Nonsymmetric Linear Systems Research Articles

Preconditioning in parallel for fractional step Navier--Stokes solvers

Preconditioning for the Pressure Poisson Equation, used with the fractional step Navier--Stokes solvers, is studied. The Pressure Poisson Equation results from the segregated calculation of the velocity and pressure in the momentum equations, with the divergence of the velocity as the source term. The coefficient matrix of the Pressure Poisson Equation is dependent only upon grid-size, and thus preconditioners need to be constructed only once initially, and used for all subsequent time steps. Several preconditioning techniques are studied, including Jacobi, incomplete matrix decomposition variants, and sparse approximate inverses. The test case is a three dimensional turbulent channel flow, with the domain discretised using a structured nonstaggered grid. Parallel computing is performed on a cluster of processors by message passing, with domain partitioning to avoid the use of global gather and scatter operations and to minimise the effect of communication bandwidth. The preconditioners are all constructed based on the local grid partition. For the sparse approximate inverse preconditioners, cell dependencies are bounded to limit communication across grid partition boundaries. The effect of a defined sparsity pattern is also investigated. The optimum sparsity pattern and the dependency on the neighbouring cells are found to be influenced by the grid ratios in each axis direction. References M. Benzi. Preconditioning techniques for large linear systems: A survey. J. of Comp. Physics , 182(2):418--477, 2002. doi:10.1006/jcph.2002.7176 Y. Saad. Iterative Methods for Sparse Linear Systems . SIAM, Philadelphia, PA, USA, 2nd edition, 2003. V. S. Djanali, S. W. Armfield, and M. P. Kirkpatrick. Comparison of approximate inverse preconditioners for the fractional step Navier--Stokes equations. ANZIAM J. , 52:C581--C595, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3890 M. Ament, G. Knittel, D. Weiskopf, and W. Strasser. A parallel preconditioned conjugate gradient solver for the Poisson problem on a multi-gpu platform. Parallel, Distributed and Network-Based Processing (PDP), 2010 18th Euromicro International Conference , 583--592, 2010. doi:10.1109/PDP.2010.51 S.M. Xu, H.X. Lin, K. Wang, and W. Xue. Utilizing cuda for preconditioned gmres solvers. Proc. 8th Int‚Aol Symp. on Distributed Computing and Applications to Business, Engineering and Sciences (DCABES 2009) . S. Armfield and R. Street. An analysis and comparison of the time accuracy of fractional‚aistep methods for the Navier--Stokes equations on staggered grid. Int. J. Numer. Methods Fluids , 38:255--282, 2002. doi:10.1002/fld.217 H. L. Stone. Iterative solution of implicit approximations of multidimensional partial differential equations. SIAM J. Numer. Anal. , 5(3):530--558, 1968. doi:10.1137/0705044 G. E. Schneider and M. Zedan. A Modified strongly implicit procedure for the numerical solution of field problems. Numer. Heat Transfer , 4(1):1--19, 1981. doi:10.1080/01495728108961775 M. J. Grote and T. Huckle. Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. , 18(3):838--853, May 1997. doi:10.1137/S1064827594276552 M. Benzi and M. Tuma. A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. , 19(3):968--994, 1998. doi:10.1137/S1064827595294691 R. D. Moser, J. Kim, and N. N. Mansour. {Direct numerical simulation of turbulent channel flow up to Re$_\tau $ = 590}. Phys. Fluids , 11(4):943--945, 1999. doi:10.1063/1.869966

The BiConjugate gradient method on GPUs

In a wide variety of applications from different scientific and engineering fields, the solution of complex and/or nonsymmetric linear systems of equations is required. To solve this kind of linear systems the BiConjugate Gradient method (BCG) is especially relevant. Nevertheless, BCG has a enormous computational cost. GPU computing is useful for accelerating this kind of algorithms but it is necessary to develop suitable implementations to optimally exploit the GPU architecture. In this paper, we show how BCG can be effectively accelerated when all operations are computed on a GPU. So, BCG has been implemented with two alternative routines of the Sparse Matrix Vector product (SpMV): the CUSPARSE library and the ELLR-T routine. Although our interest is focused on complex matrices, our implementation has been evaluated on a GPU for two sets of test matrices: complex and real, in single and double precision data. Experimental results show that BCG based on ELLR-T routine achieves the best performance, particularly for the set of complex test matrices. Consequently, this method can be useful as a tool to efficiently solve large linear system of equations (complex and/or nonsymmetric) involved in a broad range of applications.

An Efficient Variant of the GMRES(m) Method Based on the Error Equations

AbstractThe GMRES(m) method proposed by Saad and Schultz is one of the most successful Krylov subspace methods for solving nonsymmetric linear systems. In this paper, we investigate how to update the initial guess to make it converge faster, and in particular propose an efficient variant of the method that exploits an unfixed update. The mathematical background of the unfixed update variant is based on the error equations, and its potential for efficient convergence is explored in some numerical experiments.

