Krylov Subspace Iterative Methods Research Articles

Block-triangular preconditioning methods for linear third-order ordinary differential equations based on reduced-order sinc discretizations

By applying the reduced-order sinc discretization to the two-point boundary value problem of a linear third-order ordinary differential equation, we can obtain a block two-by-two system of linear equations, with each block of its coefficient matrix being a combination of Toeplitz and diagonal matrices. This class of linear systems can be effectively solved by Krylov subspace iteration methods such as GMRES and BiCGSTAB. We construct block-triangular preconditioning matrices to accelerate the convergence rates of the Krylov subspace iteration methods, and demonstrate that the eigenvalues of certain approximations to the preconditioned matrices are uniformly bounded within a rectangle, being independent of the size of the discrete linear system, on the complex plane. In addition, we use numerical examples to show the effectiveness of the proposed preconditioning methods.

Parallel Newton–Krylov–Schur Flow Solver for the Navier–Stokes Equations

The objective of the present paper is to demonstrate the effectiveness of a spatial discretization based on summation-by-parts operators with simultaneous approximation terms in combination with a parallel Newton–Krylov–Schur algorithm for solving the three-dimensional Reynolds-averaged Navier–Stokes equations coupled with the Spalart–Allmaras one-equation turbulence model. The algorithm employs second-order summation-by-parts operators on multiblock structured grids with simultaneous approximation terms to enforce block interface coupling and boundary conditions. The discrete equations are solved iteratively with an inexact-Newton method, while the linear system at each Newton iteration is solved using a flexible Krylov subspace iterative method with an approximate-Schur parallel preconditioner. The algorithm is verified and validated through the solution of two-dimensional model problems, highlighting the correspondence of the current algorithm with several established flow solvers. A transonic solution over the ONERA M6 wing on a mesh with 15.1 million nodes shows good agreement with experiment. Using 128 processors, the residual is reduced by 12 orders of magnitude in 86 min. The solution of transonic flow over the common research model wing–body geometry exhibits the expected grid convergence behavior. The algorithm performs well in solving flows around nonplanar geometries and flows with explicitly specified laminar-to-turbulent transition locations. Parallel scaling studies highlight the excellent scaling characteristics of the algorithm on cases with up to 6656 processors and grids with over 150 million nodes.

Parallelization Development Method and Realization for a FEM Simulation Program

The parallel computation of equation sets solution are crucial for finite element analysis. The paper discussed the key technology of FEM parallelization and the parallelization strategy for the FEM program is given; and then discussed the Data structure of Aztec and how to call Aztec which consists of Krylov subspace iterative methods solvers and preconditioners; finally, the realization and testing of the Aztec-based finite element program parallelization was put forward. Test results show that there are high efficiency of this method which make the SapTis software more powerful, flexible and adaptable.

Rotated block triangular preconditioning based on PMHSS

Based on the PMHSS preconditioning matrix, we construct a class of rotated block triangular preconditioners for block two-by-two matrices of real square blocks, and analyze the eigen-properties of the corresponding preconditioned matrices. Numerical experiments show that these rotated block triangular preconditioners can be competitive to and even more efficient than the PMHSS preconditioner when they are used to accelerate Krylov subspace iteration methods for solving block two-by-two linear systems with coefficient matrices possibly of nonsymmetric sub-blocks.

Robust partitioned block preconditioners for large-scale geotechnical applications with soil-structure interactions

SUMMARY Soil–structure interaction problems are commonly encountered in geotechnical practice and remarkably characterized with significant material stiffness contrast. When solving the soil–structure interaction problems, the employed Krylov subspace iterative method may converge slowly or even fail, indicating that the adopted preconditioning method may not suit for such problems. The inexact block diagonal preconditioners proposed recently have been shown effective for the soil–structure interaction problems; however, they haven't been exploited to full capabilities. By using the same partition strategy according to the structure elements and soil elements, the partitioned block symmetric successive over-relaxation preconditioners or partitioned block constraint preconditioners are proposed. Based on two pile-group foundation problems and a tunnel problem, the proposed preconditioners are evaluated and compared with the available preconditioners for the consolidation analysis and the drained analysis, respectively. In spite of one additional solve associated with the structure block and multiplications with off-diagonal blocks in the preconditioning step, numerical results reveal that the proposed preconditioners obviously possess better performance than the recently developed inexact block preconditioners. Copyright © 2013 John Wiley & Sons, Ltd.

