Parallel Iterative Methods Research Articles

Parallel iterative regularization methods for solving systems of ill-posed equations

In this note two parallel iterative regularization methods for finding a minimal-norm solution to a system of ill-posed equations involving the so-called strongly-inverse monotone operators have been investigated. Some applications of the proposed methods are considered and numerical experiments are discussed.

Parallel iterative methods for parabolic equations

For the problems of the parabolic equations in one- and two-dimensional space, the parallel iterative methods are presented to solve the fully implicit difference schemes. The methods presented are based on the idea of domain decomposition in which we divide the linear system of equations into some non-overlapping sub-systems, which are easy to solve in different processors at the same time. The iterative value is proved to be convergent to the difference solution resulted from the implicit difference schemes. Numerical experiments for both one- and two-dimensional problems show that the methods are convergent and may reach the linear speed-up.

A parallel method of solving nonlinear inverse problems in gravimetry and magnetometry

A parallel iterative method of solving nonlinear inverse logarithmic potential problems and three-dimensional problems in gravimetry and magnetometry is proposed.

On the Convergence and Stability of a Pressure–Velocity Iteration of Chorin Type with Natural Boundary Conditions

Because of the implementation of numerical solution algorithms for the nonstationary Navier–Stokes equations of an incompressible fluid on massively parallel computers iterative methods are of special interest. A red–black pressure–velocity iteration that allows an efficient parallelization based on a domain decomposition [3] will be analyzed in this paper. We prove the equivalence of the pressure–velocity iteration (PUI) by Chorin/Hirt/Cook [1, 2] with a SOR iteration to solve a Poisson equation for the pressure. We show this on a 2D rectangle with some special outflow boundary conditions and Dirichlet data for the velocity elsewhere. This equivalence allows us to prove the convergence of that iteration scheme. We also discuss the stability of the occurring discrete Laplacian in discrete Sobolev spaces.

Recovery Patterns for Iterative Methods in a Parallel Unstable Environment

Several recovery techniques for parallel iterative methods are presented. First, the implementation of checkpoints in parallel iterative methods is described and analyzed. Then a simple checkpoint-free fault-tolerant scheme for parallel iterative methods, the lossy approach, is presented. When one processor fails and all its data is lost, the system is recovered by computing a new approximate solution using the data of the nonfailed processors. The iterative method is then restarted with this new vector. The main advantage of the lossy approach over standard checkpoint algorithms is that it does not increase the computational cost of the iterative solver when no failure occurs. Experiments are presented that compare the different techniques. The fault-tolerant FT-MPI library is used. Both iterative linear solvers and eigensolvers are considered.

Development and Performance Evaluation of Parallel Iterative Method for Multi-scale Piezoelectric Analyses

Development and Performance Evaluation of Parallel Iterative Method for Multi-scale Piezoelectric Analyses

A parallel algorithm based on Galerkin theory for block-tridiagonal linear systems

The work presented in this paper focuses on parallel iterative method for solving block-tridiagonal linear systems. Being based on Galerkin theory predetermining W m = V m , and forming a suitable matrix V m , a parallel iterative method on distributed-memory multi-computer is established. Then Eq. (3.1)–(3.3) are provided for carrying out computation required by our parallel algorithm. Furthermore, convergence is proved when the coefficient matrix A is a symmetric positive definite matrix, and the sufficient condition is given. In the end, two illustrative examples implemented on HP rx2600 cluster show that our algorithm’s parallel acceleration rates and efficiency are higher.

A parallel iterative method for solving periodical block-tridiagonal linear equations

In this paper, based upon splitting the coefficient matrix, we propose a parallel iterative algorithm for periodical block-tridiagonal linear equations on distributed-memory multi-computers, which is more general applied than presented in [Lihua Chi, Jie Liu, Xiaomei Li. An effective parallel algorithm for tridiagonal linear equations, Journal of Computer 22 (2) (1999) 218–221]. Our algorithm can be easily generalized to solve the tridiagonal systems, the block-tridiagonal, and the periodical block-tridiagonal systems. The communication only needs twice between the adjacent processors per iteration. Furthermore, analysis of convergence and error about this algorithm is given. Finally, some numerical results on HP r×2600 cluster show that practice computing is consistent with theory. The algorithm has great parallel efficiency.