GMRES implementations and residual smoothing techniques for solving ill-posed linear systems

There are verities of useful Krylov subspace methods to solve nonsymmetric linear system of equations. GMRES is one of the best Krylov solvers with several different variants to solve large sparse linear systems. Any GMRES implementation has some advantages. As the solution of ill-posed problems are important. In this paper, some GMRES variants are discussed and applied to solve these kinds of problems. Residual smoothing techniques are efficient ways to accelerate the convergence speed of some iterative methods like CG variants. At the end of this paper, some residual smoothing techniques are applied for different GMRES methods to test the influence of these techniques on GMRES implementations.

A new family of global methods for linear systems with multiple right-hand sides

The global bi-conjugate gradient (Gl-BCG) method is an attractive matrix Krylov subspace method for solving nonsymmetric linear systems with multiple right-hand sides, but it often show irregular convergence behavior in many applications. In this paper, we present a new family of global A-biorthogonal methods by using short two-term recurrences and formal orthogonal polynomials, which contain the global bi-conjugate residual (Gl-BCR) algorithm and its improved version. Finally, numerical experiments illustrate that the proposed methods are highly competitive and often superior to originals.

On the block GMRES method with deflated restarting

In this paper, we consider the block-GMRES method with deflated restarting for solving nonsymmetric linear systems with multiple right-hand sides. We modify slightly the restarted block-GMRES method with deflation of eigenvalues proposed by Morgan to obtain a new one. It is shown that the modified method is mathematically equivalent to the Morgan’s original method. However, the error analysis shows the modified version minimizes the numerical errors during a restart and therefore is better suited if the linear systems have to be solved with high precision. Numerical experiments report the effectiveness of the modified method.

A variant of the IDR( [formula omitted]) method with the quasi-minimal residual strategy

The IDR( s ) method proposed by Sonneveld and van Gijzen is an effective method for solving nonsymmetric linear systems, but usually with irregular convergence behavior. In this paper, we reformulate the relations of residuals and their auxiliary vectors generated by the IDR( s ) method in matrix form. Then, using this new formulation and motivated by other QMR-type methods, we propose a variant of the IDR( s ) method, called QMRIDR( s ), for overcoming the disadvantage of its irregular convergence behavior. Both fast and smooth convergence behaviors of the QMRIDR( s ) method can be shown. Numerical experiments are reported to show the efficiency of our proposed method.

A parallel sparse algorithm targeting arterial fluid mechanics computations

Iterative solution of large sparse nonsymmetric linear equation systems is one of the numerical challenges in arterial fluid---structure interaction computations. This is because the fluid mechanics parts of the fluid + structure block of the equation system that needs to be solved at every nonlinear iteration of each time step corresponds to incompressible flow, the computational domains include slender parts, and accurate wall shear stress calculations require boundary layer mesh refinement near the arterial walls. We propose a hybrid parallel sparse algorithm, domain-decomposing parallel solver (DDPS), to address this challenge. As the test case, we use a fluid mechanics equation system generated by starting with an arterial shape and flow field coming from an FSI computation and performing two time steps of fluid mechanics computation with a prescribed arterial shape change, also coming from the FSI computation. We show how the DDPS algorithm performs in solving the equation system and demonstrate the scalability of the algorithm.

Improving the efficiency of variational methods for solving strongly nonsymmetric linear algebraic equation system received in convection-diffusion problems

An efficient algorithm for implementing the mathematical model of convection-diffusion transport with a dominant convection is proposed. Preconditioned Krylov subspace methods are used to solve strongly nonsymmetric systems of linear algebraic equations. Theoretical convergence analysis of product triangular skew-symmetric preconditioners is carried out. A series of numerical experiments have confirmed the technique’s efficiency.

GMRES with adaptively deflated restarting and its performance on an electromagnetic cavity problem

GMRES with deflated restarting, denoted by GMRES-DR( m , k ), is a variant of restarted GMRES for solving nonsymmetric linear systems, which improves the convergence of restarted GMRES by weakening the influence of the eigenvalues incurring poor numerical performance. However, it is difficult to determine the deflation parameter k. In this paper, we present a new deflation strategy which determines k dynamically in different cycles. Numerical results for an electromagnetic cavity problem show that the deflation strategy is efficient.

A note on simultaneous preconditioning and symmetrization of non-symmetric linear systems

A note on simultaneous preconditioning and symmetrization of non-symmetric linear systems

A block IDR( [formula omitted]) method for nonsymmetric linear systems with multiple right-hand sides

The IDR( s ) based on the induced dimension reduction (IDR) theorem, is a new class of efficient algorithms for large nonsymmetric linear systems. IDR(1) is mathematically equivalent to BiCGStab at the even IDR(1) residuals, and IDR( s ) with s > 1 is competitive with most Bi-CG based methods. For these reasons, we extend the IDR( s ) to solve large nonsymmetric linear systems with multiple right-hand sides. In this paper, a variant of the IDR theorem is given at first, then the block IDR( s ), an extension of IDR( s ) based on the variant IDR( s ) theorem, is proposed. By analysis, the upper bound on the number of matrix-vector products of block IDR( s ) is the same as that of the IDR( s ) for a single right-hand side in generic case, i.e., the total number of matrix-vector products of IDR( s ) may be m times that of of block IDR( s ), where m is the number of right-hand sides. Numerical experiments are presented to show the effectiveness of our proposed method.

