Block ILU Preconditioner Research Articles

Domain decomposition based parallel computing for multi-scale coronary blood flow simulations

In the present study, we demonstrated a parallel P2P1 finite element scheme with a four-step fractional splitting approach to conduct a multiscale coronary flow simulation. The three-dimensional (3D) computational fluid dynamics (CFD) for patient-specific coronary artery flow and zero-dimensional (0D) lumped-parameter network (LPN) modeling for distal coronary beds was fully coupled, and an MPI parallel algorithm based on domain decomposition was applied. A parallel conjugate gradient (CG)-LPN subroutine for the 3D-0D coupled system with a monolithic scheme was derived, and it provides a correct pressure solution that may not be obtained by the conventional CG solver, particularly when the subdomain division intersects the 3D-0D coupling outlet. The overall computing time for parallel CG-LPN does not show a noticeable difference from the conventional CG solver, despite the extra MPI calls for data transfer at the interfaces of subdomains.For the BiCGSTAB solver, the block ILU(0) preconditioner showed favorable performance for a high density mesh compared with the simple Jacobi preconditioner. MPI_COMM is a major bottleneck that saturates the overall parallel performance at high core count, but a computing time of less than 10 min per cardiac cycle on a medium density mesh could be attained for a patient-specific coronary flow simulation using 60 CPU cores run in parallel, which is in an acceptable range for clinical practice. Further tests of accuracy are needed with a large set of patients to enable wide use of the proposed technique in helping to make interventional decisions in routine clinical practice for coronary stenotic lesions.

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.

On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations

We study the convergence of iterative linear solvers for discontinuous Galerkin discretizations of systems of hyperbolic conservation laws with polygonal mesh elements compared with that of traditional triangular elements. We solve the semi-discrete system of equations by means of an implicit time discretization method, using iterative solvers such as the block Jacobi method and GMRES. We perform a von Neumann analysis to analytically study the convergence of the block Jacobi method for the two-dimensional advection equation on four classes of regular meshes: hexagonal, square, equilateral-triangular, and right-triangular. We find that hexagonal and square meshes give rise to smaller eigenvalues, and thus result in faster convergence of Jacobi's method. We perform numerical experiments with variable velocity fields, irregular, unstructured meshes, and the Euler equations of gas dynamics to confirm and extend these results. We additionally study the effect of polygonal meshes on the performance of block ILU(0) and Jacobi preconditioners for the GMRES method.

Accelerating the GMRES Solver with Block ILU (K) Preconditioner on GPUs in Reservoir Simulation

This paper studies the parallelization of the restarted GMRES solver, GMRES (m), and the block ILU (k) preconditioner on GPUs used in petroleum reservoir simulations. The difficulty is how to accelerate this preconditioner with a variable block size. In this paper, parallel solution techniques for block triangular systems are proposed, which work for matrices with an arbitrary block size. These techniques also work with an arbitrary level k for the block ILU (k) preconditioner. Numerical experiments show that the GPU-based linear solver GMRES (m) is much faster than its CPU version.

Using an ILU/Deflation Preconditioner for Simulation of a PEM Fuel Cell Cathode Catalyst Layer

AbstractNumerical aspects of a pore scale model are investigated for the simulation of catalyst layers of polymer electrolyte membrane fuel cells. Coupled heat, mass and charged species transport together with reaction kinetics are taken into account using parallelized finite volume simulations for a range of nanostructured, computationally reconstructed catalyst layer samples. The effectiveness of implementing deflation as a second stage preconditioner generally improves convergence and results in better convergence behavior than more sophisticated first stage pre-conditioners. This behavior is attributed to the fact that the two stage preconditioner updates the preconditioning matrix at every GMRES restart, reducing the stalling effects that are commonly observed in restarted GMRES when a single stage preconditioner is used. In addition, the effectiveness of the deflation preconditioner is independent of the number of processors, whereas the localized block ILU preconditioner deteriorates in quality as the number of processors is increased. The total number of GMRES search directions required for convergence varies considerably depending on the preconditioner, but also depends on the catalyst layer microstructure, with low porosity microstructures requiring a smaller number of iterations. The improved model and numerical solution strategy should allow simulations for larger computational domains and improve the reliability of the predicted transport parameters. The preconditioning strategies presented in the paper are scalable and should prove effective for massively parallel simulations of other problems involving nonlinear equations.

