Preconditioned Krylov Subspace Methods Research Articles

A note on preconditioners for complex linear systems arising from PDE-constrained optimization problems

In this note, the block-diagonal preconditioner proposed and the block triangular proposed in Krendl et al. (2013) and Pearson and Wathen (2012), respectively, are further studied and optimized. Two improved preconditioners are proposed for solving a class of complex linear systems arising from optimal control with time-periodic parabolic equation. Theoretical analyses show that the eigenvalues of the improved block-diagonal and block triangular preconditioned matrices are located in [ − 1 , − 2 / 2 ) ∪ ( 2 / 2 , 1 ] and ( 1 / 2 , 1 ] , respectively. Numerical experiments illustrate the feasibility and effectiveness of these preconditioners for Krylov subspace methods.

Fast preconditioned iterative methods for finite volume discretization of steady-state space-fractional diffusion equations

We consider the preconditioned Krylov subspace method for linear systems arising from the finite volume discretization method of steady-state variable-coefficient conservative space-fractional diffusion equations. We propose to use a scaled-circulant preconditioner to deal with such Toeplitz-like discretization matrices. We show that the difference between the scaled-circulant preconditioner and the coefficient matrix is equal to the sum of a small-norm matrix and a low-rank matrix. Numerical tests are conducted to show the effectiveness of the proposed method for one- and two-dimensional steady-state space-fractional diffusion equations and demonstrate that the preconditioned Krylov subspace method converges very quickly.

A full-space barrier method for stress-constrained discrete material design optimization

In this paper, we present a full-space barrier method designed for stress-constrained mass minimization problems with discrete material options. The advantages of the full-space barrier method are twofold. First, in the full-space the stress constraints are provably concave, which facilitates the construction of convex subproblems within the optimization algorithm. Second, by using the full-space, it is no longer necessary to employ stress constraint aggregation techniques to reduce adjoint-gradient evaluation costs. The proposed optimization algorithm uses a Newton method where an approximate linearization of the KKT conditions is solved inexactly at each iteration using a preconditioned Krylov subspace method. Sparse constraints that arise in the discrete material parametrization are treated using a null-space method. Results of the proposed algorithm are demonstrated on a series of three topology and multimaterial optimization problems with selection between isotropic and orthotropic materials, as well as discrete ply-angle selection.

Sparse approximate inverse preconditioners on high performance GPU platforms

Simulation with models based on partial differential equations often requires the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present here some experiences focused on the approximate inverse preconditioners proposed by Benzi and Tůma from 1996 and the sparsification and inversion proposed by van Duin in 1999. Computational costs, reorderings and implementation issues are considered both on conventional and innovative computing architectures like Graphics Programming Units (GPUs).

An iterative method for solving the continuous sylvester equation by emphasizing on the skew-hermitian parts of the coefficient matrices

ABSTRACTWe present an iterative method based on the Hermitian and skew-Hermitian splitting (HSS) for solving the continuous Sylvester equation. By using the HSS of the coefficient matrices A and B, we establish a method which is practically inner/outer iterations, by employing a conjugate gradient on the normal equations (CGNR)-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrices. Via this method, a Sylvester equation with coefficient matrices and (which are the skew-Hermitian part of A and B, respectively) is solved iteratively by a CGNR-like method. Convergence conditions of this method are studied and numerical examples show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods.

辺有限要素法による電磁界解析におけるブロックマルチカラーオーダリングを援用した並列化前処理付きMRTR法に基づく線形方程式求解の高速化

To realize fast electromagnetic field analysis, the parallelization technique has been often introduced into the preconditioned Krylov subspace method. When the multicolor ordering is applied to parallelization of forward and backward substitution, the elapsed time of matrix calculation might increase owing to the increment of bandwidth. Therefore, the block-multicolor ordering based on the level structure arising in reverse Cuthill-McKee ordering has been developed. The validity of developed method was demonstrated on the parallelized incomplete-Cholesky-preconditioned conjugate gradient (ICCG) method. In this paper, the parallelization performance of preconditioned minimized residual method based on the three-term recurrence formula of the CG-type (MRTR) method supported by developed ordering is investigated. Furthermore, the affinity of developed ordering or cache-cache elements technique for parallelized forward and backward substitution in Eisenstat's technique is particularly examined.

Deflation by restriction for the inverse-free preconditioned Krylov subspace method

A deflation by restriction scheme is developed for the inverse-free preconditioned Krylov subspace method for computing a few extreme eigenvalues of the definite symmetric generalized eigenvalue problem $Ax = \lambda Bx$. The convergence theory for the inverse-free preconditioned Krylov subspace method is generalized to include this deflation scheme and numerical examples are presented to demonstrate the convergence properties of the algorithm with the deflation scheme.

Fast Iterative Solvers for Linear Systems Arising from Time-Dependent Space-Fractional Diffusion Equations

In this paper, we study the linear systems arising from the discretization of time-dependent space-fractional diffusion equations. By using a finite difference discretization scheme for the time derivative and a finite volume discretization scheme for the space-fractional derivative, Toeplitz-like linear systems are obtained. We propose using the approximate inverse-circulant preconditioner to deal with such Toeplitz-like matrices, and we show that the spectra of the corresponding preconditioned matrices are clustered around 1. Experimental results on time-dependent and space-fractional diffusion equations are presented to demonstrate that the preconditioned Krylov subspace methods converge very quickly.

