A method for preconditioning matrices arising from linear integral equations for elliptic boundary value problems

The discretization of linear integral equations for elliptic boundary value problems by the boundary element method yields linear systems of simultaneous equations with filled matrices. The structure of these matrices allows Fourier methods to be used to determine preconditioning matrices such that fast iterative solution of the linear system of algebraic equations is possible. The preconditioning method is applicable to Fredholm integral equations of the first kind with non-smooth convolutional principal part as well as to Fredholm integral equations of the second kind. Numerical examples are presented.     

Distributed generic approximate sparse inverses

The need for accuracy in the solution of linear systems derived from the discretization of partial differential equations leads to large sparse linear systems. The solution of sparse linear systems requires efficient scalable methods. Iterative solvers require efficient parallel preconditioning methods to solve effectively sparse linear systems. Herewith, a new parallel algorithm for the generic approximate sparse inverse matrix method for distributed memory systems is proposed. The computation of the distributed generic approximate sparse inverse matrix is based on a column-wise approach, which allows the separation to independent problems that can be handled in parallel without synchronization points or intermediate communications. This is achieved by reforming the generic approximate sparse inverse matrix algorithm and its process of computation with a new partial solution method for the computation of the nonzero elements of each column dictated by the approximate inverse sparsity pattern. Moreover, an algorithmic scheme is proposed for the efficient distribution of data amongst the available workstations, along with a load balancing scheme for problems with large standard deviation in the number of nonzero elements per column. Numerical results are presented for the proposed schemes for various model problems.     

Parallel computation of incompressible flows in materials processing: Numerical experiments in diagonal preconditioning

Massively parallel computing is enabling dramatic advances in the simulation of three-dimensional flows in materials processing systems. This study focuses on the efficiency and robustness of parallel algorithms applied to such systems. Specifically, various diagonal preconditioning schemes are tested for the iterative solution of the linear equations arising from Newton's method applied to finite element discretizations. Two finite element discretizations are considered — the classical Galerkin and the Galerkin/least-squares method. Results show that the choice of preconditioning method can greatly influence the rate of convergence, but that no type worked uniformly well in all cases.     

Parallel methods for optimality criteria-based topology optimization

Topology optimization problems require the repeated solution of finite element problems that are often extremely ill-conditioned due to highly heterogeneous material distributions. This makes the use of iterative linear solvers inefficient unless appropriate preconditioning is used. Even then, the solution time for topology optimization problems is typically very high. These problems are addressed by considering the use of non-overlapping domain decomposition-based parallel methods for the solution of topology optimization problems. The parallel algorithms presented here are based on the solid isotropic material with penalization (SIMP) formulation of the topology optimization problem and use the optimality criteria method for iterative optimization. We consider three parallel linear solvers to solve the equilibrium problem at each step of the iterative optimization procedure. These include two preconditioned conjugate gradient (PCG) methods: one using a diagonal preconditioner and one using an incomplete LU factorization preconditioner with a drop tolerance. A third substructuring solver that employs a hybrid of direct and iterative (PCG) techniques is also studied. This solver is found to be the most effective of the three solvers studied, both in terms of parallel efficiency and in terms of its ability to mitigate the effects of ill-conditioning. In addition to examining parallel linear solvers, we consider the parallelization of the iterative optimality criteria method. To tackle checkerboarding and mesh dependence, we propose a multi-pass filtering technique that limits the number of “ghost” elements that need to be exchanged across interprocessor boundaries.     

Performance of iterative methods in ANSYS on cray parallel/vector supercomputers

This paper describes recent work using iterative methods for the solution of linear systems in the ANSYS program. The ANSYS program, a general purpose finite element code widely used in structural analysis applications, has now added an iterative solver option. The development of robust iterative solvers and their use in commercial programs is discussed. Discussion of the applicability of iterative solvers as a general purpose solver will include the topics of robustness; as well as memory requirements and CPU performance. A new iterative solver for general purpose finite element codes which functions as a “black-box” solver using element-specific information and the underlying problem physics to construct an effective and inexpensive preconditioner is described. Some results are given from realistic examples comparing the performance of the iterative solver implemented in ANSYS with the traditional parallel/vector frontal solver used in ANSYS and a robust shifted incomplete Choleski iterative solver.     

