Block Incomplete Factorization Research Articles

Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method

The main goal of the study is the approximation of the solution to the Dirichlet boundary value problem (DBVP) of the heat equation on a rectangle by developing a new difference method on a grid system of hexagons. It is proved that the given special scheme is unconditionally stable and converges to the exact solution on the grids with fourth order accuracy in space variables and second order accuracy in time variable. Secondly, an incomplete block factorization is given for symmetric positive definite block tridiagonal (SPD-BT) matrices utilizing a conservative iterative method that approximates the inverse of the pivoting diagonal blocks by preserving the symmetric positive definite property. Subsequently, by using this factorization block hybrid preconditioning of the conjugate gradient (BHP-CG) method is applied to solve the obtained algebraic system of equations at each time level.

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.

Block subspace projection preconditioned conjugate gradient method in modal structural analysis

This paper considers the preconditioned block conjugate gradient method for solving the algebraic, generalized partial eigenvalue problem, which arises when the finite element method is applied to dynamic analysis in structural and solid mechanics. The focus is on the development of an algorithm that keeps constant the iterable subspace dimension, independently of the number of eigenpairs required. Novelties in this approach include a preconditioning strategy based on block incomplete factorization of the shifted stiffness matrix, and analysis of unremovable errors, which accumulate with increasing numbers of eigenpairs.

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

SummaryMonolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared with preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this article, several techniques are applied to monolithic two‐level generalized Dryja‐Smith‐Widlund (GDSW) preconditioners to further improve the convergence speed and the computing time. In particular, reduced dimension GDSW coarse spaces, restricted and scaled versions of the first level, hybrid, and parallel coupling of the levels, and recycling strategies are investigated. Using a combination of all these improvements, for a small time‐dependent Navier‐Stokes problem on 240 message passing interface (MPI) ranks, a reduction of 86% of the time‐to‐solution can be obtained. Even without applying recycling strategies, the time‐to‐solution can be reduced by more than 50% for a larger steady Stokes problem on 4608 MPI ranks. For the largest problems with 11 979 MPI ranks, the scalability deteriorates drastically for the monolithic GDSW coarse space. On the other hand, using the reduced dimension coarse spaces, good scalability up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, could be achieved.

Three-Dimensional Wide-Band Electromagnetic Forward Modelling Using Potential Technique

The efficacy of Krylov subspace solvers is strongly dependent on the preconditioner applied to solve the large sparse linear systems of equation for electromagnetic problems. In this study, we present a three-dimensional (3-D) plane wave electromagnetic forward simulation over a broadband frequency range. The Maxwell’s equation is solved in a secondary formulation of the Lorentz gauge coupled-potential technique. A finite-volume scheme is employed for discretizing the system of equations on a structured rectilinear mesh. We employed a block incomplete lower-upper factorization (ILU) preconditioner that is suitable for our potential formulation to enhance the computing time and convergence of the systems of equation by comparing with other preconditioners. Furthermore, we observe their effect on the iterative solvers such as the quasi-minimum residual and bi-conjugate gradient stabilizer. Several applications were used to validate and test the effectiveness of our method. Our scheme shows good agreement with the analytical solution. Notably, from the marine hydrocarbon and the crustal model, the utilisation of the bi-conjugate gradient stabilizer with block ILU preconditioner is the most appropriate. Thus, our approach can be incorporated to optimize the inversion process.

A parallel monolithic approach for fluid-structure interaction in a cerebral aneurysm

Abstract A parallel fully-coupled numerical algorithm has been developed for the fluid-structure interaction problem in a cerebral artery with aneurysm. For the fluid part of the problem, an Arbitrary Lagrangian-Eulerian formulation based on the side-centered unstructured finite volume method is employed for the governing incompressible Navier–Stokes equations. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant-Kirchhoff material and the classical Galerkin finite element method is used to discretise the governing equations in a Lagrangian frame. The time integration method for the structure domain is based on the Newmark type generalized − α method while the second-order backward difference (BDF2) is used in the fluid domain. A special attention is given to construct an algorithm obeying the local/global discrete geometric conservation laws (DGCL) in order to conserve fluid volume at machine precision when the fluid domain is entirely enclosed by solid domain boundary. Therefore, a compatible kinematic boundary condition is applied at the interface between the solid and fluid domains. The parallel implementation of the present fully coupled unstructured fluid-structure solver is based on the PETSc library and a one-level restricted additive Schwarz preconditioner with a block-incomplete factorization within each partitioned sub-domains is utilized for the resulting fully coupled system. The proposed algorithm is initially validated for a pressure pulse propagating in a flexible tube and the mass conservation accuracy is tested for a thin elastic sphere filled with an incompressible fluid in a circular tube. Then the numerical method is applied to a complicated problem involving unsteady pulsatile blood flow in a cerebral artery with aneurysm as a realistic fluid-structure interaction problem encountered in biomechanics. Various hemodynamic quantities of interest like fluid velocities, blood pressure and wall shear stresses (WSS) are computed as well as the time dependent artery wall deformations.

