A Comparison of Subspace Methods for Sylvester Equations

Implicit schemes are a popular approach to the discretization of linear partial differential equations by finite differences. They require to solve a linear set of equations in each time step. Since finite difference discretizations lead to a local coupling, these systems of equations are sparse and can be effectively solved by iterative procedures. Numerical procedures of this type are known in control theory as repetitive processes. They have mostly been used to describe control algorithms for processes where passes of finite length are repeated over and over. This contribution shows how the implicit discretization of partial differential equations can be cast into the framework of repetitive processes. Thus it establishes a link between yet unrelated results in numerical mathematics and control theory.

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations.

Most of researches for large-scale parallel structural analysis using finite element method have focused on iterative solution methods since direct solution methods generally have many difficulties and disadvantages for large-scale problems. However, due to the numerical robustness of direct methods that guarantees the solution to be obtained within estimated time, direct methods are much more desirable for general application of large-scale structural analysis, as well as due to ease of use which makes most of structural engineers prefer direct methods. In this research, we propose a parallel direct solver for large-scale finite element analysis that can overcome most of difficulties and disadvantages of direct methods by using domain-wise approach for direct methods. By using the proposed solver with our own structural analysis code, we can show good scalability as a direct method and can solve the largest problem ever solved by direct solvers.

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

Hardware trends have motivated the development of mixed precision algorithms in numerical linear algebra, which aim to decrease runtime while maintaining acceptable accuracy. One recent development is the development of an adaptive precision sparse matrix–vector produce routine, which may be used to accelerate the solution of sparse linear systems by iterative methods. This approach is also applicable to the application of inexact preconditioners, such as sparse approximate inverse preconditioners used in Krylov subspace methods. In this work, we develop an adaptive precision sparse approximate inverse preconditioner and demonstrate its use within a five-precision GMRES-based iterative refinement method. We call this algorithm variant BSPAI-GMRES-IR. We then analyze the conditions for the convergence of BSPAI-GMRES-IR, and determine settings under which BSPAI-GMRES-IR will produce similar backward and forward errors as the existing SPAI-GMRES-IR method, the latter of which does not use adaptive precision in preconditioning. Our numerical experiments show that this approach can potentially lead to a reduction in the cost of storing and applying sparse approximate inverse preconditioners, although a significant reduction in cost may comes at the expense of increasing the number of GMRES iterations required for convergence.

Stochastic Galerkin finite element discretizations of partial differential\nequations with coefficients characterized by arbitrary distributions lead, in\ngeneral, to fully block dense linear systems. We propose two novel strategies\nfor constructing preconditioners for these systems to be used with Krylov\nsubspace iterative solvers. In particular, we present a variation on of the\nhierarchical Schur complement preconditioner, developed recently by the\nauthors, and an adaptation of the symmetric block Gauss-Seidel method. Both\npreconditioners take advantage of the hierarchical structure of global\nstochastic Galerkin matrices, and also, when applicable, of the decay of the\nnorms of the stiffness matrices obtained from the polynomial chaos expansion of\nthe coefficients. This decay allows to truncate the matrix-vector\nmultiplications in the action of the preconditioners. Also, throughout the\nglobal matrix hierarchy, we approximate solves with certain submatrices by the\nassociated diagonal block solves. The preconditioners thus require only a\nlimited number of stiffness matrices obtained from the polynomial chaos\nexpansion of the coefficients, and a preconditioner for the diagonal blocks of\nthe global matrix. The performance is illustrated by numerical experiments.\n

Partial tensor decomposition for decoupling isogeometric Galerkin discretizations

Solution of a large sparse or dense linear systems often constitutes the most time consuming step when solving large multidimensional industrial problems on modern high performance supercomputers. In most practical cases these problems produce linear systems with ill-conditioned and off-diagonal dominant coefficient matrices. As the problem size grows, especially in the 3-dimensional case, direct methods for solving these systems (such as triangular decomposition) begin to lose their effectiveness due to the large arithmetic costs and memory requirements. These difficulties motivate resorting to iterative solution methods which are, at least theoretically, more efficient with respect to the arithmetic cost and memory requirements. On March 12, 1993 Qasim Sheikh from Cray Research presented numerical experiments with iterative solvers that are capable of solving large linear systems which arise in industrial applications. He then compared the effectiveness and some performance aspects of these iterative methods with other state of the art, highly optimized, iterative as well as direct solvers. The main results of his talk are based on a recent paper [1] written jointly with Kharchenko, Nikishin, Yeremin and Heroux.KeywordsIterative SolverPreconditioned MatrixLarge Linear SystemNonsymmetric Linear SystemTurbine PumpThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in. The systems that have efficient compression of the fill-in mostly arise from discretization of partial differential equations. We show that the resulting factorization can be used as an efficient preconditioner and compare the proposed approach with state-of-art direct and iterative solvers.

