On the GPGPU parallelization issues of finite element approximate inverse preconditioning

During the past decades, explicit finite element approximate inverse preconditioning methods have been extensively used for efficiently solving sparse linear systems on multiprocessor systems. The effectiveness of explicit approximate inverse preconditioning schemes relies on the use of efficient preconditioners that are close approximants to the coefficient matrix and are fast to compute in parallel. New parallel computational techniques are proposed for the parallelization of the Optimized Banded Generalized Approximate Inverse Finite Element Matrix (OBGAIFEM) algorithm, based on the concept of the “fish bone” computational approach, and for the Explicit Preconditioned Conjugate Gradient type methods on a General Purpose Graphics Processing Unit (GPGPU). The proposed parallel methods have been implemented using Compute Unified Device Architecture (CUDA) developed by NVIDIA. Finally, numerical results for the performance of the finite element explicit approximate inverse preconditioning for solving characteristic two dimensional boundary value problems on a massive multiprocessor interface on a GPU are presented. The CUDA implementation issues of the proposed methods are also discussed.     

Local and parallel finite element method for solving the biharmonic eigenvalue problem of plate vibration

In this paper, we establish a new local and parallel finite element discrete scheme based on the shifted‐inverse power method for solving the biharmonic eigenvalue problem of plate vibration. We prove the local error estimation of finite element solution for the biharmonic equation/eigenvalue problem and prove the error estimation of approximate solution obtained by the local and parallel scheme. When the diameters of three grids satisfy H⁴ = ?(w²) = ?(h), the approximate solutions obtained by our schemes can achieve the asymptotically optimal accuracy. The numerical experiments show that the computational schemes proposed in this paper are effective to solve the biharmonic eigenvalue problem of plate vibration.     

A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS

1. IntroductionConsider the large sparse system of linear equationsAx = b, (1.1)where, for a fixed positive integer cr, A e L(R") is a symmetric positive definite (SPD) matrir,having the bloCked formx,b E R" are the uDknwn and the known vectors, respectively, having the correspondingblocked formsni(ni S n, i = 1, 2,', a) are a given positthe integers, satisfying Z ni = n. This systemi= 1of linear equations often arises in sultable finite element discretizations of many secondorderseifad…     

Domain Decomposition–Finite Difference Approximate Inverse Preconditioned Schemes for Solving Fourth-Order Equations

A new class of explicit approximate inverse preconditioning is introduced for solving fourth-order equations, based on the coupled equation approach, by the domain decomposition method in conjunction with various finite difference approximation schemes. Explicit approximate inverse arrow-type matrix techniques, based on the concept of sparse L U-type factorization procedures, are introduced for computing a class of approximate inverses. Explicit preconditioned conjugate gradient-type schemes are presented for the efficient solution of linear systems. Applications of the method to a biharmonic problem are discussed and numerical results are given.     

An inexact parallel splitting augmented Lagrangian method for large system of linear equations

Parallel iterative methods are powerful in solving large systems of linear equations (LEs). The existing parallel computing research results focus mainly on sparse systems or others with particular structure. Most are based on parallel implementation of the classical relaxation methods such as Gauss-Seidel, SOR, and AOR methods which can be efficiently carried out on multiprocessor system. In this paper, we propose a novel parallel splitting operator method in which we divide the coefficient matrix into two or three parts. Then we convert the original problem (LEs) into a monotone (linear) variational inequality problem (VI) with separable structure. Finally, an inexact parallel splitting augmented Lagrangian method is proposed to solve the variational inequality problem (VI). To avoid dealing with the matrix inverse operator, we introduce proper inexact terms in subproblems such that the complexity of each iteration of the proposed method is O(n²). In addition, the proposed method does not require any special structure of system of LEs under consideration. Convergence of the proposed methods in dealing with two and three separable operators respectively, is proved. Numerical computations are provided to show the applicability and robustness of the proposed methods.     

Spectral optimization of explicit iterative methods. I

Methods of constructing preconditioning of explicit iterative methods of solving systems of linear, algebraic equations with sparse matrices are considered in the work. The techniques considered can, first of all, be realized within the framework of the simplest data structures; secondly, the graph structures of the corresponding algorithms are well adapted to realization on parallel computers; thirdly, in conjunction with modifications of Chebyshev methods they make it possible to construct rather effective computational algorithms. Experimental data are presented which demonstrate the effect of the proposed techniques of preconditioning of the distribution of the eigenvalues of matrices of systems arising in discretization of two-dimensional elliptic boundary-value problems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 51–60, 1984.     

