Incomplete Factorization Methods Research Articles

A high performance level-block approximate LU factorization preconditioner algorithm

Many application problems that lead to solving linear systems make use of preconditioned Krylov subspace solvers to compute their solution. Among the most popular preconditioning approaches are incomplete factorization methods either as single-level approaches or within a multilevel framework. We will present a block incomplete factorization that is based on skillfully blocking the system initially and throughout the factorization. Our objective is to develop algebraic block preconditioners for the efficient solution of such systems by Krylov subspace methods. We will demonstrate how our level-block approximate algorithm outperforms a single level-block scalar method often by orders of magnitude on modern architectures, thus paving the way for its prospective use inside various multilevel incomplete factorization approaches or other applications where the core part relies on an efficient incomplete factorization algorithms.

Nested Alternating - Triangular Incomplete Factorization Methods

We consider several versions of incomplete nested factorization methods for solving the large systems of linear algebraic equations (SLAEs) with sparse matrices which arise in grid approximations of the multi-dimensional boundary value problems. Our approach is based on the two-level iterative process in the Krylov subspaces in 3D case. Corresponding hierarchical incomplete factorization is applied to the block tridiagonal matrix structure. At the upper level, the diagonal blocks correspond to 2D grid subproblems which are factorized in the line-by-line framework. Instead of the low and upper triangular matrix factors, the alternating triangular matrices are used, which allows to apply the parallel counter sweeping approaches. The improvement of preconditioners is made by means of generalized compensation principles. To solve SLAE iterative conjugate direction methods in Krylov subspaces are applied. The efficiency of the proposed methods are demonstrated on the set of representative test problems.

Performance Evaluation of Parallel Finite-Element Eddy-Current Analysis Using Direct Method as Subdomain Solver

An eddy-current analysis (A-φ method) based on a domain decomposition method is proposed. A conjugate orthogonal conjugate gradient with an incomplete Cholesky factorization method is generally applied to solve the subdomain problems because the matrices become a singular system. However, it is difficult to achieve high-accuracy solutions using iterative methods because the solutions contain truncation errors. Therefore, the authors propose a method of improving the convergence by applying a direct method based on a singular value decomposition as the subdomain solver. As a result, it is thus confirmed that the improvement of convergence can be realized with the direct method. This article describes the proposed method and the performance evaluation with some examinations.

On a Parallel Version of a Second-Order Incomplete Triangular Factorization Method

We consider a parallel version of the stabilized second-order incomplete triangular factorization–conjugate gradient method in which the reordering of the coefficient matrix that corresponds to the ordering based on splitting into subdomains with separators is used. The incomplete triangular factorization is constructed using the truncation of the fill-in chosen “by value” at the internal nodes of the subdomains and “by value” and ‘by positions” on the separators. The reliability of the method under consideration is theoretically proved. The reliability and convergence rate of the parallel method are numerically analyzed. The developed algorithms are implemented using a Message Passing Interface (MPI); the computational results are presented for benchmark problems with matrices from the collection of the University of Florida.

A scalable solution strategy for high-order stabilized finite-element solvers using an implicit line preconditioner

