Iterative realization of finite difference schemes in the fictitious domain method for elliptic problems with mixed derivatives

Development of efficient finite difference schemes and iterative methods for solving anisotropic diffusion problems in an arbitrary geometry domain is considered. To simplify the formulation of the Neumann boundary conditions, the method of fictitious domains is used. On the example of a two-dimensional model problem of potential distribution in an isolated anisotropic ring conductor a comparative efficiency analysis of some promising finite-difference schemes and iterative methods in terms of their compatibility with the fictitious domain method is carried out. On the basis of numerical experiments empirical estimates of the asymptotic dependence of the convergence rate of the biconjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners on the step size and the value of the small parameter determining the continuation of the conductivity coefficient in the fictitious domain method are obtained. It is shown, that for one of the considered schemes the Fourier – Jacobi preconditioner is spectrally optimal and allows to eliminate the asymptotical dependence of the iterations number to achieve a given accuracy both on the value of the step size and the value of the small parameter in the fictitious domain method.

Performance comparison of various parallel incomplete LU factorization preconditioners for domain decomposition method

A finite element code is parallelized by vertex-oriented domain decomposition method which utilizes one- or multi-dimensional partitioning in structured mesh and METIS Library in unstructured mesh. For obtaining the domain-decomposed solution, iterative solvers like conjugate gradient method are used. To accelerate the convergence of iterative solvers, parallel incomplete LU factorization preconditioners are employed, and their performances are compared. For the communication between processors, Message Passing Interface Library is used. The speedups of parallel preconditioned iterative solvers are estimated through computing 2- and 3-dimensional Laplace equations. The effects of mesh and partitioning method on the speedup of parallel preconditioners are also examined.

An investigation on HOBEM in evaluating ship wave of high speed displacement ship

A practical numerical tool is developed to evaluate ship waves of high speed displacement ships on the basis of potential flow theory, in which high order boundary element method (HOBEM) based on biquadratic shape functions is applied to solve the boundary value problem. Since the sinkage and trim of ship at high speeds are notable, influences of ship attitude on wave drag are investigated and three kinds of models are used to evaluate them. To make the numerical approach highly efficient, an incomplete LU factorization preconditioner is adopted and incorporated with the restarted generalized minimal residual method GMRES (m) to solve the boundary integral equation. A corresponding Fortran code is developed and applied to evaluate ship waves of the Wigley hull and 4a model, a transom stern ship. Computations are performed for both monohulls and catamarans over a wide range of Froude numbers (Fr = 0.10 - 1.00) . Numerical issues including mesh convergence and computational efficiency are investigated at first. Computed results of the wave drag, sinkage and trim show generally good agreement with experimental data. Reasonable wave patterns are obtained and physical phenomena that wake angle φmax, where the largest waves occur, would become narrow at high speeds is also captured by the present computations. Numerical results indicate the proposed method would be accurate and efficient to evaluate resistance for hull design of high speed displacement ship.

Efficient CPR-type preconditioner and its adaptive strategies for large-scale parallel reservoir simulations

With the rapid development of high performance parallel computers and the increasing demands of large-scale reservoir simulation, the development of effective and fast solvers and preconditioners for parallel reservoir simulation has become increasingly important. The incomplete LU factorization (ILU) preconditioners are the most commonly used preconditioners in reservoir simulations due to their low computational cost. They have been implemented in commercial simulators as the default preconditioners. However, for reservoirs with highly heterogeneous permeability, the CPR (constrained pressure residual)-type preconditioners are more effective than the ILU preconditioners, especially when parallel computing is employed. The CPR-type preconditioners consist of multi-stages of preconditioning processes, which lead to huge computational costs. Employing the algebraic multigrid (AMG) method to solve a pressure system and the ILU method to solve the whole system has two indispensable stages. For parallel computation, due to the serial nature of the ILU method, the ILU method is usually combined with the restricted additive Schwarz (RAS) method and used as a subdomain solver. Different settings of the AMG and RAS methods, such as a grid coarsening strategy, interpolation and smoothing in the AMG method, and a subdomain solver and an overlap level in the RAS method, as well as different CPR-type preconditioners, yield significantly different performance. This paper first gives an overview of the CPR-type preconditioners and the AMG and RAS methods. Then a detailed comparison between different CPR-type preconditioners with different settings of the AMG and RAS methods is presented. Based on the comparison, the efficient settings of the AMG and RAS methods and the most efficient CPR-type preconditioner are given. Furthermore, an adaptive preconditioning strategy is designed to save the computational time. The adaptive preconditioning strategy introduces an indicator to measure the difficulty level of solving a linear system. According to difficulties, a RAS–ILU preconditioner or a CPR-type preconditioner is automatically and dynamically selected to achieve both efficiency and effectiveness.

