Large scale petroleum reservoir simulation and parallel preconditioning algorithms research

Solving large scale linear systems efficiently plays an important role in a petroleum reservoir simulator, and the key part is how to choose an effective parallel preconditioner. Properly choosing a good preconditioner has been beyond the pure algebraic field. An integrated preconditioner should include such components as physical background, characteristics of PDE mathematical model, nonlinear solving method, linear solving algorithm, domain decomposition and parallel computation. We first discuss some parallel preconditioning techniques, and then construct an integrated preconditioner, which is based on large scale distributed parallel processing, and reservoir simulation-oriented. The infrastructure of this preconditioner contains such famous preconditioning construction techniques as coarse grid correction, constraint residual correction and subspace projection correction. We essentially use multi-step means to integrate totally eight types of preconditioning components in order to give out the final preconditioner. Million-grid cell scale industrial reservoir data were tested on native high performance computers. Numerical statistics and analyses show that this preconditioner achieves satisfying parallel efficiency and acceleration effect.     

A Comparative Study on Dynamic and Static Sparsity Patterns in Parallel Sparse Approximate Inverse Preconditioning

Sparse approximate inverse (SAI) techniques have recently emerged as a new class of parallel preconditioning techniques for solving large sparse linear systems on high performance computers. The choice of the sparsity pattern of the SAI matrix is probably the most important step in constructing an SAI preconditioner. Both dynamic and static sparsity pattern selection approaches have been proposed by researchers. Through a few numerical experiments, we conduct a comparable study on the properties and performance of the SAI preconditioners using the different sparsity patterns for solving some sparse linear systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

A survey of preconditioned iterative methods for linear systems of algebraic equations

We survey preconditioned iterative methods with the emphasis on solving large sparse systems such as arise by discretization of boundary value problems for partial differential equations.We discuss shortly various acceleration methods but the main emphasis is on efficient preconditioning techniques. Numerical simulations on practical problems have indicated that an efficient preconditioner is the most important part of an iterative algorithm. We report in particular on the state of the art of preconditioning methods for vectorizable and/or parallel computers.Dedicated to Carl-Erik Fröberg, a pioneer in Numerical Methods.     

A High-Quality Preconditioning Technique for Multi-Length-Scale Symmetric Positive Definite Linear Systems

We study preconditioning techniques used in conjunction with the conjugate gradient method for solving multi-length-scale symmetric positive definite linear systems originating from the quantum Monte Carlo simulation of electron interaction of correlated materials. Existing preconditioning techniques are not designed to be adaptive to varying numerical properties of the multi-length-scale systems. In this paper, we propose a hybrid incomplete Cholesky (HIC) preconditioner and demonstrate its adaptivity to the multi-length-scale systems. In addition, we propose an extension of the compressed sparse column with row access (CSCR) sparse matrix storage format to efficiently accommodate the data access pattern to compute the HIC preconditioner. We show that for moderately correlated materials, the HIC preconditioner achieves the optimal linear scaling of the simulation. The development of a linear-scaling preconditioner for strongly correlated materials remains an open topic.     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

A family of constrained pressure residual preconditioners for parallel reservoir simulations

Large‐scale reservoir simulations are extremely time‐consuming because of the solution of large‐scale linear systems arising from the Newton or Newton–Raphson iterations. The problem becomes even worse when highly heterogeneous geological models are employed. This paper introduces a family of multi‐stage preconditioners for parallel black oil simulations, which are based on the famous constrained pressure residual preconditioner. Numerical experiments demonstrate that our preconditioners are robust, efficient, and scalable. Copyright © 2015 John Wiley & Sons, Ltd.     

Parallel preconditioners for large scale partial difference equation systems

We propose a new preconditioner DASP (discrete approximate spectral preconditioner), based on the existing well-known preconditioners and our computational experience. Parallel preconditioning strategies for large scale partial difference equation systems arising from partial differential equations are investigated. Numerical results are given to show the efficiency and effectiveness of the new preconditioners for both model problems and real applications in petroleum reservoir simulation.     

Efficient parallel solution to large‐size sparse eigenproblems with block FSAI preconditioning

The choice of the preconditioner is a key factor to accelerate the convergence of eigensolvers for large‐size sparse eigenproblems. Although incomplete factorizations with partial fill‐in prove generally effective in sequential computations, the efficient preconditioning of parallel eigensolvers is still an open issue. The present paper describes the use of block factorized sparse approximate inverse (BFSAI) preconditioning for the parallel solution of large‐size symmetric positive definite eigenproblems with both a simultaneous Rayleigh quotient minimization and the Jacobi–Davidson algorithm. BFSAI coupled with a block diagonal incomplete decomposition proves a robust and efficient parallel preconditioner in a number of test cases arising from the finite element discretization of 3D fluid‐dynamical and mechanical engineering applications, outperforming FSAI even by a factor of 8 and exhibiting a satisfactory scalability. Copyright © 2011 John Wiley & Sons, Ltd.     