Block ILU preconditioners for block-tridiagonal systems

In this paper, two new block ILU preconditioners for block-tridiagonal M-matrices and H-matrices are proposed. Some theoretical properties for the preconditioners are studied and how to construct preconditioners effectively is also discussed. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.

A numerical study on preconditioning and partitioning schemes for reactive transport in a PEMFC catalyst layer

A numerical method used to simulate PEMFC catalyst layer transport and electrochemistry is described. The set of nonlinear equations is discretized using the finite volume method and solved using an inexact Newton method. A block “ILU porous” preconditioner is used to precondition the linear system. A “porous partitioning” scheme is used to partition the domain for parallel processors. Geometries with low porosities are found to require a much smaller number of iterations for convergence compared to geometries with high porosities. The porous partitioning scheme is shown to outperform the standard partitioning scheme for cases run with more than 4 processors. The block “ILU porous” preconditioner was not found to be more effective than the block ILU(1) preconditioner. Finally, the importance of using rigorous convergence criteria in these simulations is demonstrated by comparing the computed total consumption values of different species at different nonlinear rms tolerance values.

Block ILU-preconditioners for problems of filtration of a multicomponent mixture in a porous medium

ILU class preconditioners (ILU(0), ILU(1), ILUT) employed for iterative algorithms for asymmetric linear systems with sparse matrices are considered. Test matrices used in this study originate from discretization of systems of partial differential equations describing a multicomponent fluid flow in porous media. New algorithms for block storage of matrices and block based ILU-factorization are described. This new integrated approach was tested on a wide range of matrices resulted from actual hydrodynamic simulations of oil fields in Western Siberia and had demonstrated significant reduction of computational time.

Preconditioning methods for discontinuous Galerkin solutions of the Navier–Stokes equations

A Newton–Krylov method is developed for the solution of the steady compressible Navier–Stokes equations using a discontinuous Galerkin (DG) discretization on unstructured meshes. Steady-state solutions are obtained using a Newton–Krylov approach where the linear system at each iteration is solved using a restarted GMRES algorithm. Several different preconditioners are examined to achieve fast convergence of the GMRES algorithm. An element Line-Jacobi preconditioner is presented which solves a block-tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling. This reordering is shown to be superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill–Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and an algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. Additionally, a linear p-multigrid preconditioner is also considered, where Block-Jacobi, Line-Jacobi and Block-ILU(0) are used as smoothers. The linear multigrid preconditioner is shown to significantly improve convergence in term of number of iterations and CPU time compared to a single-level Block-Jacobi or Line-Jacobi preconditioner. Similarly the linear multigrid preconditioner with Block-ILU smoothing is shown to reduce the number of linear iterations to achieve convergence over a single-level Block-ILU(0) preconditioner, though no appreciable improvement in CPU time is shown.

On the preconditioning of the block tridiagonal linear system of equations

Two algorithms for computing the inverse factors of general tridiagonal and pentadiagonal matrices are obtained. Then, these algorithms are used for computing a block ILU preconditioner for the block tridiagonal linear system of equations. Some numerical results are given to show the robustness and efficiency of the preconditioner. The performance of the proposed preconditioner is compared with a recently proposed method.

On computing of block ILU preconditioner for block tridiagonal systems

We present the recurrence formulas for computing the approximate inverse factors of tridiagonal and pentadiagonal matrices using bordering technique. Resulting algorithms are used to approximate the inverse of pivot blocks needed for constructing block ILU preconditioners for solving the block tridiagonal linear systems, arising from discretization of partial differential equations. Resulting preconditioners are suitable for parallel implementation. Comparison with other methods are also included.

The effect of block red-black ordering on blockILU preconditioner for sparse matrices

It is well known that the ordering of the unknowns can have a significant effect on the convergence of a preconditioned iterative method and on its implementation on a parallel computer. To do so, we introduce a block red-black coloring to increase the degree of parallelism in the application of the blockILU preconditioner for solving sparse matrices, arising from convection-diffusion equations discretized using the finite difference scheme (five-point operator). We study the preconditioned PGMRES iterative method for solving these linear systems.

Ainv and bilum preconditioning techniques

It was proposed that a robust and efficient parallelizable preconditioner for solving general sparse linear systems of equations, in which the use of sparse approximate inverse (AINV) techniques in a multi-level block ILU (BILUM) preconditioner were investigated. The resulting preconditioner retains robustness of BILUM preconditioner and has two advantages over the standard BILUM preconditioner: the ability to control sparsity and increased parallelism. Numerical experiments are used to show the effectiveness and efficiency of the new preconditioner.