Incomplete LU Preconditioner Based on Max-Plus Approximation of LU Factorization

We present a new method for the a priori approximation of the orders of magnitude of the entries in the LU factors of a complex or real matrix $A$. This approximation is used to determine the positions of the largest entries in the LU factors of $A$, and these positions are used as the sparsity pattern for an incomplete LU factorization preconditioner. Our method uses max-plus algebra and is based solely on the moduli of the entries of $A$. We also present techniques for predicting which permutation matrices will be chosen by Gaussian elimination with partial pivoting. We exploit the strong connection between the field of Puiseux series and the max-plus semiring to prove properties of the max-plus LU factors. Experiments with a set of test matrices from the University of Florida Sparse Matrix Collection show that our max-plus LU preconditioners outperform traditional level of fill methods and have similar performance to those preconditioners computed with more expensive threshold-based methods.

A multi-continuum multi-phase parallel simulator for large-scale conventional and unconventional reservoirs

Reservoir simulation is considered as the core of data analysis during the development of an oilfield. With higher requirements from reservoir engineers, high-resolution geological model problems are commonly used in reservoir simulation. To simulate this kind of large-scale reservoir problems, based on black-oil models and compositional model, a parallel simulator is developed in this paper. Cartesian and corner point grids are supported by our simulator to describe complex geology, including faults and pinch-outs. The multi-continuum models (including a dual-porosity model, a dual-porosity dual-permeability model, and a multiple interacting continua model) are used to simulate fractured reservoirs. Geomechanics effects are successfully involved through a flexible and convenient interface provided by our simulator. Krylov subspace linear solvers, restricted additive Schwarz method, incomplete LU factorization (ILU) preconditioner, and a family of CPR (constrained pressure residual)-type preconditioners are implemented to provide flexibility of the linear system solution processes. Moreover, a new adaptive preconditioning strategy, which can automatically select and change preconditioner between a CPR-type preconditioner and an ILU preconditioner, is designed to accelerate the solution processes and is considered as the most efficient preconditioning technique to our knowledge. MPI-based parallel implementation is employed for each module of the simulator. With the simulator we developed, various conventional and unconventional reservoir simulation problems with large-scale geological models can be achieved on parallel computers within practical computational time. The computational efficiency and parallel scalability are achieved by our high-quality parallel implementation and efficient nonlinear and linear solution techniques.

P-FFT Accelerated EFIE with a Novel Diagonal Perturbed ILUT Preconditioner for Electromagnetic Scattering by Conducting Objects in Half Space

A highly efficient and robust scheme is proposed for analyzing electromagnetic scattering from electrically large arbitrary shaped conductors in a half space. This scheme is based on the electric field integral equation (EFIE) with a half-space Green’s function. The precorrected fast Fourier transform (p-FFT) is first extended to a half space for general three-dimensional scattering problems. A novel enhanced dual threshold incomplete LU factorization (ILUT) is then constructed as an effective preconditioner to improve the convergence of the half-space EFIE. Inspired by the idea of the improved electric field integral operator (IEFIO), the geometrical-optics current/principle value term of the magnetic field integral equation is used as a physical perturbation to stabilize the traditional ILUT perconditioning matrix. The high accuracy of EFIE is maintained, yet good calculating efficiency comparable to the combined field integral equation (CFIE) can be achieved. Furthermore, this approach can be applied to arbitrary geometrical structures including open surfaces and requires no extra types of Sommerfeld integrals needed in the half-space CFIE. Numerical examples are presented to demonstrate the high performance of the proposed solver among several other approaches in typical half-space problems.

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.

Effective block diagonal preconditioners for Biot's consolidation equations in piled‐raft foundations

SUMMARYThe finite element (FE) simulation of large‐scale soil–structure interaction problems (e.g. piled‐raft, tunnelling, and excavation) typically involves structural and geomaterials with significant differences in stiffness and permeability. The symmetric quasi‐minimal residual solver coupled with recently developed generalized Jacobi, modified symmetric successive over‐relaxation (MSSOR), or standard incomplete LU factorization (ILU) preconditioners can be ineffective for this class of problems. Inexact block diagonal preconditioners that are inexpensive approximations of the theoretical form are systematically evaluated for mitigating the coupled adverse effects because of such heterogeneous material properties (stiffness and permeability) and because of the percentage of the structural component in the system in piled‐raft foundations. Such mitigation led the proposed preconditioners to offer a significant saving in runtime (up to more than 10 times faster) in comparison with generalized Jacobi, modified symmetric successive over‐relaxation, and ILU preconditioners in simulating piled‐raft foundations. Copyright © 2012 John Wiley & Sons, Ltd.