Study on performance of Krylov subspace methods at solving large-scale contact problems

This paper discusses the performance of preconditioned Krylov subspace methods at solving large-scale finite-element-formulated contact problems based on a finite element analysis program (FEAP) and portable, extensible toolkit for scientific computation (PETSc). Different combinations of the common preconditioners and Krylov subspace methods, namely conjugate gradient, generalized minimal residual, conjugate residual, biconjugate gradient, conjugate gradient square, biconjugate gradient stabilized, and transpose-free quasi-minimal residual method, are tested on the problems with rising size. Numerical experiments are carried out in respect of time consumption, parallelism, memory usage, and stability. The results show time consumption for most of combinations increases linearly with the growth of problem size, and block Jacobi and Jacobi-preconditioned conjugate gradient and conjugate residual methods have better potential for large-scale problems in the given and similar case.

Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver

We focus on robust and efficient iterative solvers for the pressure Poisson equation in incompressible Navier–Stokes problems. Preconditioned Krylov subspace methods are popular for these problems, with BiCGStab and GMRES(m) most frequently used for nonsymmetric systems. BiCGStab is popular because it has cheap iterations, but it may fail for stiff problems, especially early on as the initial guess is far from the solution. Restarted GMRES is better, more robust, in this phase, but restarting may lead to very slow convergence. Therefore, we evaluate the rGCROT method for these systems. This method recycles a selected subspace of the search space (called recycle space) after a restart. This generally improves the convergence drastically compared with GMRES(m). Recycling subspaces is also advantageous for subsequent linear systems, if the matrix changes slowly or is constant. However, rGCROT iterations are still expensive in memory and computation time compared with those of BiCGStab. Hence, we propose a new, hybrid approach that combines the cheap iterations of BiCGStab with the robustness of rGCROT. For the first few time steps the algorithm uses rGCROT and builds an effective recycle space, and then it recycles that space in the rBiCGStab solver.We evaluate rGCROT on a turbulent channel flow problem, and we evaluate both rGCROT and the new, hybrid combination of rGCROT and rBiCGStab on a porous medium flow problem. We see substantial performance gains for both the problems.

Parallel iterative solution of the incompressible Navier–Stokes equations with application to rotating wings

We discuss aspects of implementation and performance of parallel iterative solution techniques applied to low Reynolds number flows around fixed and moving rigid bodies. The incompressible Navier–Stokes equations are discretised with Taylor-Hood finite elements in combination with a semi-implicit pressure-correction method. The resulting sequence of convection–diffusion and Poisson equations are solved with preconditioned Krylov subspace methods. To achieve overall scalability we consider new auxiliary algorithms for mesh handling and assembly of the system matrices. We compute the flow around a translating plate and a rotating insect wing to establish the scaling properties of the developed solver. The largest meshes have up to 132 × 106 hexahedral finite elements leading to around 3.3 × 109 unknowns. For the scalability runs the maximum core count is around 65.5 × 103. We find that almost perfect scaling can be achieved with a suitable Krylov subspace iterative method, like conjugate gradients or GMRES, and a block Jacobi preconditioner with incomplete LU factorisation as a subdomain solver. In addition to parallel performance data, we provide new highly-resolved computations of flow around a rotating insect wing and examine its vortex structure and aerodynamic loading.

A fast method for a generalized nonlocal elastic model

We develop a numerical method for a generalized nonlocal elastic model, which is expressed as a composition of a Riesz potential operator with a fractional differential operator, by composing a collocation method with a finite difference discretization.By carefully exploring the structure of the coefficient matrix of the numerical method, we develop a preconditioned fast Krylov subspace method, which reduces the computations to (Nlog⁡N) per iteration and the memory to O(N). The use of the preconditioner significantly reduces the number of iterations, and the preconditioner can be inverted in O(Nlog⁡N) computations. Numerical results show the utility of the method.

A block alternating splitting iteration method for a class of block two-by-two complex linear systems

Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering. In this paper, we focus on solving a class of block two-by-two complex linear systems arising from the distributed optimal control with time-periodic parabolic equations. A new block alternating splitting (BAS) iteration method is presented for solving the class of complex linear systems. The convergence theory and the spectral properties of the BAS iteration method are discussed. Numerical experiments are presented to illustrate the efficiency of the BAS iteration as a solver as well as a preconditioner for Krylov subspace methods.

Speedup of Iterative Solver for Electromagnetic Analysis Using Many Integrated Core Architecture

Speedup of Krylov subspace method for the linear system obtained from an electromagnetic analysis using Many Integrated Core (MIC) architecture is investigated. In recent years, MIC architecture appears on the scene of high-performance computing research fields. Since Xeon Phi coprocessor is x86 architecture, the ordinal numerical code can be implemented on the devices without transcribing. In this paper, the linear systems obtained by extended element-free Galerkin method and edge element are solved by mixed precision variable preconditioned (VP) Krylov subspace method, and the methods are implemented on MIC to get high performance. The results of computation show that speedup of VP generalized minimal residual on MIC is three times faster than that of sequential calculation. In addition, the speedup of VP conjugate gradient (CG) is 40.47 times faster than that of sequential CG method.