Multigrid algorithm from cyclic reduction for Markovian queueing networks

A multigrid method based on cyclic reduction strategy is proposed to solve huge, nonsymmetric singular linear systems arising from Markovian queueing networks. A simple way to construct the matrix-dependent prolongation and restriction operators is presented in this paper. Numerical results for multiple queues are given to illustrate the efficiency and robustness of our methods.

Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations

This paper deals with boundary-value methods (BVMs) for ordinary and neutral differential-algebraic equations. Different from what has been done in Lei and Jin (Lecture Notes in Computer Science, vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large-sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang-type block-circulant preconditioners are suggested: one is for ordinary differential-algebraic equations (ODAEs), and the other is for neutral differential-algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.

Minimizing synchronization in IDR (s )

IDR (s) is a family of fast algorithms for iteratively solving large nonsymmetric linear systems. With cluster computing and in particular with Grid computing, the inner product is a bottleneck operation. In this paper, three techniques are investigated for alleviating this bottleneck. First, a recently proposed IDR (s) algorithm that is highly efficient and stable is reformulated in such a way that it has a single global synchronization point per iteration step. Second, the so-called test matrix is chosen so that the work, communication, and storage involving this matrix is minimized in multi-cluster environments. Finally, a methodology is presented for a-priori estimation of the optimal value of s using only problem and machine-based parameters. Numerical experiments applied to a 3D convection–diffusion problem are performed on the DAS-3 Grid computer, demonstrating the effectiveness of our approach. Copyright © 2011 John Wiley & Sons, Ltd.

Quasi-minimal residual smoothing technique for the IDR($s$) method

The IDR( s) proposed by Sonneveld and Gijzen is an efficient method for solving large nonsymmetric linear systems. In this paper, QMRIDR( s), a new variant of the IDR( s) method is presented. In this method, the irregular convergence behavior of IDR( s) is remedied and both the fast and smooth convergence behaviors are expected. Numerical experiments are reported to show the performance of our method.

The BiCOR and CORS Iterative Algorithms for Solving Nonsymmetric Linear Systems

We present two iterative algorithms for solving real nonsymmetric and complex non-Hermitian linear systems of equations and that were developed from variants of the nonsymmetric Lanczos method. In this paper, we give the theoretical background of the two iterative methods and discuss their main computational aspects. Using a large number of numerical experiments, we analyze their convergence properties, and we also compare them with other popular nonsymmetric iterative solvers in use today.

On the convergence of the Bl-LSQR algorithm for solving matrix equations

Karimi and Toutounian [The block least squares method for solving nonsymmetric linear system with multiple right-hand sides, Appl. Math. Comput. 177 (2006), pp. 852–862], proposed a block version of the LSQR algorithm, say Bl-LSQR, for solving linear system of equations with multiple right-hand sides. In this paper, the convergence of the Bl-LSQR algorithm is studied. We deal with some computational aspects of the Bl-LSQR algorithm for solving matrix equations. Some numerical experiments on test matrices are presented.

Estimates of the Norm of the Error in Solving Linear Systems with FOM and GMRES

We provide formulas for the norm of the error when solving nonsymmetric linear systems with the full orthogonalization method (FOM) and the generalized minimum residual method (GMRES) as well as relations between the error norm and the residual norm. From these formulas we are able to compute estimates of the norm of the error during the iterations. Since stopping criteria based on the norm of the residual may sometimes be misleading, such estimates could lead to a more robust way to stop the iterations. Numerical experiments show that the proposed norm estimates work nicely on difficult linear systems.

Preconditioner updates for solving sequences of linear systems in matrix-free environment

We present two new ways of preconditioning sequences of nonsymmetric linear systems in the special case where the implementation is matrix free. Both approaches are fully algebraic, they are based on the general updates of incomplete LU decompositions recently introduced in (SIAM J. Sci. Comput. 2007; 29(5):1918–1941), and they may be directly embedded into nonlinear algebraic solvers. The first of the approaches uses a new model of partial matrix estimation to compute the updates. The second approach exploits separability of function components to apply the updated factorized preconditioner via function evaluations with the discretized operator. Experiments with matrix-free implementations of test problems show that both new techniques offer useful, robust and black-box solution strategies. In addition, they show that the new techniques are often more efficient in matrix-free environment than either recomputing the preconditioner from scratch for every linear system of the sequence or than freezing the preconditioner throughout the whole sequence. Copyright © 2010 John Wiley & Sons, Ltd.