Robust incomplete factorization for nonsymmetric matrices

In this paper, a new incomplete LU factorization preconditioner for nonsymmetric matrices is being considered which is also breakdown-free (no zero pivots occurs) for positive definite matrices. To construct this preconditioner, only the information of matrix A is used and just one of the factors of the AINV process is computed. The L factor is extracted as a by-product of the AINV process. The pivots of the AINV process are used as diagonal entries of U. The new preconditioner has left and right-looking versions. To improve the efficiency of the preconditioner, we have used the inverse-based dropping strategies for both L and U factors. Numerical experiments show that the left-looking version of the preconditioner is significantly faster than its right-looking version in terms of preconditioning time and both are equally effective to reduce the number of iterations. Comparisons of the new preconditioner with AINV and ILUT preconditioners are also presented.

Symmetric Permutations for I-matrices to Delay and Avoid Small Pivots During Factorization

In this article, we present several new permutations for I-matrices making these more suitable for incomplete LU-factorization preconditioners used in solving linear systems by iterative methods. A general matrix can be transformed by row permutation as well as row and columns scaling into an I-matrix, i.e., a matrix having elements of modulus 1 on the diagonal and elements of modulus of no more than 1 elsewhere. Reordering rows and columns by the same permutation clearly preserves I-matrices. In this article, we consider such reordering techniques which make the permuted matrix more suitable for an incomplete LU-factorization preconditioner than the original I-matrix. We use a multilevel ILUC, an incomplete LU-factorization preconditioner using Crout's implementation of Gaussian elimination without pivoting to test these reorderings. The combination of I-matrix preprocessing with the various algorithms presented here and the multilevel incomplete LU-factorizations forms a powerful preconditioning method for unsymmetric, highly indefinite problems. The C++ code has been made available in the software package ILU++.

THE EFFICIENT ANALYSIS OF ELECTROMAGNETIC SCATTERING FROM COMPOSITE STRUCTURES USING HYBRID CFIE-IEFIE

The efficient algorithm is presented for the analysis of electromagnetic scattering from composite structures with coexisting open and closed conductors. A hybrid combined-field integral equation-the improved electric-field integral equation (CFIE-IEFIE) formulation with the incomplete LU factorization (ILU) preconditioner is proposed. Numerical results are given to demonstrate that the efficiency of our algorithm can be significantly improved when compared with the conventional EFIE formulation and the hybrid CFIE-EFIE formulation.

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 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.

Parallel methods for optimality criteria-based topology optimization

Topology optimization problems require the repeated solution of finite element problems that are often extremely ill-conditioned due to highly heterogeneous material distributions. This makes the use of iterative linear solvers inefficient unless appropriate preconditioning is used. Even then, the solution time for topology optimization problems is typically very high. These problems are addressed by considering the use of non-overlapping domain decomposition-based parallel methods for the solution of topology optimization problems. The parallel algorithms presented here are based on the solid isotropic material with penalization (SIMP) formulation of the topology optimization problem and use the optimality criteria method for iterative optimization. We consider three parallel linear solvers to solve the equilibrium problem at each step of the iterative optimization procedure. These include two preconditioned conjugate gradient (PCG) methods: one using a diagonal preconditioner and one using an incomplete LU factorization preconditioner with a drop tolerance. A third substructuring solver that employs a hybrid of direct and iterative (PCG) techniques is also studied. This solver is found to be the most effective of the three solvers studied, both in terms of parallel efficiency and in terms of its ability to mitigate the effects of ill-conditioning. In addition to examining parallel linear solvers, we consider the parallelization of the iterative optimality criteria method. To tackle checkerboarding and mesh dependence, we propose a multi-pass filtering technique that limits the number of “ghost” elements that need to be exchanged across interprocessor boundaries.

Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems

We consider the preconditioned iterative solution of large dense linear systems, where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering. For some scattering structures this matrix can be poorly conditioned. The main purpose of this study is to evaluate the efficiency of a class of incomplete LU (ILU) factorization preconditioners for solving this type of matrices. We solve the electromagnetic wave equations using the BiCG method with an ILU preconditioner in the context of a multilevel fast multipole algorithm (MLFMA). The novelty of this work is that the ILU preconditioner is constructed using the near part block diagonal submatrices generated from the MLFMA. Experimental results show that the ILU preconditioner reduces the number of BiCG iterations substantially, compared to the block diagonal preconditioner. The preconditioned iteration scheme also maintains the computational complexity of the MLFMA, and consequently reduces the total CPU time.

Parallel performance of block ILU preconditioners for a block-tridiagonal matrix

The parallelizable block ILU (incomplete LU) factorization preconditioners for a block-tridiagonal matrix have been recently proposed by the author. In this paper, we describe a parallelization of Krylov subspace methods with the block ILU factorization preconditioners on distributed-memory computers such as the Cray T3E, and then parallel performance results of a preconditioned Krylov subspace method are provided to evaluate the effectiveness and efficiency of the block ILU preconditioners on the Cray T3E.