GPU-accelerated iterative solutions for finite element analysis of soil–structure interaction problems

Soil–structure interaction problems are commonly encountered in engineering practice, and the resulting linear systems of equations are difficult to solve due to the significant material stiffness contrast. In this study, a novel partitioned block preconditioner in conjunction with the Krylov subspace iterative method symmetric quasiminimal residual is proposed to solve such linear equations. The performance of these investigated preconditioners is evaluated and compared on both the CPU architecture and the hybrid CPU–graphics processing units (GPU) computing environment. On the hybrid CPU–GPU computing platform, the capability of GPU in parallel implementation and high-intensity floating point operations is exploited to accelerate the iterative solutions, and particular attention is paid to the matrix–vector multiplications involved in the iterative process. Based on a pile-group foundation example and a tunneling example, numerical results show that the partitioned block preconditioners investigated are very efficient for the soil–structure interaction problems. However, their comparative performances may apparently depend on the computer architecture. When the CPU computer architecture is used, the novel partitioned block symmetric successive over-relaxation preconditioner appears to be the most efficient, but when the hybrid CPU–GPU computer architecture is adopted, it is shown that the inexact block diagonal preconditioners embedded with simple diagonal approximation to the soil block outperform the others.

Exploring large macromolecular functional motions on clusters of multicore processors

Normal modes in internal coordinates (IC) furnish an excellent way to model functional collective motions of macromolecular machines, but exhibit a high computational cost when applied to large-sized macromolecules. In this paper, we significantly extend the applicability of this approach towards much larger systems by effectively solving the computational bottleneck of these methods, the diagonalization step and associated large-scale eigenproblem, on a small cluster of nodes equipped with multicore technology. Our experiments show the superior performance of iterative Krylov-subspace methods for the solution of the dense generalized eigenproblems arising in these biological applications over more traditional direct solvers implemented on top of state-of-the-art libraries. The presented approach expedites the study of the collective conformational changes of large macromolecules opening a new window for exploring the functional motions of such relevant systems.

Communication Avoiding ILU0 Preconditioner

In this paper we present a communication avoiding ILU0 preconditioner for solving large linear systems of equations by using iterative Krylov subspace methods. Recent research has focused on communication avoiding Krylov subspace methods based on so-called s-step methods. However, there are not many communication avoiding preconditioners yet, and this represents a serious limitation of these methods. Our preconditioner allows us to perform $s$ iterations of the iterative method with no communication, through ghosting some of the input data and performing redundant computation. To avoid communication, an alternating reordering algorithm is introduced for structured and well partitioned unstructured matrices, which requires the input matrix to be ordered by using a graph partitioning technique such as k-way or nested dissection. We show that the reordering does not affect the convergence rate of the ILU0 preconditioned system as compared to k-way or nested dissection ordering, while it reduces data movement and is expected to reduce the time needed to solve a linear system. In addition to communication avoiding Krylov subspace methods, our preconditioner can be used with classical methods such as GMRES to reduce communication.

Online-ABFT

Soft errors are one-time events that corrupt the state of a computing system but not its overall functionality. Large supercomputers are especially susceptible to soft errors because of their large number of components. Soft errors can generally be detected offline through the comparison of the final computation results of two duplicated computations, but this approach often introduces significant overhead. This paper presents Online-ABFT, a simple but efficient online soft error detection technique that can detect soft errors in the widely used Krylov subspace iterative methods in the middle of the program execution so that the computation efficiency can be improved through the termination of the corrupted computation in a timely manner soon after a soft error occurs. Based on a simple verification of orthogonality and residual, Online-ABFT is easy to implement and highly efficient. Experimental results demonstrate that, when this online error detection approach is used together with checkpointing, it improves the time to obtain correct results by up to several orders of magnitude over the traditional offline approach.

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

For a class of block two-by-two systems of linear equations with certain skew-Hamiltonian coefficient matrices, we construct additive block diagonal preconditioning matrices and discuss the eigen-properties of the corresponding preconditioned matrices. The additive block diagonal preconditioners can be employed to accelerate the convergence rates of Krylov subspace iteration methods such as MINRES and GMRES. Numerical experiments show that MINRES preconditioned by the exact and the inexact additive block diagonal preconditioners are effective, robust and scalable solvers for the block two-by-two linear systems arising from the Galerkin finite-element discretizations of a class of distributed control problems.

On order-reducible sinc discretizations and block-diagonal preconditioning methods for linear third-order ordinary differential equations

By introducing a variable substitution, we transform the two-point boundary value problem of a third-order ordinary differential equation into a system of two second-order ordinary differential equations (ODEs). We discretize this order-reduced system of ODEs by both sinc-collocation and sinc-Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order-reduced system of ODEs. The coefficient matrix of the linear system is of block two-by-two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block-diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.

Improving the Communication Pattern in Matrix-Vector Operations for Large Scale-Free Graphs by Disaggregation

Matrix-vector multiplication is the key operation in any Krylov-subspace iteration method. We are interested in Krylov methods applied to problems associated with the graph Laplacian arising from large scale-free graphs. Computations with graphs of this type on parallel distributed-memory computers are challenging. This is due to the fact that scale-free graphs have a degree distribution that follows a power law, and currently available graph partitioners are not efficient for such an irregular degree distribution. The lack of a good partitioning leads to excessive interprocessor communication requirements during every matrix-vector product. We present an approach to alleviate this problem based on embedding the original irregular graph into a more regular one by disaggregating (splitting up) vertices in the original graph. The matrix-vector operations for the original graph are performed via a factored triple matrix-vector product involving the embedding graph. Even though the latter graph is larger, we ...