Multistep matrix splitting iteration preconditioning for singular linear systems

Multistep matrix splitting iterations serve as preconditioning for Krylov subspace methods for solving singular linear systems. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES) method. We present theoretical and practical justifications for using this approach. Numerical experiments show that the multistep generalized shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) iteration preconditioning are more robust and efficient compared to standard preconditioners for some test problems of large sparse singular linear systems.     

Multicolor low‐rank preconditioner for general sparse linear systems

This article presents a multilevel parallel preconditioning technique for solving general large sparse linear systems of equations. Subdomain coloring is invoked to reorder the coefficient matrix by multicoloring the adjacency graph of the subdomains, resulting in a two‐level block diagonal structure. A full binary tree structure $\mathrm{??}$ is then built to facilitate the construction of the preconditioner. A key property that is exploited is the observation that the difference between the inverse of the original matrix and that of its block diagonal approximation is often well approximated by a low‐rank matrix. This property and the block diagonal structure of the reordered matrix lead to a multicolor low‐rank (MCLR) preconditioner. The construction procedure of the MCLR preconditioner follows a bottom‐up traversal of the tree $\mathrm{??}$. All irregular matrix computations, such as ILU factorizations and related triangular solves, are restricted to leaf nodes where these operations can be performed independently. Computations in nonleaf nodes only involve easy‐to‐optimize dense matrix operations. In order to further reduce the number of iteration of the Preconditioned Krylov subspace procedure, we combine MCLR with a few classical block‐relaxation techniques. Numerical experiments on various test problems are proposed to illustrate the robustness and efficiency of the proposed approach for solving large sparse symmetric and nonsymmetric linear systems.     

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.     

A study on preconditioning iterative methods and relativity of pressure in the numerical calculation of fluid flow

This article presents the effect of preconditioning iterative methods on boundary conditions of the pressure‐correction in the numerical computation of fluid flow with known velocity components on all boundaries using the SIMPLE algorithm. In such computation, a set of solutions of the pressure‐correction is indefinite, because only the Neumann condition is imposed on all boundaries. However, solutions become unique if the value of pressure‐correction is fixed at least on one boundary point, and the Dirichlet condition is additionally imposed. Though both conditions must give exactly the same velocity and temperature fields, this problem arises from the relativity of the pressure. The mathematical illustration for this problem is provided using the numerical computation of the natural convection in an enclosure. It is concluded that the preconditioner adopted and the condition that only the Neumann condition on all boundaries is given are effective to reduce the number of iterations in solving the linear system of equations of the pressure‐correction at the computation of the natural convection. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Post-filtering of IC2-factors for load balancing in parallel preconditioning

A modification is proposed for the second order incomplete Cholesky decomposition (IC2). It makes possible to design a preconditioning procedure for the conjugate gradient method (CGM) with a controllable fill-in in the preconditioner. The modified algorithm is used to develop a load-balancing parallel preconditioning for CGM as applied to linear systems with symmetric positive definite matrices. Numerical results obtained using a multiprocessor computer system are presented.     

Convergence of nonstationary multisplitting methods using ILU factorizations

In this paper, we first study convergence of nonstationary multisplitting methods associated with a multisplitting which is obtained from the ILU factorizations for solving a linear system whose coefficient matrix is a large sparse H-matrix. We next study a parallel implementation of the relaxed nonstationary two-stage multisplitting method (called Algorithm 2 in this paper) using ILU factorizations as inner splittings and an application of Algorithm 2 to parallel preconditioner of Krylov subspace methods. Lastly, we provide parallel performance results of both Algorithm 2 using ILU factorizations as inner splittings and the BiCGSTAB with a parallel preconditioner which is derived from Algorithm 2 on the IBM p690 supercomputer.     