Enhancing Performance and Robustness of ILU Preconditioners by Blocking and Selective Transposition

Incomplete factorization is one of the most effective general-purpose preconditioning methods for Krylov subspace solvers for large sparse systems of linear equations. Two techniques for enhancing the robustness and performance of incomplete LU factorization for sparse unsymmetric systems are described. A block incomplete factorization algorithm based on the Crout variation of LU factorization is presented. The algorithm is suitable for incorporating threshold-based dropping as well as unrestricted partial pivoting, and it overcomes several limitations of existing incomplete LU factorization algorithms with and without blocking. It is shown that blocking has a three-pronged impact: it speeds up the computation of incomplete factors and the solution of the associated triangular systems, it permits denser and more robust factors to be computed economically, and it permits a trade-off with the restart parameter of GMRES to further improve the overall speed and robustness. A highly effective heuristic for imp...

Overlapping for preconditioners based on incomplete factorizations and nested arrow form

In this paper, we discuss the usage of overlapping techniques for improving the convergence of preconditioners based on incomplete factorizations. To enable parallelism, these preconditioners are usually applied after the input matrix is permuted into a nested arrow form using k-way nested dissection. This graph partitioning technique uses k-way partitionning by vertex separator to recursively partition the graph of the input matrix into k subgraphs using a subset of its vertices called a separator. The overlapping technique is then based on algebraically extending the associated subdomains of these subgraphs and their corresponding separators obtained from k-way nested dissection by their direct neighbours. A similar approach is known to accelerate the convergence of domain decomposition methods, where the input matrix is partitioned into a number of independent subdomains using k-way vertex partitioning of a graph by edge separators, a different graph decomposition technique. We discuss the effect of the overlapping technique on the convergence of two classes of preconditioners, on the basis of nested factorization and block incomplete LDU factorization.

An advancement in iterative solution schemes for three-dimensional, two-fluid modeling of two-phase flow in PWR fuel bundles

This paper outlines a fully three-dimensional two-fluid one-pressure model with a semi-implicit finite difference scheme coupled with heat conduction which can be applicable to thermal non-equilibrium two-phase flow field in subchannel geometry of Pressurized Water Reactors (PWR). The system of equations was linearized using the Newton–Raphson method and was collapsed into the pressure equations forming a system of the Poisson type. Then, two-phase flow modeling was combined with Krylov methods as advanced computing techniques to investigate the feasibility of implementing preconditioned Krylov subspace solvers as the numerical scheme to solve pressure equations. Six popular Krylov subspace solvers were considered: GMRES, FGMRES, DQGMRES, CGNR, BCG, and TFQMR combined with the block incomplete LU factorization with a dual truncation strategy (BILUT) preconditioner. These proposed iterative solvers were applied to the constructed linear pressure equations in the inner iteration in combination with the outer-Raphson iteration loop. Evaluation was performed in two stages. First, two-fluid numerical scheme capability was evaluated against OECD/NRC NUPEC PWR Bundle tests (PSBT Benchmark). The results for steady-state (PSBT) bundle show that an overall agreement can be found. At the second stage, convergency, stability, and accuracy of the proposed schemes were studied based on PSBT steady-state data through a comparison of utilized Krylov solvers and the direct inversion method as the pressure solution modeling options. It was found that the Krylov solvers do not introduce deviations to the two-phase flow results. Moreover, the convergence history of candidate Krylov solvers demonstrated that GMRES, FGMRES, and DQGMRES have a more efficient and stable convergence performance.