Hierarchical Preconditioning for PTAP‐Systems arising from particulate flows

AbstractParticulate flows, i.e. flow of an incompressible, Newtonian carrier fluid loaded with (many) rigid bodies, play an important role in diverse technical applications. In [1] a finite element method was introduced to simulate such flows. The method relies on a splitting method and a subspace projection method to incorporate the rigid body motion of the particles in a so called one domain approach [2]. The resulting systems arising after discretization are ill conditioned in general. Thus preconditioning is mandatory for instance when using an iterative Krylov subspace methods. In this paper we present a Schur complement based preconditioner well suited for this type of application. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Comparative study on order-reduced methods for linear third-order ordinary differential equations

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.

Алгоритмы решения СЛАУ на системах с распределенной памятью в применении к задачам электромагнетизма

Paper presents various aspects of harmonic electromagnetic fields simulation on clusters. The major computational complexity comes from the solution of the systems of linear algebraic equations (SLAEs) arising from the approximations of corresponding electromagnetic boundary value problems by Nedelec elements of various orders. Effective and efficient approaches to the decomposition of the computational domain and the matrix of the system are considered. Distributed SLAEs are solved using iterative Krylov subspace methods preconditioned by additive Schwarz method. In order to increase the effectiveness of the algorithms iterations are performed in the trace space. Implementation of the solvers is based on MPI for data transfers. The solution of the systems in subdomains is performed by PARDISO direct solver from IntelR MKL library. Numerical experiments results on a series of model and real-life problems show the effectiveness of the presented algorithms.

An implicit meshless method for application in computational fluid dynamics

SUMMARYAn implicit meshless scheme is developed for solving the Euler equations, as well as the laminar and Reynolds‐averaged Navier–Stokes equations. Spatial derivatives are approximated using a least squares method on clouds of points. The system of equations is linearised, and solved implicitly using approximate, analytical Jacobian matrices and a preconditioned Krylov subspace iterative method. The details of the spatial discretisation, linear solver and construction of the Jacobian matrix are discussed; and results that demonstrate the performance of the scheme are presented for steady and unsteady two dimensional fluid flows. Copyright © 2012 John Wiley & Sons, Ltd.

Solving dense generalized eigenproblems on multi-threaded architectures

We compare two approaches to compute a fraction of the spectrum of dense symmetric definite generalized eigenproblems: one is based on the reduction to tridiagonal form, and the other on the Krylov-subspace iteration. Two large-scale applications, arising in molecular dynamics and material science, are employed to investigate the contributions of the application, architecture, and parallelism of the method to the performance of the solvers. The experimental results on a state-of-the-art 8-core platform, equipped with a graphics processing unit (GPU), reveal that in realistic applications, iterative Krylov-subspace methods can be a competitive approach also for the solution of dense problems.

Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems

We construct a preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration scheme for solving and preconditioning a class of block two-by-two linear systems arising from the Galerkin finite element discretizations of a class of distributed control problems. The convergence theory of this class of PMHSS iteration methods is established and the spectral properties of the PMHSS-preconditioned matrix are analysed. Numerical experiments show that the PMHSS preconditioners can be quite competitive when used to precondition Krylov subspace iteration methods such as GMRES.

On iterative techniques for computing flow in large two-dimensional discrete fracture networks

Computation of flow in discrete fracture networks often involves solving for hydraulic head values at all intersection points of a large number of stochastically generated fractures inside a bounded domain. For large systems, this approach leads to the generation of problems involving highly sparse matrices which must be solved iteratively. Distributions of fracture lengths spanning over several orders of magnitude, and the randomness of fracture orientations and locations, lead to coefficient matrices that are devoid of any regular structure in the sparsity pattern. In addition to the rapid increase in computational effort with increase in the size of the fracture network, the spread in the distribution of fracture parameters, such as length and transmissivity, dramatically influences the convergence behavior of the system of linear equations. An overview of the discrete fracture network (DFN) methodology for computation of flow is presented along with a comparative study of various Krylov subspace iterative methods for the resulting class of sparse matrices. The rate of convergence of the iterative techniques is found to exhibit a systematic pattern with respect to changes in statistical parameters of the stochastically generated fracture networks. Salient features of the observed trends in the convergence pattern are discussed and guidelines for design of DFN algorithms are provided.

Iterative schemes for high order compact discretizations to the exterior Helmholtz equation

We consider high order finite difference approximations to the Helmholtz equation in an exterior domain. We include a simplified absorbing boundary condition to approximate the Sommerfeld radiation condition. This yields a large, but sparse, complex system, which is not self-adjoint and not positive definite. We discretize the equation with a compact fourth or sixth order accurate scheme. We solve this large system of linear equations with a Krylov subspace iterative method. Since the method converges slowly, a preconditioner is introduced, which is a Helmholtz equation but with a modified complex wavenumber. This is discretized by a second or fourth order compact scheme. The system is solved by BICGSTAB with multigrid used for the preconditioner. We study, both by Fourier analysis and computations this preconditioned system especially for the effects of high order discretizations.