Parallel iterative methods using factorized preconditioning matrices for solving elliptic equations on triangular grids

Parallel analogs of the variants of the incomplete Cholesky-conjugate gradient method and the modified incomplete Cholesky-conjugate gradient method for solving elliptic equations on uniform triangular and unstructured triangular grids on parallel computer systems with the MIMD architecture are considered. The construction of parallel methods is based on the use of various variants of ordering the grid points depending on the decomposition of the computation domain. Results of the theoretic and experimental studies of the convergence rate of these methods are presented. The solution of model problems on a moderate number processors is used to examine the efficiency of the proposed parallel methods.

309 マルチスケール圧電解析のための並列反復計算法(MEMSの計測・モデリング・マルチフィジックスシミュレーション(3),OS03 MEMSの計測・モデリング・マルチフィジックスシミュレーション)

309 マルチスケール圧電解析のための並列反復計算法(MEMSの計測・モデリング・マルチフィジックスシミュレーション(3),OS03 MEMSの計測・モデリング・マルチフィジックスシミュレーション)

The performance of parallel iterative solvers

A performance model is constructed for parallel iterative numerical methods under the assumption of a message-passing computing system. Arguments are given for the fact that the speedup of parallel iterative methods is mainly influenced by the speedup at one iterative step. Using the theoretical model, it is proved why explicit iterative methods for ordinary differential equations are inefficient in implementation on distributed memory multiprocessor systems. Numerical tests on parallel and distributed computing environments confirm the correctness of the theoretical model at least in the case of iterative methods for ordinary differential equations and time-dependent partial differential equations.

Parallel iterative solvers for finite-element methods using an OpenMP/MPI hybrid programming model on the Earth Simulator

An efficient parallel iterative method for finite-element method has been developed for symmetric multiprocessor (SMP) cluster architectures with vector processors such as the Earth Simulator. The method is based on a three-level hybrid parallel programming model, including message passing for inter-SMP node communication, loop directives by OpenMP for intra-SMP node parallelization and vectorization for each processing element (PE). Simple 3D linear elastic problems with more than 2.2 × 10 9 DOF have been solved using 3 × 3 block ICCG(0) method with additive Schwarz domain decomposition and PDJDS/CM-RCM reordering on 176 nodes of the Earth Simulator, achieving performance of 3.80 TFLOPS. Furthermore, effect of color number in reordering has been evaluated on various types of computers.

Speedup in solving differential equations on clusters of workstations

A performance model is constructed for parallel iterative numerical methods under the assumption of a message-passing computing system. Arguments are given for the fact that the speedup of parallel iterative methods is mainly influenced by the speedup at one iterative step including computations and communications. Using the theoretical model, it is proved why explicit iterative methods for ordinary differential equations are inefficient in implementation on distributed memory multiprocessor systems. Numerical tests on a cluster of dedicated machines confirm the correctness of the theoretical model at least in the case of iterative methods for ordinary differential equations and time-dependent partial differential equations.

Three-level hybrid vs. flat MPI on the Earth Simulator: Parallel iterative solvers for finite-element method

An efficient parallel iterative method for finite element method has been developed for symmetric multiprocessor (SMP) cluster architectures with vector processors such as the Earth Simulator. The method is based on a three-level hybrid parallel programming model, including message passing for inter-SMP node communication, loop directives by OpenMP for intra-SMP node parallelization and vectorization for each processing element (PE). Simple 3D linear elastic problems with more than 2.2 × 10 9 DOF have been solved using 3 × 3 block ICCG(0) method with additive Schwarz domain decomposition and PDJDS/CM-RCM reordering on 176 nodes of the Earth Simulator, achieving performance of 3.80 TFLOPS. The PDJDS/CM-RCM reordering method provides excellent vector and parallel performance in SMP nodes. A three-level hybrid parallel programming model outperforms flat MPI in the problems involving large numbers of SMP nodes.

On parallel multisplitting iterative methods for singular linear systems

In this paper, the semiconvergence of parallel multisplitting iterative methods and extrapolated parallel methods for singular linear systems is discussed. Some sufficient conditions for semiconvergence are presented whenever the multisplittings are P-regular, sub-regular or nonnegative. As special cases the semiconvergence of parallel generalized AOR, block AOR, AOR, parallel generalized SOR, block SOR, SOR, extrapolated parallel (generalized, block) AOR and extrapolated parallel (generalized, block) SOR methods are proved.