Iterative solvers by substructuring for the p-version finite element method

We study a class of substructuring methods well-suited for iterative solution of large systems of linear equations arising from the p-version finite element method. The p-version offers a natural decomposition with every element treated as a substructure. We use the preconditioned conjugate gradient method with preconditioning constructed by a decomposition of the local function space on each element. We develop an elementary theory giving bounds on the condition numbers which do not depend on the number of elements if a sparse system with only few variables per element is solved in each iteration. This bound can be evaluated considering one element at a time and we compute such condition numbers numerically for various elements.     

Uniformly Convergent Iterative Methods for Discontinuous Galerkin Discretizations

We present iterative and preconditioning techniques for the solution of the linear systems resulting from several discontinuous Galerkin (DG) Interior Penalty (IP) discretizations of elliptic problems. We analyze the convergence properties of these algorithms for both symmetric and non-symmetric IP schemes. The iterative methods are based on a “natural” decomposition of the first order DG finite element space as a direct sum of the Crouzeix-Raviart non-conforming finite element space and a subspace that contains functions discontinuous at interior faces. We also present numerical examples confirming the theoretical results.     

Explicit approximate inverse preconditioning techniques

Summary  The numerical treatment and the production of related software for solving large sparse linear systems of algebraic equations, derived mainly from the discretization of partial differential equation, by preconditioning techniques has attracted the attention of many researchers. In this paper we give an overview of explicit approximate inverse matrix techniques for computing explicitly various families of approximate inverses based on Choleski and LU—type approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference, finite element and the domain decomposition discretization of elliptic and parabolic partial differential equations. Composite iterative schemes, using inner-outer schemes in conjunction with Picard and Newton method, based on approximate inverse matrix techniques for solving non-linear boundary value problems, are presented. Additionally, isomorphic iterative methods are introduced for the efficient solution of non-linear systems. Explicit preconditioned conjugate gradient—type schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear system of algebraic equations. Theoretical estimates on the rate of convergence and computational complexity of the explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.     

Preconditioned iterative solver on the coarsest level of a multi-grid method for high frequency time harmonic electromagnetic field analyses

A multi-grid method is one of the most powerful linear solvers for finite element electromagnetic field analysis. However, as the discretized model has recently been enlarged, a solution process for a linear system arising on the coarsest level tends to be problematic in a complete multi-grid solution process. Whereas a linear system on the coarsest level is generally solved by a direct solver, we solve it here by means of an iterative solver to reduce the memory requirements. Since a conventional preconditioning technique is not effective for such a linear system, we introduce preconditioning techniques based on Arnold, Falk, and Winther’s and on Hiptmair’s smoothers. Numerical tests show that the newly installed preconditioning technique greatly improves the convergence rate.     

Explicit preconditioned domain decomposition schemes for solving nonlinear boundary value problems

A new class of inner-outer iterative procedures in conjunction with Picard-Newton methods based on explicit preconditioning iterative methods for solving nonlinear systems is presented. Explicit preconditioned iterative schemes, based on the explicit computation of a class of domain decomposition generalized approximate inverse matrix techniques are presented for the efficient solution of nonlinear boundary value problems on multiprocessor systems. Applications of the new composite scheme on characteristic nonlinear boundary value problems are discussed and numerical results are given.     

Comparing direct and iterative equation solvers in a large structural analysis software system

This paper describes two vectorized implementations of preconditioned conjugate gradient (PCG) solvers. Sparse and diagonal matrix storage schemes are described and compared. A vectorized incomplete Choleski preconditioning is described and compared with Jacobi preconditioning. A modification to the basic no-fill incomplete Choleski method to improve performance and robustness is described. The two PCG solvers are compared with direct Choleski methods using a sparse Choleski solver from SPARSPAK and a vectorized variable-band Choleski solver developed at NASA Langley Research Center. All of the linear equation solvers are implemented in a large structural analysis finite element software system called the Computational Structural Mechanics (CSM) Testbed. The CSM Testbed is used to provide a common software system in which new methods are developed and tested. Several representative two- and three-dimensional structural analysis problems are solved using the various equation solvers. Results are given from runs made on the CONVEX C220 and CRAY 2 computer systems. Comparisons of the convergence rates for the iterative solvers as well as the computation rates, number of operations, and overall CPU time required by all of the equation solvers are given.     

