Parallel Preconditioned Conjugate Gradient Square Method Based on Normalized Approximate Inverses

High performance finite element approximate inverse preconditioning

An efficient strategy for the solution of N-S Equations using collocated, non-orthogonal grids is presented. The governing equations have been discretized in the physical plane itself without co-ordinate transformation, thereby retaining the lucidity of the basic finite volume method. The non-orthogonal terms and QUICK type corrections for the convective terms in the momentum equations are treated explicitly, while the other terms are taken in implicit form. In the pressure correction equation, the non-orthogonal terms have been dropped altogether. The discretized equations have been solved by the preconditioned conjugate gradient square method. The specific combination of above steps has resulted in better convergence properties as compared to those of existing algorithms, even for highly skewed grids. The scheme has been validated against benchmark solutions such as lid-driven flow in square and skewed cavities and experi mental results of flow over a single cylinder. Its applicability has also been illustrated for flow through a bank of staggered cylinders, with anti-symmetric inlet and outlet boundary conditions. Copyright © 1999 John Wiley & Sons, Ltd.

SIERRA is a 3-D general-purpose semiconductor-device simulation program which serves as a foundation for investigating integrated-circuit (IC) device and reliability issues. This program solves the Poisson and continuity equations in silicon under DC, transient, and small-signal conditions. Executing on a vector/parallel minisupercomputer, SIERRA utilizes a matrix solver which uses an incomplete LU (ILU) preconditioned conjugate gradient square (CGS, BCG) method. The ILU-CGS method provides a good compromise between memory size and convergence rate. The authors have observed a 5* to 7* speedup over standard direct methods in simulations of transient problems containing highly coupled Poisson and continuity equations such as those found in reliability-oriented simulations. The application of SIERRA to parasitic CMOS latchup and DRAM (dynamic random-access memory) single-event-upset studies is described. >

PurposeTo propose novel parallel/distributed normalized explicit finite element (FE) approximate inverse preconditioning for solving sparse FE linear systems.Design/methodology/approachThe design of suitable methods was the main objective for which several families of the normalized approximate inverse, based on sparse normalized approximate factorization, are produced. The main motive for the derivation of the new normalized approximate inverse FE matrix algorithmic techniques is that they can be efficiently used in conjunction with normalized explicit preconditioned conjugate gradient (NEPCG) – type schemes on parallel and distributed systems. Theoretical estimates on the rate of convergence and computational complexity of the NEPCG method are also derived.FindingsApplication of the proposed method on a three‐dimensional boundary value problem is discussed and numerical results for uniprocessor systems along with speed‐ups and efficiency for multicomputer systems are given. These results tend to become optimum, which are in qualitative agreement with the theoretical results presented for uniprocessor and distributed memory systems, using message passing interface (MPI) communication library.Research limitations/implicationsFurther parallel algorithmic techniques will be investigated in order to improve the speed‐ups and the computational complexity of the parallel normalized explicit approximate inverse preconditioning.Originality/valueThe proposed parallel/distributed normalized explicit approximate inverse preconditioning, using approximate factorization and approximate inverse algorithms, is an efficient computational method that is valuable for computer scientists and for scientists and engineers in engineering computations.

A new class of explicit preconditioning methods based on the concept of sparse approximate factorization procedures and inverse matrix techniques is introduced for solving biharmonic equations. Isomorphic methods in conunction with explicit preconditioned schemes based on approximate inverse matrix techniques are presented for the efficient solution of biharmonic equations. Application of the proposed method on linear systems is discussed and numerical results are given.

The preconditioned conjugate gradient (PCG) algorithm is a well-known iterative method for solving sparse linear systems in scientific computations. GPU-accelerated PCG algorithms for large-sized problems have attracted considerable attention recently. However, on a specific multi-GPU platform, producing a highly parallel PCG implementation for any large-sized problem requires significant time because several manual steps are involved in adjusting the related parameters and selecting an appropriate storage format for the matrix block that is assigned to each GPU. This motivates us to propose adaptive optimization modeling of PCG on multi-GPUs, which mainly involves the following parts: (1) an optimization multi-GPU parallel framework of PCG and (2) the profile-based optimization modeling for each one of the main components of the PCG algorithm, including vector operation, inner product, and sparse matrix-vector multiplication (SpMV). Our model does not construct a new storage format or kernel but automatically and rapidly generates an optimal parallel PCG algorithm for any problem on a specific multi-GPU platform by integrating existing storage formats and kernels. We take a vector operation kernel, an inner-product kernel, and five popular SpMV kernels for an example to present the idea of constructing the model. Given that our model is general, independent of the problems, and dependent on the resources of devices, this model is constructed only once for each type of GPU. The experiments validate the high efficiency of our proposed model.