Developing object-oriented parallel iterative methods

In this paper, we describe our work developing an object-oriented parallel toolkit on OO-MPI. We are developing a parallelised implementation of the multi-view tomography toolbox, an iterative solver for a range of tomography problems. The code is developed in object-oriented C++. The MVT toolbox is presently used by researchers in the field of tomography to solve linear and non-linear forward modelling problems. The performance of the toolbox is heavily dependent on the performance of a small number of classes present in the IML++ library. This paper describes our experience parallelising a sparse matrix algorithm provided in the IML++ object-oriented numerical library. This library comprises a number of iterative algorithms. In this work, we present a parallel version of BiCGSTAB algorithm and the block-ILU preconditioner. These two algorithms are implemented in C++ using OOMPI (object-oriented MPI), and run on a 32-node Beowulf cluster. These two routines are also fundamental to obtaining good performance when using the parallelised version of the MVT toolbox. We also demonstrate the importance of using threads to overlap communication and computation, as an effective path to obtain improved speedup.

Parallel iterative methods for dense linear systems in inductance extraction

Accurate estimation of the inductive coupling between interconnect segments of a VLSI circuit is critical to the design of high-end microprocessors. This paper presents a class of parallel iterative methods for solving the linear systems of equations that arise in the inductance extraction process. The coefficient matrices are made up of dense and sparse submatrices where the dense structure is due to the inductive coupling between current filaments and the sparse structure is due to Kirchoff’s constraints on current. By using a solenoidal basis technique to represent current, the problem is transformed to an unconstrained one that is subsequently solved by an iterative method. A dense preconditioner resembling the inductive coupling matrix is used to increase the rate of convergence of the iterative method. Multipole-based hierarchical approximations are used to compute products with the dense coefficient matrix as well as the preconditioner. A parallel formulation of the preconditoned iterative solver is outlined along with parallelization schemes for the hierarchical approximations. A variety of experiments is presented to show the parallel efficiency of the algorithms on shared-memory multiprocessors.

Parallel iterative methods for Navier–Stokes equations and application to eigenvalue computation

AbstractWe describe the construction of parallel iterative solvers for finite‐element approximations of the Navier–Stokes equations on unstructured grids using domain decomposition methods. The iterative method used is FGMRES, preconditioned by a parallel adaptation of a block preconditioner recently proposed by Kay et al. The parallelization is achieved by adapting the technology of our domain decomposition solver DOUG (previously used for scalar problems) to block‐systems. The iterative solver is applied to shifted linear systems that arise in eigenvalue calculations. To illustrate the performance of the solver, we compare several strategies both theoretically and practically for the calculation of the eigenvalues of large sparse non‐symmetric matrices arising in the assessment of the stability of flow past a cylinder. Copyright © 2003 John Wiley & Sons, Ltd.

Parallel-Iterated RK-Type PC Methods With Continuous Output Formulas * This work was partly supported by N.R.P.F.S.

This paper investigates parallel predictor-corrector iteration schemes (PC iteration schemes) based on collocation Runge-Kutta corrector methods (RK corrector methods) with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. The resulting parallel-iterated RK-type PC methods are also provided with continuous output formulas. The continuous numerical approximations are used for predicting the stage values in the PC iteration processes. In this way, we obtain parallel PC methods with continuous output formulas and high-accurate predictions. Applications of the resulting parallel PC methods to a few widely-used test problems reveal that these new parallel PC methods are much more efficient when compared with the parallel and sequential explicit RK methods from the literature.

Parallel iterative solution of large‐scale acoustic scattering problems using exact non reflecting conditions on distributed memory computer systems

Parallel iterative methods for fast solution of large‐scale acoustic radiation and scattering problems are developed using exact Dirichlet‐to‐Neumann (DtN) nonreflecting boundaries. For elongated scatterers such as submarines, it is shown that the generalization of the DtN to elliptical/spheroidal artificial boundaries improves significantly the computational efficiency of accurate finite element methods for the solution of acoustic scattering problems. The outer‐product structure of the DtN map is exploited as a low‐rank update of the system matrix to efficiently compute the matrix‐by‐vector products found in Krylov subspace based iterative methods. For the complex non‐Hermitian matrices resulting from the Helmholtz equation, a distributed‐memory parallel BICG‐STAB iterative method is used in conjunction with a hybrid parallel SSOR/Jacobi preconditioner. The domain decomposition with interface minimization was performed to ensure optimal inter‐processor communication. For the distributed memory architectures tested, including Linux/Intel Beowulf clusters, when implemented as a low‐rank update, the nonlocal character of the DtN map shows little impact on the scale up or parallel efficiency compared to approximate local boundary conditions. [Work supported by NSF.]