Analysis of Schwarz Methods for a Hybridizable Discontinuous Galerkin Discretization

Schwarz methods are attractive parallel solvers for large scale linear systems obtained when partial differential equations are discretized. For hybridizable discontinuous Galerkin (HDG) methods, this is a relatively new field of research, because HDG methods impose continuity across elements using a Robin condition, while classical Schwarz solvers use Dirichlet transmission conditions. Robin conditions are used in optimized Schwarz methods to get faster convergence compared to classical Schwarz methods, and this even without overlap, when the Robin parameter is well chosen. We present in this paper a rigorous convergence analysis of Schwarz methods for the concrete case of hybridizable interior penalty (IPH) method. We show that the penalization parameter needed for convergence of IPH leads to slow convergence of the classical additive Schwarz method, and propose a modified solver which leads to much faster convergence. Our analysis is entirely at the discrete level, and thus holds for arbitrary interfaces between two subdomains. We then generalize the method to the case of many subdomains, including cross points, and obtain a new class of preconditioners for Krylov subspace methods which exhibit better convergence properties than the classical additive Schwarz preconditioner. We illustrate our results with numerical experiments.

Convergence analysis of the two preconditioned iterative methods for [formula omitted]-matrix linear systems

In this paper, we present the two preconditioners I+Sˆ and I+Sˆ+R for solving M-matrix linear systems and discuss the convergence of the two preconditioned iterative methods. Meanwhile, we obtain comparison theorems between the two preconditioned iterative methods and consider the solution of M-matrix linear systems by preconditioned Krylov subspace methods. Numerical experiments are given to validate the performance of the preconditioners.

Convergence of relaxation iterative methods for saddle point problem

In this paper, we first provide convergence results of three relaxation iterative methods for solving saddle point problem. Next, we propose how to find near optimal parameters for which preconditioned Krylov subspace method performs nearly best when the relaxation iterative methods are applied to the preconditioners of Krylov subspace method. Lastly, we provide efficient implementation for the relaxation iterative methods and efficient computation for the preconditioner solvers. Numerical experiments show that the MIAOR method and the BiCGSTAB with MAOR preconditioner using near optimal parameters perform more than twice faster than the GSOR method.

Generalized Kronecker product splitting iteration for the solution of implicit Runge–Kutta and boundary value methods

SummaryThis paper is concerned with a generalization of the Kronecker product splitting (KPS) iteration for solving linear systems arising in implicit Runge–Kutta and boundary value methods discretizations of ordinary differential equations. It is shown that the new scheme can outperform the standard KPS method in some situations and can be used as an effective preconditioner for Krylov subspace methods. Numerical experiments are presented to demonstrate the effectiveness of the methods. Copyright © 2014 John Wiley & Sons, Ltd.

An efficient preconditioned Krylov subspace method for large-scale finite element equations with MPC using Lagrange multiplier method

Purpose– The purpose of this paper is to propose an efficient iterative method for large-scale finite element equations of bad numerical stability arising from deformation analysis with multi-point constraint using Lagrange multiplier method.Design/methodology/approach– In this paper, taking warpage analysis of polymer injection molding based on surface model as an example, the performance of several popular Krylov subspace methods, including conjugate gradient, BiCGSTAB and generalized minimal residual (GMRES), with diffident Incomplete LU (ILU)-type preconditions is investigated and compared. For controlling memory usage, GMRES(m) is also considered. And the ordering technique, commonly used in the direct method, is introduced into the presented iterative method to improve the preconditioner.Findings– It is found that the proposed preconditioned GMRES method is robust and effective for solving problems considered in this paper, and approximate minimum degree (AMD) ordering is most beneficial for the reduction of fill-ins in the ILU preconditioner and acceleration of the convergence, especially for relatively accurate ILU-type preconditioning. And because of concerns about memory usage, GMRES(m) is a good choice if necessary.Originality/value– In this paper, for overcoming difficulties of bad numerical stability resulting from Lagrange multiplier method, together with increasing scale of problems in engineering applications and limited hardware conditions of computer, a stable and efficient preconditioned iterative method is proposed for practical purpose. Before the preconditioning, AMD reordering, commonly used in the direct method, is introduced to improve the preconditioner. The numerical experiments show the good performance of the proposed iterative method for practical cases, which is implemented in in-house and commercial codes on PC.

Preconditioned Hss-Like Iterative Method for Saddle Point Problems

A new HSS-like iterative method is first proposed based on HSS-like splitting of nonHermitian (1,1) block for solving saddle point problems. The convergence analysis for the new method is given. Meanwhile, we consider the solution of saddle point systems by preconditioned Krylov subspace method and discuss some spectral properties of the preconditioned saddle point matrices. Numerical experiments are given to validate the performances of the preconditioners. Mathematics subject classification: 65F10, 65F50.