Sparse Approximations of the Schur Complement for Parallel Algebraic Hybrid Solvers in 3D

<正>In this paper we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems.In earlier works,the local Schur complements were computed exactly using a sparse direct solver.The robustness of the preconditioner comes at the price of this memory and time intensive computation that is the main bottleneck of the approach for tackling huge problems.In this work we investigate the use of sparse approximation of the dense local Schur complements.These approximations are computed using a partial incomplete LU factorization.Such a numerical calculation is the core of the multi-level incomplete factorization such as the one implemented in pARMS. The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems;preliminary experiments on linear systems arising from structural mechanics are also reported.     

Block Preconditioners for Saddle Point Problems

A general purpose block LU preconditioner for saddle point problems is presented. A major difference between the approach presented here and that of other studies is that an explicit, accurate approximation of the Schur complement matrix is efficiently computed. This is used to obtain a preconditioner to the Schur complement matrix which in turn defines a preconditioner for the global system. A number of variants are developed and results are reported for a few linear systems arising from CFD applications.     

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two‐by‐two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean‐value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix–vector multiplications for the off‐diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen–Loève expansion and a good preconditioned for the mean‐value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Improving an interior-point approach for large block-angular problems by hybrid preconditioners

The computational time required by interior-point methods is often dominated by the solution of linear systems of equations. An efficient specialized interior-point algorithm for primal block-angular problems has been used to solve these systems by combining Cholesky factorizations for the block constraints and a conjugate gradient based on a power series preconditioner for the linking constraints. In some problems this power series preconditioner resulted to be inefficient on the last interior-point iterations, when the systems became ill-conditioned. In this work this approach is combined with a splitting preconditioner based on LU factorization, which works well for the last interior-point iterations. Computational results are provided for three classes of problems: multicommodity flows (oriented and nonoriented), minimum-distance controlled tabular adjustment for statistical data protection, and the minimum congestion problem. The results show that, in most cases, the hybrid preconditioner improves the performance and robustness of the interior-point solver. In particular, for some block-angular problems the solution time is reduced by a factor of 10.     

On the use of dense matrix techniques within sparse simplex

In this paper we discuss some instances where dense matrix techniques can be utilized within a sparse simplex linear programming solver. The main emphasis is on the use of the Schur complement matrix as a part of the basis matrix representation. This approach enables to represent the basis matrix as an easily invertible sparse matrix and one or more dense Schur complement matrices. We describe our variant of this method which uses updating of the QR factorization of the Schur complement matrix. We also discuss some implementation issues of the LP software package which is based on this approach.     

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.     

Iterative solver for systems of linear equations with a sparse stiffness matrix on clusters

In this paper, a package of programs for solving systems of linear equations with a sparse matrix for computers with distributed memory is proposed. The package is based on an iterative algorithm for solving the initial system of equations with a preconditioner constructed using an algebraic domain decomposition. Such an approach makes it possible to simultaneously multiply the preconditioner and the stiffness matrix by a vector on a cluster. Also, to improve the efficiency of computation, the functionalities PARDISO and Sparse BLAS of the Intel®MKL library are used on each process. In addition to processes parallelization, the package uses OpenMP parallelization on each of these processes, as well as Intel®MKL internal functional parallelization.     

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.     

FSAI-based parallel Mixed Constraint Preconditioners for saddle point problems arising in geomechanics

In this paper we propose and describe a parallel implementation of a block preconditioner for the solution of saddle point linear systems arising from Finite Element (FE) discretization of 3D coupled consolidation problems. The Mixed Constraint Preconditioner developed in [L. Bergamaschi, M. Ferronato, G. Gambolati, Mixed constraint preconditioners for the solution to FE coupled consolidation equations, J. Comput. Phys., 227(23) (2008), 9885–9897] is combined with the parallel FSAI preconditioner which is used here to approximate the inverses of both the structural (1, 1) block and an appropriate Schur complement matrix. The resulting preconditioner proves effective in the acceleration of the BiCGSTAB iterative solver. Numerical results on a number of test cases of size up to 2×10⁶ unknowns and 1.2×10⁸ nonzeros show the perfect scalability of the overall code up to 256 processors.     

FSAI-based parallel Mixed Constraint Preconditioners for saddle point problems arising in geomechanics

In this paper we propose and describe a parallel implementation of a block preconditioner for the solution of saddle point linear systems arising from Finite Element (FE) discretization of 3D coupled consolidation problems. The Mixed Constraint Preconditioner developed in [L. Bergamaschi, M. Ferronato, G. Gambolati, Mixed constraint preconditioners for the solution to FE coupled consolidation equations, J. Comput. Phys., 227(23) (2008), 9885-9897] is combined with the parallel FSAI preconditioner which is used here to approximate the inverses of both the structural (1, 1) block and an appropriate Schur complement matrix. The resulting preconditioner proves effective in the acceleration of the BiCGSTAB iterative solver. Numerical results on a number of test cases of size up to 2×10⁶ unknowns and 1.2×10⁸ nonzeros show the perfect scalability of the overall code up to 256 processors.     