On the two classes of high‐order convergent methods of approximate inverse preconditioners for solving linear systems

Two classes of methods for approximate matrix inversion with convergence orders p =3?2^k +1 (Class 1) and p =5?2^k ?1 (Class 2), k ≥1 an integer, are given based on matrix multiplication and matrix addition. These methods perform less number of matrix multiplications compared to the known hyperpower method or p th‐order method for the same orders and can be used to construct approximate inverse preconditioners for solving linear systems. Convergence, error, and stability analyses of the proposed classes of methods are provided. Theoretical results are justified with numerical results obtained by using the proposed methods of orders p =7,13 from Class 1 and the methods with orders p =9,19 from Class 2 to obtain polynomial preconditioners for preconditioning the biconjugate gradient (BICG) method for solving well‐ and ill‐posed problems. From the literature, methods with orders p =8,16 belonging to a family developed by the effective representation of the p th‐order method for orders p =2^k , k is integer k ≥1, and other recently given high‐order convergent methods of orders p =6,7,8,12 for approximate matrix inversion are also used to construct polynomial preconditioners for preconditioning the BICG method to solve the considered problems. Numerical comparisons are given to show the applicability, stability, and computational complexity of the proposed methods by paying attention to the asymptotic convergence rates. It is shown that the BICG method converges very quickly when applied to solve the preconditioned system. Therefore, the cost of constructing these preconditioners is amortized if the preconditioner is to be reused over several systems of same coefficient matrix with different right sides.     

A family of explicit parallel Runge-Kutta-Nyström methods

A procedure for the construction of high-order explicit parallel Runge-Kutta-Nyström (RKN) methods for solving second-order nonstiff initial value problems (IVPs) is analyzed. The analysis reveals that starting the procedure with a reference symmetric RKN method it is possible to construct high-order RKN schemes which can be implemented in parallel on a small number of processors. These schemes are defined by means of a convex combination of k disjoint s_i-stage explicit RKN methods which are constructed by connecting s_i steps of a reference explicit symmetric method. Based on the reference second-order Störmer-Verlet methods we derive a family of high-order explicit parallel schemes which can be implemented in variable-step codes without additional cost. The numerical experiments carried out show that the new parallel schemes are more efficient than some sequential and parallel codes proposed in the scientific literature for solving second-order nonstiff IVPs.     

Optimal server resource reservation policies for priority classes of users under cyclic non-homogeneous markov modeling

Resource availability optimization is studied on a server–client system where different users are partitioned into priority classes. The aim is to provide higher resource availability according to the priority of each class. For this purpose, resource reservation is modeled by a homogeneous continuous time Markov chain (CTMC), but also by a cyclic non-homogeneous Markov chain (CNHMC) as there is a cyclic behavior of the users’ requests for resources. The contribution of the work presented consists in the formulation of a multiobjective optimization problem for both the above cases that aims to determine the optimal resource reservation policy providing higher levels of resource availability for all classes. The optimization problem is solved either with known methods or with a proposed kind of heuristic algorithm. Finally, explicit generalized approximate inverse preconditioning methods are adopted for solving efficiently sparse linear systems that are derived, in order to compute resource availability.     

Diagonally compensated reduction and related preconditioning methods

When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on M-matrices (H-matrices). In some application areas, however, the arising coefficient matrices are not H-matrices. This is the case, for instance, when higher-order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an M-matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.     

On the partial synchronization of iterative methods

The rapidly growing field of parallel computing systems promotes the study of parallel algorithms, with the Monte Carlo method and asynchronous iterations being among the most valuable types. These algorithms have a number of advantages. There is no need for a global time in a parallel system (no need for synchronization), and all computational resources are efficiently loaded (the minimum processor idle time). The method of partial synchronization of iterations for systems of equations was proposed by the authors earlier. In this article, this method is generalized to include the case of nonlinear equations of the form x = F(x), where x is an unknown column vector of length n, and F is an operator from ?ⁿ into ?ⁿ. We consider operators that do not satisfy conditions that are sufficient for the convergence of asynchronous iterations, with simple iterations still converging. In this case, one can specify such an incidence of the operator and such properties of the parallel system that asynchronous iterations fail to converge. Partial synchronization is one of the effective ways to solve this problem. An algorithm is proposed that guarantees the convergence of asynchronous iterations and the Monte Carlo method for the above class of operators. The rate of convergence of the algorithm is estimated. The results can prove useful for solving high-dimensional problems on multiprocessor computational systems.     

A framework of parallel algebraic multilevel preconditioning iterations

1.IntroductionConsiderthesyllUnetricpositivedeflate(SPD)systemsoflinearequationsthatariseinfiniteelementdiscretisstionsofmanysecond-orderself-adjointellipticboundaryvalueproblems.Tosolvethisclassoflinearsystemsiteratively,AxelssonandVassilevski[1--4]preselltedthealgebraicmultileveliteration(AMLI)methodsbyreasonablyutilizingthemultigridtechniqueandthepolynomialaccelerationstrategy.Thesemethodsareamongthemostefficientiterativesolversbecausetheirpreconditioningmatricesarespectrallyequlvalellt…     

A Comparative Study on Dynamic and Static Sparsity Patterns in Parallel Sparse Approximate Inverse Preconditioning