A recursive block incomplete factorization preconditioner for adaptive filtering problem

A recursive block incomplete factorization preconditioner for adaptive filtering problem

A supercoarsening multigrid method for poroelasticity in 3D coupled flow and geomechanics modeling

The Galerkin finite-element discretization of the force balance equation typically leads to large linear systems for geomechanical problems with realistic dimensions. In iteratively coupled flow and geomechanics modeling, a large linear system is solved at every timestep often multiple times during coupling iterations. The iterative solution of the linear system stemming from the poroelasticity equations constitutes the most time-consuming and memory-intensive component of coupled modeling. Block Jacobi, LSOR, and Incomplete LU factorization are popular preconditioning techniques used for accelerating the iterative solution of the poroelasticity linear systems. However, the need for more effective, efficient, and robust iterative solution techniques still remains especially for large coupled modeling problems requiring the solution of the poroelasticity system for a large number of timesteps. We developed a supercoarsening multigrid method (SCMG) which can be multiplicatively combined with commonly used preconditioning techniques. SCMG has been tested on a variety of coupled flow and geomechanics problems involving single-phase depletion and multiphase displacement of in-situ hydrocarbons, CO2 injection, and extreme material property contrasts. Our analysis indicates that the SCMG consistently improves the convergence properties of the linear systems arising from the poroelasticity equations, and thus, accelerates the coupled simulations for all cases subject to investigation. The joint utilization of the two-level SCMG with the ILU1 preconditioner emerges as the most optimal preconditioning/iterative solution strategy in a great majority of the problems evaluated in this work. The BiCGSTAB iterative solver converges more rapidly compared to PCG in a number of test cases, in which various SCMG-accelerated preconditioning strategies are applied to both iterators.

Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier–Stokes Equations

We study preconditioners for the iterative solution of the linear systems arising in the implicit time integration of the compressible Navier-Stokes equations. The spatial discretization is carried out using a discontinuous Galerkin method with fourth order polynomial interpolations on triangular elements. The time integration is based on backward difference formulas resulting in a nonlinear system of equations which is solved at each timestep. This is accomplished using Newton's method. The resulting linear systems are solved using a preconditioned GMRES iterative algorithm. We consider several existing preconditioners such as block Jacobi and Gauss-Seidel combined with multilevel schemes which have been developed and tested for specific applications. While our results are consistent with the claims reported, we find that these preconditioners lack robustness when used in more challenging situations involving low Mach numbers, stretched grids, or high Reynolds number turbulent flows. We propose a preconditioner based on a coarse scale correction with postsmoothing based on a block incomplete LU factorization with zero fill-in (ILU0) of the Jacobian matrix. The performance of the ILU0 smoother is found to depend critically on the element numbering. We propose a numbering strategy based on minimizing the discarded fill-in in a greedy fashion. The coarse scale correction scheme is found to be important for diffusion dominated problems, whereas the ILU0 preconditioner with the proposed ordering is effective at handling the convection dominated case. While little can be said in the way of theoretical results, the proposed preconditioner is shown to perform remarkably well for a broad range of representative test problems. These include compressible flows ranging from very low Reynolds numbers to fully turbulent flows using the Reynolds averaged Navier-Stokes equations discretized on highly stretched grids. For low Mach number flows, the proposed preconditioner is more than one order of magnitude more efficient than the other preconditioners considered.

Preconditioning of Boundary Value Problems Using Elementwise Schur Complements

This paper deals with an efficient technique for computing high-quality approximations of Schur complement matrices to be used in various preconditioners for the iterative solution of finite element discretizations of elliptic boundary value problems. The Schur complements are based on a two-by-two block decomposition of the matrix, and their approximations are computed by assembly of local (macroelement) Schur complements. The block partitioning is done by imposing a particular node ordering following the grid refinement hierarchy in the discretization mesh. For the theoretical derivation of condition number bounds, but not for the actual application of the method, we assume that the corresponding differential operator is self-adjoint and positive definite. The numerical efficiency of the proposed Schur complement approximation is illustrated in the framework of block incomplete factorization preconditioners of multilevel type, which require approximations of a sequence of arising Schur complement matrices. The behavior of the proposed approximation is compared with that of the coarse mesh finite element matrix, commonly used as an approximation of the Schur complement in the context of the above preconditioning methods. Moreover, the influence of refining a coarse mesh using a higher refinement number (m) than the customary $m=2$ is analyzed and its efficiency is also illustrated by numerical tests.