Incorporation of linear multipoint constraints in domain-decomposition-based iterative solvers - Part II: Blending FETI-DP and mortar methods and assembling floating substructures

Two approaches are presented for extending the FETI-DP domain-decomposition-based iterative method to the solution of problems with Linear Multipoint Constraints (LMPCs). Both cases of arbitrary LMPCs and LMPCs generated by the mortar method for enforcing a weak continuity of the solution across non-matching finite element interfaces are considered. In the first approach, the LMPCs are addressed during the preconditioning step but enforced only at convergence. In the second approach, the LMPCs are enforced at each iteration through the solution of an auxiliary coarse problem. Both extended FETI-DP solvers are benchmarked on a parallel processor for a wide range of constrained structural mechanics problems. The obtained results reveal that both solvers are numerically scalable and complement each other from the memory and CPU performance viewpoints.     

Adaptive multigrid for finite element computations in plasticity

The solution of the system of equilibrium equations is the most time-consuming part in large-scale finite element computations of plasticity problems. The development of efficient solution methods are therefore of utmost importance to the field of computational plasticity. Traditionally, direct solvers have most frequently been used. However, recent developments of iterative solvers and preconditioners may impose a change. In particular, preconditioning by the multigrid technique is especially favorable in FE applications.The multigrid preconditioner uses a number of nested grid levels to improve the convergence of the iterative solver. Prolongation of fine-grid residual forces is done to coarser grids and computed corrections are interpolated to the fine grid such that the fine-grid solution successively is improved. By this technique, large 3D problems, invincible for solvers based on direct methods, can be solved in acceptable time at low memory requirements. By means of a posteriori error estimates the computational grid could successively be refined (adapted) until the solution fulfils a predefined accuracy level. In contrast to procedures where the preceding grids are erased, the previously generated grids are used in the multigrid algorithm to speed up the solution process.The paper presents results using the adaptive multigrid procedure to plasticity problems. In particular, different error indicators are tested.     

H-矩阵方程组的预条件迭代法

针对系数矩阵A为H-矩阵的线性方程组Ax=b,引入了预条件矩阵I＋S_α~β,通过对系数矩阵施行初等行变换,提出了求解线性方程组Ax=b的一种新的预条件Gauss-Seidel方法.论文中首先证明了若A为H-矩阵,则（I＋S_α~β）A仍然是H-矩阵;其次,以定理的形式给出了新的预条件Gauss-Seidel方法收敛的充分条件,即给出了为保证新的预条件Gauss-Seidel方法收敛时参数所需满足的条件;然后从理论上证明了新的预条件Gauss-Seidel迭代方法较经典的Gauss-Seidel迭代方法收敛速度快,论文中提出的新的预条件Gauss-Seidel迭代方法推广了文[1-2]中提出的预条件方法;最后又通过数值算例说明了新的预条件Gauss-Seidel迭代方法的有效性.     

A partitioning scheme and iterative solution for sparse bordered systems

Sparse bordered matrix systems arise in certain classes of problems. An example is the Jacobian system for the finite difference or finite element approximation and continuation solution of nonlinear PDE's. We develop a partitioning scheme that can be exploited in residual-based iterative methods and permits easy implementation in existing software. The scheme has been examined for several iterative methods, and particularly the Lanczos algorithms which proves very effective for the representative nonlinear test problems examined. The scheme vectorizes well and performance studies on vectorized supercomputers are given.     

Algebraic multigrid methods based on element preconditioning

This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/Stübens algorithm is robust for M-matrices, but unfortunately the “region of robustness“ between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy.

For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem.

The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements.     