On a Toeplitz Representation of the H1/2 Norm

Some representations of the H1/2 norm are used as Schur complement preconditioner in PCG based domain decomposition algorithms for elliptic problems. These norm representations are efficient preconditioners but the corresponding matrices are dense, so they need FFT algorithm for matrix-vector multiplications. Here we give a new matrix representation of this norm by a special Toeplitz matrix. It contains only O(log(n)) different entries at each row, where n is the number of rows and so a matrix-vector computation can be done by O(nlog(n)) arithmetic operation without using FFT algorithm. The special properties of this matrix assure that it can be used as preconditioner. This is proved by estimating spectral equivalence constants and this fact has also been verified by numerical tests.     

Stability and preconditioning for a hybrid approximation on the sphere

This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. In principle the resulting linear system can be preconditioned by the block-diagonal preconditioner of Murphy, Golub and Wathen. Making use of a recently derived inf–sup condition and the Brezzi stability and convergence theorem for this approximation scheme, we show that in this context the Schur complement in the above preconditioner is spectrally equivalent to a certain non-constant diagonal matrix. Numerical experiments with a non-uniform distribution of data points support the theoretically proved quality of the new preconditioner.     

Multigrid transfers for nonsymmetric systems based on Schur complements and Galerkin projections

A framework is proposed for constructing algebraic multigrid transfer operators suitable for nonsymmetric positive definite linear systems. This framework follows a Schur complement perspective as this is suitable for both symmetric and nonsymmetric systems. In particular, a connection between algebraic multigrid and approximate block factorizations is explored. This connection demonstrates that the convergence rate of a two‐level model multigrid iteration is completely governed by how well the coarse discretization approximates a Schur complement operator. The new grid transfer algorithm is then based on computing a Schur complement but restricting the solution space of the corresponding grid transfers in a Galerkin‐style so that a far less expensive approximation is obtained. The final algorithm corresponds to a Richardson‐type iteration that is used to improve a simple initial prolongator or a simple initial restrictor. Numerical results are presented illustrating the performance of the resulting algebraic multigrid method on highly nonsymmetric systems. Copyright © 2013 John Wiley & Sons, Ltd.     

A preconditioning technique for a class of PDE-constrained optimization problems

We investigate the use of a preconditioning technique for solving linear systems of saddle point type arising from the application of an inexact Gauss?CNewton scheme to PDE-constrained optimization problems with a hyperbolic constraint. The preconditioner is of block triangular form and involves diagonal perturbations of the (approximate) Hessian to insure nonsingularity and an approximate Schur complement. We establish some properties of the preconditioned saddle point systems and we present the results of numerical experiments illustrating the performance of the preconditioner on a model problem motivated by image registration.     

Correlative Sparsity in Primal-Dual Interior-Point Methods for LP,SDP, and SOCP

Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, second-order cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of the coefficient matrix of the equation which is determined by the sparsity of an optimization problem (linear program, semidefinite program or second-order cone program). We show if an optimization problem is correlatively sparse, then the coefficient matrix of the Schur complement equation inherits the sparsity, and a sparse Cholesky factorization applied to the matrix results in no fill-in. S. Kim’s research was supported by Kosef R01-2005-000-10271-0 and KRF-2006-312-C00062.     

Fast inexact subspace iteration for generalized eigenvalue problems with spectral transformation

We study inexact subspace iteration for solving generalized non-Hermitian eigenvalue problems with spectral transformation, with focus on a few strategies that help accelerate preconditioned iterative solution of the linear systems of equations arising in this context. We provide new insights into a special type of preconditioner with “tuning” that has been studied for this algorithm applied to standard eigenvalue problems. Specifically, we propose an alternative way to use the tuned preconditioner to achieve similar performance for generalized problems, and we show that these performance improvements can also be obtained by solving an inexpensive least squares problem. In addition, we show that the cost of iterative solution of the linear systems can be further reduced by using deflation of converged Schur vectors, special starting vectors constructed from previously solved linear systems, and iterative linear solvers with subspace recycling. The effectiveness of these techniques is demonstrated by numerical experiments.     

Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations

We consider an abstract parameter dependent saddle-point problem and present a general framework for analyzing robust Schur complement preconditioners. The abstract analysis is applied to a generalized Stokes problem, which yields robustness of the Cahouet-Chabard preconditioner. Motivated by models for two-phase incompressible flows we consider a generalized Stokes interface problem. Application of the general theory results in a new Schur complement preconditioner for this class of problems. The robustness of this preconditioner with respect to several parameters is treated. Results of numerical experiments are given that illustrate robustness properties of the preconditioner.     

Preconditioning Techniques for Newton's Method for the Incompressible Navier–Stokes Equations

Newton's method for the incompressible Navier—Stokes equations gives rise to large sparse non-symmetric indefinite matrices with a so-called saddle-point structure for which Schur complement preconditioners have proven to be effective when coupled with iterative methods of Krylov type. In this work we investigate the performance of two preconditioning techniques introduced originally for the Picard method for which both proved significantly superior to other approaches such as the Uzawa method. The first is a block preconditioner which is based on the algebraic structure of the system matrix. The other approach uses also a block preconditioner which is derived by considering the underlying partial differential operator matrix. Analysis and numerical comparison of the methods are presented.This revised version was published online in October 2005 with corrections to the Cover Date.