Sparse approximate inverse (SAI) techniques have recently emerged as a new class of parallel preconditioning techniques for solving large sparse linear systems on high performance computers. The choice of the sparsity pattern of the SAI matrix is probably the most important step in constructing an SAI preconditioner. Both dynamic and static sparsity pattern selection approaches have been proposed by researchers. Through a few numerical experiments, we conduct a comparable study on the properties and performance of the SAI preconditioners using the different sparsity patterns for solving some sparse linear systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Solving Symmetric Arrowhead and Special Tridiagonal Linear Systems by Fast Approximate Inverse Preconditioning

A new class of approximate inverses for arrowhead and special tridiagonal linear systems, based on the concept of sparse approximate Choleski-type factorization procedures, are introduced for computing fast explicit approximate inverses. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of symmetric linear systems. A theorem on the rate of convergence of the explicit preconditioned conjugate gradient scheme is given and estimates of the computational complexity are presented. Applications of the proposed method on linear and nonlinear systems are discussed and numerical results are given.     

Stagnation in the p-version of the finite element method

A special feature of the p-version of the finite element method for solving a differential boundary value problem stated in the form of minimizing a quadratic functional on a certain set is studied. This special feature results in approximate solutions remaining unchanged on finite numbers of increasing finite-dimensional subsets of increasing dimension, in which solutions are sought. Necessary and sufficient conditions for the existence of this feature are found, and the stagnation effect is interpreted for a specially constructed example. For the adaptive p-version of the finite element approach, a modified strategy is proposed that takes this feature into account and thus improves the reliability of the method.     

Linear models for the approximate solution of the problem of packing equal circles into a given domain

The linear models for the approximate solution of the problem of packing the maximum number of equal circles of the given radius into a given closed bounded domain G are proposed. We construct a grid in G; the nodes of this grid form a finite set of points T, and it is assumed that the centers of circles to be packed can be placed only at the points of T. The packing problems of equal circles with the centers at the points of T are reduced to 0–1 linear programming problems. A heuristic algorithm for solving the packing problems based on linear models is proposed. This algorithm makes it possible to solve packing problems for arbitrary connected closed bounded domains independently of their shape in a unified manner. Numerical results demonstrating the effectiveness of this approach are presented.     

A Two-Phase Cyclic Nonhomogeneous Markov Chain Performability Evaluation by Explicit Approximate Inverses Applied to a Replicated Database System

The need for a more accurate modeling of the performance of systems whose functioning mainly dependant on external time parameters such as the number of requests during a particular time phase, led us to a novel approach, taking into account the time parameters involved. This is achieved through the evaluation of a performability indicator modeled by means of a two-phase cyclic nonhomogenous Markov chain considering periodical time-dependant arrival request probabilities and applied to a replicated database system. The computation of the performability indicator modeled by cyclic nonhomogeneous Markov chain requires the use of efficient computational methods by using explicit approximate inverse preconditioning methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Parallel multiobjective evolutionary algorithms for batch scheduling in heterogeneous computing and grid systems

This article presents six parallel multiobjective evolutionary algorithms applied to solve the scheduling problem in distributed heterogeneous computing and grid systems. The studied evolutionary algorithms follow an explicit multiobjective approach to tackle the simultaneous optimization of a system-related (i.e. makespan) and a user-related (i.e. flowtime) objectives. Parallel models of the proposed methods are developed in order to efficiently solve the problem. The experimental analysis demonstrates that the proposed evolutionary algorithms are able to efficiently compute accurate results when solving standard and new large problem instances. The best of the proposed methods outperforms both deterministic scheduling heuristics and single-objective evolutionary methods previously applied to the problem.     

A survey of multilevel preconditioned iterative methods

We survey multilevel iterative methods applied for solving large sparse systems with matrices, which depend on a level parameter, such as arise by the discretization of boundary value problems for partial differential equations when successive refinements of an initial discretization mesh is used to construct a sequence of nested difference or finite element meshes.We discuss various two-level (two-grid) preconditioning techniques, including some for nonsymmetric problems. The generalization of these techniques to the multilevel case is a nontrivial task. We emphasize several ways this can be done including classical multigrid methods and a recently proposed algebraic multilevel preconditioning method. Conditions for which the methods have an optimal order of computational complexity are presented.On leave from the Institute of Mathematics, and Center for Informatics and Computer Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria. The research of the second author reported here was partly supported by the Stichting Mathematisch Centrum, Amsterdam.     

Parallel Computing for Long-Time Simulations of Calcium Waves in a Heart Cell

Calcium waves are modeled by parabolic partial differential equations, whose simulation codes contain Krylov subspace methods as computational kernels. This paper presents GPU-based parallel computations for the conjugate gradient method applied to the finite difference discretization of a Poisson equation as prototype problem for the computational kernel. The CUDA algorithm tests the three memory systems of global memory, texture memory, and shared memory of a CUDA-enabled GPU. Due to the caching mechanism and coalesced read/write operations, the CUDA algorithm using global memory and single precision floating point numbers outperforms algorithms accessing texture memory and the shared memory. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)