A simple ILU preconditioning coupled to Orthomin method

Conjugate gradient type methods with preconditionings based on incompleteLU factorization (ILU) are known to be very useful for solving nonsymmetric linear systems of equations. In particular, the effectiveness of such iterative methods has been demonstrated on sparse linear systems which arise from discretizing non-self adjoint elliptic problems. In this paper, we present a simplifiedILU preconditioning coupled to a variational iterative method (Orthomin) on a model problem. It is shown that this simplified version ofILU is easy to implement and when coupled to Orthomin method, is as competitive or better than the standardILU with Orthomin method.     

A Matrix-free Two-grid Preconditioner for Solving Boundary Integral Equations in Electromagnetism

In this paper, we describe a matrix-free iterative algorithm based on the GMRES method for solving electromagnetic scattering problems expressed in an integral formulation. Integral methods are an interesting alternative to differential equation solvers for this problem class since they do not require absorbing boundary conditions and they mesh only the surface of the radiating object giving rise to dense and smaller linear systems of equations. However, in realistic applications the discretized systems can be very large and for some integral formulations, like the popular Electric Field Integral Equation, they become ill-conditioned when the frequency increases. This means that iterative Krylov solvers have to be combined with fast methods for the matrix-vector products and robust preconditioning to be affordable in terms of CPU time. In this work we describe a matrix-free two-grid preconditioner for the GMRES solver combined with the Fast Multipole Method. The preconditioner is an algebraic two-grid cycle built on top of a sparse approximate inverse that is used as smoother, while the grid transfer operators are defined using spectral information of the preconditioned matrix. Experiments on a set of linear systems arising from real radar cross section calculation in industry illustrate the potential of the proposed approach for solving large-scale problems in electromagnetism.     

A parallel solving method for block-tridiagonal equations on CPU–GPU heterogeneous computing systems

Solving block-tridiagonal systems is one of the key issues in numerical simulations of many scientific and engineering problems. Non-zero elements are mainly concentrated in the blocks on the main diagonal for most block-tridiagonal matrices, and the blocks above and below the main diagonal have little non-zero elements. Therefore, we present a solving method which mixes direct and iterative methods. In our method, the submatrices on the main diagonal are solved by the direct methods in the iteration processes. Because the approximate solutions obtained by the direct methods are closer to the exact solutions, the convergence speed of solving the block-tridiagonal system of linear equations can be improved. Some direct methods have good performance in solving small-scale equations, and the sub-equations can be solved in parallel. We present an improved algorithm to solve the sub-equations by thread blocks on GPU, and the intermediate data are stored in shared memory, so as to significantly reduce the latency of memory access. Furthermore, we analyze cloud resources scheduling model and obtain ten block-tridiagonal matrices which are produced by the simulation of the cloud-computing system. The computing performance of solving these block-tridiagonal systems of linear equations can be improved using our method.     

PARALLEL SOLUTION OF SYMMETRIC POSITIVE DEFINITE TOEPLITZ SYSTEMS

Initially, parallel algorithms were designed by parallelising the existing sequential algorithms for frequently occurring problems on available parallel architectures.

More recently, parallel strategies have been identified and utilised resulting in many new parallel algorithms. However, the analysis of such techniques reveals that further strategies can be applied to increase the parallelism. One of these strategies, i.e., increasing the computational work in each processing node, can reduce the memory accesses and hence congestion in a shared memory multiprocessor system. Similarly, when network message passing is minimised in a distributed memory processor system, dramatic improvements in the performance of the algorithm ensue.

A frequently occurring computational problem in digital signal processing (DSP) is the solution of symmetric positive definite Toeplilz linear systems. The Levinson algorithm for solving such linear equations is where the Toeplitz matrix property is utilised in the elimination process of each element to advantage. However, it can be shown that in the Parallel Implicit Elimination (PIE) method where more than one element is eliminated simultaneously, the Toeplitz structure can again be utilised to advantage. This relatively simple strategy yields a reduction in accesses to shared memory or network message passing, resulting in a significant improvement in the performance of the algorithm [2],