This paper presents a robust, efficient, and strongly scalable solution methodology for simulation of complex turbulent flows on unstructured grids. The compressible Reynolds averaged Navier–Stokes (RANS) equations and the negative Spalart–Allmaras (SA) turbulence model are discretized, in coupled form, using a Streamline Upwind Petrov–Galerkin (SUPG) scheme. The time integration is fully implicit, and the discretized equations are advanced towards a steady-state solution using a pseudo-transient continuation (PTC). For solution of the linearized systems, a preconditioned Krylov solver is used. Seeking robustness, Krylov solvers are commonly preconditioned using incomplete factorization methods such as ILU(k). However, these methods are neither memory efficient, nor strongly scalable. To provide a better alternative, the implicit line solution method, which has been traditionally used in finite-volume methods, is revised and enhanced to solve stiffer linear systems. In the developed method, the lines are generated using a matrix-based approach, which connects the strongly-coupled unknowns. In addition, to improve the robustness of the line solver for high-CFL systems, a dual-CFL strategy, with a lower CFL number in the preconditioner matrix, is developed. Also, it is shown that for high-order continuous finite-element discretizations, the interconnections of the degrees of freedom on a line form a banded matrix which is wider than tridiagonal, but still can be factorized completely without generating any fill-ins. The developed line preconditioner is strongly scalable and, in contrast to the ILU factorization, its convergence behavior does not depend on the number of partitions. Two three-dimensional numerical examples are presented in which the performance of the line preconditioner is compared with that of the ILU(k) preconditioner. This comparison shows that, in addition to robustness improvements, the line preconditioner offers significant benefits in terms of memory efficiency.

A study of direct and Krylov iterative sparse solver techniques to approach linear scaling of the integration of chemical kinetics with detailed combustion mechanisms

The integration of the stiff ODE systems associated with chemical kinetics is the most computationally demanding task in most practical combustion simulations. The introduction of detailed reaction mechanisms in multi-dimensional simulations is limited by unfavorable scaling of the stiff ODE solution methods with the mechanism’s size. In this paper, we compare the efficiency and the appropriateness of direct and Krylov subspace sparse iterative solvers to speed-up the integration of combustion chemistry ODEs, with focus on their incorporation into multi-dimensional CFD codes through operator splitting. A suitable preconditioner formulation was addressed by using a general-purpose incomplete LU factorization method for the chemistry Jacobians, and optimizing its parameters using ignition delay simulations for practical fuels. All the calculations were run using a same efficient framework: SpeedCHEM, a recently developed library for gas-mixture kinetics that incorporates a sparse analytical approach for the ODE system functions. The solution was integrated through direct and Krylov subspace iteration implementations with different backward differentiation formula integrators for stiff ODE systems: LSODE, VODE, DASSL. Both ignition delay calculations, involving reaction mechanisms that ranged from 29 to 7171 species, and multi-dimensional internal combustion engine simulations with the KIVA code were used as test cases. All solvers showed similar robustness, and no integration failures were observed when using ILUT-preconditioned Krylov enabled integrators. We found that both solver approaches, coupled with efficient function evaluation numerics, were capable of scaling computational time requirements approximately linearly with the number of species. This allows up to three orders of magnitude speed-ups in comparison with the traditional dense solution approach. The direct solvers outperformed Krylov subspace solvers at mechanism sizes smaller than about 1000 species, while the Krylov approach allowed more than 40% speed-up over the direct solver when using the largest reaction mechanism with 7171 species.

Сверхнеявная схема неполной факторизации для решения трехмерных разностных уравнений эллиптического типа

Сверхнеявная схема неполной факторизации для решения трехмерных разностных уравнений эллиптического типа

A Supernodal Approach to Incomplete LU Factorization with Partial Pivoting

We present a new supernode-based incomplete LU factorization method to construct a preconditioner for solving sparse linear systems with iterative methods. The new algorithm is primarily based on the ILUTP approach by Saad, and we incorporate a number of techniques to improve the robustness and performance of the traditional ILUTP method. These include new dropping strategies that accommodate the use of supernodal structures in the factored matrix and an area-based fill control heuristic for the secondary dropping strategy. We present numerical experiments to demonstrate that our new method is competitive with the other ILU approaches and is well suited for modern architectures with memory hierarchy.

A CONSTRAINT PRECONDITIONER FOR SOLVING SYMMETRIC POSITIVE DEFINITE SYSTEMS AND APPLICATION TO THE HELMHOLTZ EQUATIONS AND POISSON EQUATIONS

In this paper, first, by using the diagonally compensated reduction and incomplete Cholesky factorization methods, we construct a constraint preconditioner for solving symmetric positive definite linear systems and then we apply the preconditioner to solve the Helmholtz equations and Poisson equations. Second, according to theoretical analysis, we prove that the preconditioned iteration method is convergent. Third, in numerical experiments, we plot the distribution of the spectrum of the preconditioned matrix M−1A and give the solution time and number of iterations comparing to the results of [5, 19].