Developing object-oriented parallel iterative methods

In this paper, we describe our work developing an object-oriented parallel toolkit on OO-MPI. We are developing a parallelised implementation of the multi-view tomography toolbox, an iterative solver for a range of tomography problems. The code is developed in object-oriented C++. The MVT toolbox is presently used by researchers in the field of tomography to solve linear and non-linear forward modelling problems. The performance of the toolbox is heavily dependent on the performance of a small number of classes present in the IML++ library. This paper describes our experience parallelising a sparse matrix algorithm provided in the IML++ object-oriented numerical library. This library comprises a number of iterative algorithms. In this work, we present a parallel version of BiCGSTAB algorithm and the block-ILU preconditioner. These two algorithms are implemented in C++ using OOMPI (object-oriented MPI), and run on a 32-node Beowulf cluster. These two routines are also fundamental to obtaining good performance when using the parallelised version of the MVT toolbox. We also demonstrate the importance of using threads to overlap communication and computation, as an effective path to obtain improved speedup.

Multilevel block ILU preconditioner for sparse nonsymmetric M-matrices

We develop in this paper a multilevel block ILU preconditioner for solving sparse nonsymmetric M-matrices, arising from convection–diffusion equations, discretized by finite difference scheme. This preconditioner is based on recursive block ILU factorization, where the Schur complement matrices are successively approached by some sparse matrices satisfying some hypothesis. We study the preconditioned GMRES iterative method and multigrid method for solving linear systems arising from this scheme.

Parallel two level block ILU preconditioning techniques for solving large sparse linear systems

We discuss issues related to domain decomposition and multilevel preconditioning techniques which are often employed for solving large sparse linear systems in parallel computations. We implement a parallel preconditioner for solving general sparse linear systems based on a two level block ILU factorization strategy. We give some new data structures and strategies to construct a local coefficient matrix and a local Schur complement matrix on each processor. The preconditioner constructed is fast and robust for solving certain large sparse matrices. Numerical experiments show that our domain based two level block ILU preconditioners are more robust and more efficient than some published ILU preconditioners based on Schur complement techniques for parallel sparse matrix solutions.

New block ILU preconditioner scheme for numerical analysis of very large electromagnetic problems

Large electromagnetic scattering and radiation problems are tackled by iterative solvers, which require the use of huge preconditioners. Most often, the incomplete LU decomposition (ILU) of the preconditioner is applied to the system matrix at each iteration. However, the preconditioner ILU cannot be done in-core when the size of the preconditioning matrix exceeds the available memory. This paper presents a new preconditioning scheme to do the preconditioner ILU in small blocks that fit in core memory. The resulting approach allows the solution of very large problems in small computers.

Block and full matrix ILU preconditioners for parallel finite element solvers

Parallel finite element solvers based on ILU preconditionings are developed, implemented and tested in two- and three-dimensional Laplace problems. The computational domain is decomposed into N subdomains for parallel processing. The structure of the parallel computer system consists of the main processor and N satellite processors. Two algorithms are developed: a block ILU preconditioner at the subdomain level, without communication between the satellite processors, and a full matrix ILU preconditioner coupling the subdomain degrees of freedom and requiring communication between the satellite processors. Different node orderings, mesh sizes and number of satellite processors are tested. The efficiency of both block and full matrix ILU preconditioners is strongly dependent on the node ordering inside each subdomain. The finite elements in each subdomain must be connected.

AILU FOR HELMHOLTZ PROBLEMS: A NEW PRECONDITIONER BASED ON THE ANALYTIC PARABOLIC FACTORIZATION

We investigate a new type of preconditioner which is based on the analytic factorization of the operator into two parabolic factors. Approximate analytic factorizations lead to new block ILU preconditioners. We analyze the preconditioner at the continuous level where it is possible to optimize its performance. Numerical experiments illustrate the effectiveness of the new approach.

AILU for Helmholtz Problems A New Preconditioner Based on the Analytic Parabolic Factorization

We investigate a new type of preconditioner which is based on the analytic factorization of the operator into two parabolic factors. Approximate analytic factorizations lead to new block ILU preconditioners. We analyze the preconditioner at the continuous level where it is possible to optimize its performance. Numerical experiments illustrate the effectiveness of the new approach.