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.

Preconditioning methods for linear systems arising in constrained optimization problems

AbstractPreconditioning methods for matrices on saddle point form, as typically arising in equality constrained optimization problems, are surveyed. Special consideration is given to two methods: a nearly symmetric block incomplete factorization preconditioning method and a preconditioner on the same saddle point form as the given matrix. Both methods result in eigenvalues with positive real parts and small or zero imaginary parts. The behaviour of the methods are illustrated by solving a regularized Stokes problem. Copyright © 2002 John Wiley & Sons, Ltd.

Robust parameter‐free algebraic multilevel preconditioning

AbstractTo precondition large sparse linear systems resulting from the discretization of second‐order elliptic partial differential equations, many recent works focus on the so‐called algebraic multilevel methods. These are based on a block incomplete factorization process applied to the system matrix partitioned in hierarchical form. They have been shown to be both robust and efficient in several circumstances, leading to iterative solution schemes of optimal order of computational complexity. Now, despite the procedure is essentially algebraic, previous works focus generally on a specific context and consider schemes that use classical grid hierarchies with characteristic mesh sizes h,2h,4h, etc. Therefore, these methods require some extra information besides the matrix of the linear system and lack of robustness in some situations where semi‐coarsening would be desirable. In this paper, we develop a general method that can be applied in a black box fashion to a wide class of problems, ranging from 2D model Poisson problems to 3D singularly perturbed convection–diffusion equations. It is based on an automatic coarsening process similar to the one used in the AMG method, and on coarse grid matrices computed according to a simple and cheap aggregation principle. Numerical experiments illustrate the efficiency and the robustness of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd.

On parallel solution of linear elasticity problems. Part II: Methods and some computer experiments

AbstractThis is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coefficients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method.In the first part of the trilogy block‐diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg‐method, i.e. inner iterations were performed. As preconditioner, we used modified incomplete factorization MIC(0), where possibly the element matrices were modified in order to give M‐matrices, i.e. in order to guarantee the existence of the MIC(0) factorization.In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner/outer iterations we also study a variant of the block‐diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)‐factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system.In the third part of the trilogy, we will focus on the use of higher‐order finite elements. Copyright © 2002 John Wiley & Sons, Ltd.

A variant of block incomplete factorization preconditioners for a symmetricH-matrix

We propose a variant of parallel block incomplete factorization preconditioners for a symmetric block-tridiagonalH-matrix. Theoretical properties of these block preconditioners are compared with those of block incomplete factorization preconditioners for the corresponding comparison matrix. Numerical results of the preconditioned CG(PCG) method using these block preconditioners are compared with those of PCG using other types of block incomplete factorization preconditioners. Lastly, parallel computations of the block incomplete factorization preconditioners are carried out on the Cray C90.

A class of multilevel recursive incomplete LU preconditioning techniques

We introduce a class of multilevel recursive incomplete LU preconditioning techniques (RILUM) for solving general sparse matrices. This technique is based on a recursive two by two block incomplete LU factorization on the coefficient matrix. The coarse level system is constructed as an (approximate) Schur complement. A dynamic preconditioner is obtained by solving the Schur complement matrix approximately. The novelty of the proposed techniques is to solve the Schur complement matrix by a preconditioned Krylov subspace method. Such a reduction process is repeated to yield a multilevel recursive preconditioner.

Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems

This paper introduces several strategies to deal with pivot blocks in multi-level block incomplete LU factorization (BILUM) preconditioning techniques. These techniques are aimed at increasing the robustness and controlling the amount of fill-ins of BILUM for solving large sparse linear systems when large-size blocks are used to form block-independent set. Techniques proposed in this paper include double-dropping strategies, approximate singular-value decomposition, variable size blocks and use of an arrowhead block submatrix. We point out the advantages and disadvantages of these strategies and discuss their efficient implementations. Numerical experiments are conducted to show the usefulness of the new techniques in dealing with hard-to-solve problems arising from computational fluid dynamics. In addition, we discuss the relation between multi-level ILU preconditioning methods and algebraic multi-level methods.