Finite volume method analysis of heat transfer problem using adapted strongly implicit procedure

In most issues representing physical problems, the complex geometry cannot be represented by a Cartesian grid. The multi-block grid technique allows artificially reducing the complexity of the geometry by breaking down the real domain into a number of sub-domains with simpler geometry. The main aim of this article is to show the usefulness of simple solvers in complex geometry problems, when using curvilinear coordinates combined with multi-block grids. This requires adapted solvers to a nine nodes computational cell instead of the five nodes computational cell used with Cartesian coordinates for two-dimensional cases. These developments are presented for the simple iterative methods Jacobi and Gauss-Seidel and also for the incomplete factorization method strongly implicit procedure (SIP). These adapted solvers are tested in two cases: a simple geometry (heat transfer in a circular cross-section) and a complex geometry (solidification case). Results of the simple geometry case show that all the adapted solvers have good performance with a slight advantage for the SIP solver. For increasing the complexity of the geometry, the results showed that Jacobi and Gauss-Seidel solvers are not suitable. However, the SIP method has a reasonable performance. A conclusion could be drawn that the SIP method could be used in complex geometry problems using multi-block grid technique when high precision results are not required.

On the semi-conjugate direction methods with dynamic preconditioning

The semi-conjugate residual methods and semi-conjugate gradient methods with dynamic preconditioners in Krylov subspaces are considered for solving the systems of linear algebraic equations whose matrices are not symmetric. Their orthogonal and variational properties are under study. New algorithms are proposed for choosing the inner iteration parameters in the preconditioning matrices corresponding to incomplete factorization methods. The efficiency of the resulting iterative processes is demonstrated by a set of numerical experiments for finite difference diffusion-convection equations.

Numerical study of a double preconditioning strategy

PurposeThe aim of this paper is to accelerate the convergence of iterative methods on ill‐conditioned linear systems of equations.Design/methodology/approachFirst a brief numerical analysis is given of left preconditioners on ill‐conditioned linear systems of equations. From this result, it is deduced that a double preconditioning approach may be better. Then, a double preconditioner based on an iterative diagonal scaling method and an incomplete factorization method is proposed. The efficiency of this approach is illustrated on two finite element models produced by computational electromagnetism.FindingsThe double preconditioning approach is efficient for 2D and 3D finite element problems. The bi‐conjugate gradient algorithm always converges when it is double preconditioned. This is not the case when a simple incomplete factorization method is applied. Furthermore, when the two preconditioning techniques lead to the convergence of the iterative solving method, the double preconditioner significantly reduces the number of iterations in comparison with the simple preconditioner. On the proposed 2D problem, the speed‐up is between 6 and 32. On the proposed 3D problem, the speed‐up is between 13 and 20. Finally, the approach seems to reduce the growth of the condition number when higher‐order finite elements are used.Research limitations/implicationsThe paper proposes a particular double preconditioning approach which can be applied to any invertible linear system of equations. A numerical evaluation on a singular linear system is also provided but no proof or analysis of stability is given for this case.Originality/valueThe paper presents a new preconditioning technique based on the combination of two very simple and elementary methods: a diagonal scaling method and an incomplete factorization process. Acceleration obtained from this approach is quite impressive.