嵌套简单ILU分解代数预处理方法

In this paper, we propose a nested simple incomplete LU decomposition (NSILU) method for preconditioning iterative methods for solving largely scale and sparse ill-conditioned hnear systems. NSILU consists of some numerical techniques such as simple modification of Schur complement, compression of ill-condition structure by permutation, nested simple ILU, and inner-outer iteration. We give detailed error analysis of NSILU and estimations of condition number of the preconditioned coefficient matrix, together with numerical comparisons. We also show an analysis of inner accuracy strategies for the inner-outer iteration approach. Our new approach NSILU is very efficient for linear systems from a kind of two-dimensional nonlinear energy equations with three different temperature variables, where most of the calculations centered around solving large number of discretized and illconditioned linear systems in large scale. Many numerical experiments are given and compared in costs of flops, CPU times, and storages to show the efficiency and effectiveness of the NSILU preconditioning method. Numerical examples include middle-scale real matrices of size n = 3180 or n = 6360, a real apphcation of solving about 755418 linear systems of size n = 6360, and a simulation of order n=814080 with structures and properties similar as the real ones.     

Massively parallel preconditioners for symmetric positive definite linear systems

In this paper, we address the problem of solving sparse symmetric linear systems on parallel computers. With further restrictive assumptions on the matrix (e.g., bidiagonal or tridiagonal structure), several direct methods may be used. These methods give ideas for constructing efficient data parallel preconditioners for general positive definite symmetric matrices. We describe two examples of such preconditioners for which the factorization (i.e., the construction of the preconditioning matrix) turns out to be parallel. This revised version was published online in June 2006 with corrections to the Cover Date.     

A parallel block overlap preconditioning with inexact submatrix inversion for linear elasticity problems

We present a parallel preconditioned iterative solver for large sparse symmetric positive definite linear systems. The preconditioner is constructed as a proper combination of advanced preconditioning strategies. It can be formally seen as being of domain decomposition type with algebraically constructed overlap. Similar to the classical domain decomposition technique, inexact subdomain solvers are used, based on incomplete Cholesky factorization. The proper preconditioner is shown to be near optimal in minimizing the so‐called K‐condition number of the preconditioned matrix. The efficiency of both serial and parallel versions of the solution method is illustrated on a set of benchmark problems in linear elasticity. Copyright © 2002 John Wiley & Sons, Ltd.     

A preconditioner for a kind of coupled FEM-BEM variational inequality

In this paper we are concerned with a kind of nonlinear transmission problem with Signorini contact conditions. This problem can be described by a coupled FEM-BEM variational inequality. We first develop a preconditioning gradient projection method for solving the variational inequality. Then we construct an effective domain decomposition preconditioner for the discrete system. The preconditioner makes the coupled inequality problem be decomposed into an equation problem and a “small” inequality problem, which can be solved in parallel. We give a complete analysis to the convergence speed of this iterative method.     

Approximate inverse preconditioning in the parallel solution of sparse eigenproblems

A preconditioned scheme for solving sparse symmetric eigenproblems is proposed. The solution strategy relies upon the DACG algorithm, which is a Preconditioned Conjugate Gradient algorithm for minimizing the Rayleigh Quotient. A comparison with the well established ARPACK code shows that when a small number of the leftmost eigenpairs is to be computed, DACG is more efficient than ARPACK. Effective convergence acceleration of DACG is shown to be performed by a suitable approximate inverse preconditioner (AINV). The performance of such a preconditioner is shown to be safe, i.e. not highly dependent on a drop tolerance parameter. On sequential machines, AINV preconditioning proves a practicable alternative to the effective incomplete Cholesky factorization, and is more efficient than Block Jacobi. Owing to its parallelizability, the AINV preconditioner is exploited for a parallel implementation of the DACG algorithm. Numerical tests account for the high degree of parallelization attainable on a Cray T3E machine and confirm the satisfactory scalability properties of the algorithm. A final comparison with PARPACK shows the (relative) higher efficiency of AINV‐DACG. Copyright © 2000 John Wiley & Sons, Ltd.     

Schur complement‐based domain decomposition preconditioners with low‐rank corrections

This paper introduces a robust preconditioner for general sparse matrices based on low‐rank approximations of the Schur complement in a Domain Decomposition framework. In this ‘Schur Low Rank’ preconditioning approach, the coefficient matrix is first decoupled by a graph partitioner, and then a low‐rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface unknowns. The method avoids explicit formation of the Schur complement. We show the feasibility of this strategy for a model problem and conduct a detailed spectral analysis for the relation between the low‐rank correction and the quality of the preconditioner. We first introduce the SLR preconditioner for symmetric positive definite matrices and symmetric indefinite matrices if the interface matrices are symmetric positive definite. Extensions to general symmetric indefinite matrices as well as to nonsymmetric matrices are also discussed. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.