Simulators of semiconductor devices have to solve systems of equations generated by the discretization of partial differential equations, which are the most time-consuming part of the simulation process. Therefore, the use of an effective method to solve these linear systems is essential. In this work we have evaluated the efficiency of different parallel direct and iterative solvers used for the solution of the drift–diffusion equations in semiconductor device simulation. Several preconditioning techniques have been applied in order to minimize the execution times. We have found that FGMRES and BCGSTAB solvers preconditioned with Additive Schwarz are the most suitable for these types of problems. The results were obtained in an HP Superdome cluster with 128 Itanium2 1.5 GHz. Copyright © 2005 John Wiley & Sons, Ltd.

A new iteration method with cubic convergence to solve nonlinear algebraic equations

After discretization of the partial differential equations from mechanics, one usually obtains large systems of (non)linear equations. Their efficient solution requires the use of fast iterative methods. Multigrid iterations are able to solve linear and nonlinear systems with a rather fast rate of convergence, provided the problem is of elliptic type. The contribution describes the basic construction of multigrid methods, their ingredients, related methods, and their application to various problem classes.Although most of the applications concern FEM discretizations of partial differential equations, there are also applications to integral equations as they occur in boundary element methods (BEM).

Krylov subspace methods are crucial techniques for solving linear systems. To effectively tackle large-scale linear systems, parallelism techniques are frequently utilized. However, the use of parallelism can increase the non-associativity of floating-point operations, potentially resulting in non-reproducibility of the computations. In order to address the issue of non-reproducibility in parallel preconditioned Krylov subspace iterative methods, we investigated the sources of irreproducibility and solutions to achieve reproducibility. The main reason for the irreproducibility in iterative methods is attributed to distributed dot products. To tackle this, we investigated three reproducible Basic Linear Algebra Subprograms (BLAS) as replacements for standard distributed dot products. We utilized OzBLAS, ExBLAS, and ReproBLAS to ensure reproducibility in iterative methods. We combined Jacobi preconditioners and tested and verified on three iterative methods: CG, BiCGSTAB and GMRES. Numerical experiments demonstrate that the iterative method based on OzBLAS is reproducible in a multi-threaded environment. The iterative method based on ExBLAS and ReproBLAS achieves reproducibility in a multi-process environment. Our evaluation indicated that ReproBLAS performed well in terms of performance, robustness, and parallelism. Therefore, we employ a reproducible iterative method based on ReproBLAS to solve the large-scale algebraic equations resulting from the discretization of the two-dimensional three-temperature heat conduction equation, which arises from practical application problems in the field of laser fusion radiation hydrodynamics. Numerical experiments demonstrate the reproducible solution of the three-temperature heat conduction algebraic equations.

5 - Iterative schemes and convergence analysis

An iterative solution method is presented and illustrated to analyse the dynamic response of bridge–vehicle systems. The method consists in dividing the whole system into 2 subsystems at the interface of the bridge and vehicles; these 2 subsystems are solved separately; their compatibility at the interface is achieved by an iterative procedure with under-relaxation or with Aitken acceleration. The characteristics of this method are explained on a simplified system with 2 degrees of freedom (DOF). The numerical results for a simple example demonstrate the high performances of the proposed method: good convergence rate and high accuracy. Finally, the method is applied to a practical example: the linear dynamic response of the Yangtze-River Bridge at Wuhan under a moving train with 2 locomotives and 4 freight cars. The efficiency is attained because neither formation nor factorisation of the coefficient matrices for the equations of the system are needed at every time step in linear analysis. The Aitken acceleration technique is more efficient in systems with multi-degrees of freedom than the relaxation technique. The proposed method will be even more efficient in non-linear dynamic response because, in this case, the iterations are necessary whether the system is solved as a whole or not.