Stochastic Preconditioning for Diagonally Dominant Matrices

This paper presents a new stochastic preconditioning approach for large sparse matrices. For the class of matrices that are rowwise and columnwise irreducibly diagonally dominant, we prove that an incomplete $\rm{LDL^T}$ factorization in a symmetric case or an incomplete LDU factorization in an asymmetric case can be obtained from random walks and used as a preconditioner. It is argued that our factor matrices have better quality, i.e., better accuracy-size trade-offs, than preconditioners produced by existing incomplete factorization methods. Therefore a resulting preconditioned Krylov-subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance. The advantage increases for larger and denser matrices. These claims are verified by numerical tests, and we provide techniques that can potentially extend the theory to non-diagonally-dominant matrices.

Incomplete factorization methods with semi-conjugate residuals

Iterative solving SLAE with nonsymmetric square real matrices by means of the robust modified method of generalized conjugate residuals and the hierarchical family of algorithms for incomplete matrix decomposition into triangular multipliers is considered. Special features of software algorithms based on symbolic factorization of matrices stored in a sparse row format are described. Experimental numerical results are given for a representative series of model problems. They demonstrate comparative efficiency of the methods.

A Parallel Multistage ILU Factorization Based on a Hierarchical Graph Decomposition

PHIDAL (parallel hierarchical interface decomposition algorithm) is a parallel incomplete factorization method which exploits a hierarchical interface decomposition of the adjacency graph of the coefficient matrix. The idea of the decomposition is similar to that of the well‐known wirebasket techniques used in domain decomposition. However, the method is devised for general, irregularly structured, sparse linear systems. This paper describes a few algorithms for obtaining good quality hierarchical graph decompositions and discusses the parallel implementation of the factorization procedure. Numerical experiments are reported to illustrate the scalability of the algorithm and its effectiveness as a general purpose parallel linear system solver.

Improving the forward solver for the complete electrode model in EIT using algebraic multigrid

Image reconstruction in electrical impedance tomography is an ill-posed nonlinear inverse problem. Linearization techniques are widely used and require the repeated solution of a linear forward problem. To account correctly for the presence of electrodes and contact impedances, the so-called complete electrode model is applied. Implementing a standard finite element method for this particular forward problem yields a linear system that is symmetric and positive definite and solvable via the conjugate gradient method. However, preconditioners are essential for efficient convergence. Preconditioners based on incomplete factorization methods are commonly used but their performance depends on user-tuned parameters. To avoid this deficiency, we apply black-box algebraic multigrid, using standard commercial and freely available software. The suggested solution scheme dramatically reduces the time cost of solving the forward problem. Numerical results are presented using an anatomically detailed model of the human head.

A Class of Incomplete Orthogonal Factorization Methods. II: Implementation and Results

We present, implement and test several incomplete QR factorization methods based on Givens rotations for sparse square and rectangular matrices. For square systems, the approximate QR factors are used as right-preconditioners for GMRES, and their performance is compared to standard ILU techniques. For rectangular matrices corresponding to linear least-squares problems, the approximate R factor is used as a right-preconditioner for CGLS. A comprehensive discussion is given about the uses, advantages and shortcomings of the preconditioners.

A multilevel block incomplete cholesky preconditioner for solving normal equations in linear least squares problems

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.

불완전분해법을 전처리로 하는 공액구배법의 안정화에 대한 연구

The preconditioned conjugate gradient method is an efficient iterative solution scheme for large size finite element problems. As preconditioning method, we choose an incomplete Cholesky factorization which has efficiency and easiness in implementation in this paper. The incomplete Cholesky factorization method sometimes leads to breakdown of the computational procedure that means pivots in the matrix become minus during factorization. So, it is inevitable that a reduction process for stabilizing and this process will guarantee robustness of the algorithm at the cost of a little computation. Recently incomplete factorization that enhances robustness through increasing diagonal dominancy instead of reduction process has been developed. This method has better efficiency for the problem that has rotational degree of freedom, but is sensitive to parameters and the breakdown can be occurred occasionally. Therefore, this paper presents new method that guarantees robustness for this method. Numerical experiment shows that the present method guarantees robustness without further efficiency loss.

Preconditioning Techniques for Large Linear Systems: A Survey

This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.