Groundwater flow simulation has become one of the top international issues in new generation of environmental applications. When managing large-scale groundwater flow problems, the intensive computational ability and large amounts of memory space required for modeling are the main bottlenecks for researchers. In order to solve three-dimensional large-scale groundwater flow problems more rapidly, the Open MP was adopted to parallelize the preconditioned conjugate gradient (PCG) algorithm in this paper. And this paper carried out a numerical experiment of the three-dimensional groundwater flow model on a computer with four cores. Based on the numerical experiment, it is found that the execution time of the original serial PCG program is about 1.74 to 2.86 times of the parallel PCG program executed with different number of threads. The experimental results also demonstrate that the PCG solver based on Open MP is an effective way for solving large-scale three-dimensional groundwater flow problem.

A multi-GPU parallel optimization model for the preconditioned conjugate gradient algorithm

The Complex neuronal network models require support from simulation environment for efficient network simulations. To compute the models increasing complexity necessitated the efforts to parallelize the NEURON simulation environment. The computational neuroscientists have extended NEURON by dividing the equations for its subnet among multiple processors for increasing the competence of hardware. For spiking neuronal networks inter-processor spikes exchange consume significant portion of overall simulation time on parallel machines. In NEURON Message Passing Interface (MPI) is used for inter processor spikes exchange, MPI_Allgather collective operation is used for spikes exchange generated after each interval across distributed memory systems. However, as the number of processors become larger and larger MPI_Allgather method become bottleneck and needs efficient exchange method to reduce the spike exchange time. This work has improved MPI_Allgather method to Remote Memory Access (RMA) based on MPI-3.0 for NEURON simulation environment, MPI based on RMA provides significant advantages through increased communication concurrency in consequence enhances efficiency of NEURON and scaling the overall run time for the simulation of large network models.1

Under highly mobile environments, the bit error rate (BER) performance of DFT spreading-orthogonal frequency division multiplexing (DFTS-OFDM) signal with a frequency domain equalization method would be degraded relatively due to the loss of orthogonality among subcarriers. To solve this problem, this paper proposes an iterative based time domain equalization (TDE) technique with a time domain channel impulse response (CIR) estimation method for the DFTS-OFDM signal. The salient features of proposed method are to employ a time domain training sequence (TS) in the estimation of CIR instead of using the conventional pilot subcarriers method and to employ the TDE technique with a maximum likelihood (ML) estimation method instead of using the conventional minimum mean square error frequency domain equalization (MMSE-FDE) method. This paper also proposes a low-complexity iterative method for the TDE method by using the preconditioned conjugate gradient squared (PCGS) algorithm for solving the simultaneous equations instead of using a direct calculation of inverse matrix. This paper presents various simulation results under highly mobile environments to demonstrate the effectiveness of proposed iterative based TDE with the CIR estimation method for the DFTS-OFDM signal as comparing with the conventional MMSE-FDE method.

A parallel preconditioned conjugate gradient package for solving sparse linear systems on a Cray Y-MP

As distributed memory systems have become common, the de facto standard for communication is still the Message Passing Interface (MPI). pupyMPI is a pure Python implementation of a broad subset of the MPI 1.3 specifications that allows Python programmers to utilize multiple CPUs with datatypes and memory handled transparently. pupyMPI also implements a few non-standard extensions such as non-blocking collectives and the option of suspending, migrating and resuming the distributed computation of a pupyMPI program. This paper introduces pupyMPI and presents benchmarks against C implementations of MPI, which show acceptable performance.

Parallel PCG Algorithm On Distributed Network By PVM

Investigating the performance of parallel applications at scale on future high-performance computing (HPC) architectures and the performance impact of different architecture choices is an important component of HPC hardware/software co-design. The Extreme-scale Simulator (xSim) is a simulation-based toolkit for investigating the performance of parallel applications at scale. xSim scales to millions of simulated Message Passing Interface (MPI) processes. The overhead introduced by a simulation tool is an important performance and productivity aspect. This paper documents two improvements to xSim: (1) a new deadlock resolution protocol to reduce the parallel discrete event simulation management overhead and (2) a new simulated MPI message matching algorithm to reduce the oversubscription management overhead. The results clearly show a significant performance improvement, such as by reducing the simulation overhead for running the NAS Parallel Benchmark suite inside the simulator from 1,020% to 238% for the conjugate gradient (CG) benchmark and from 102% to 0% for the embarrassingly parallel (EP) and benchmark, as well as, from 37,511% to 13,808% for CG and from 3,332% to 204% for EP with accurate process failure simulation.

In this study, we analyze the real-time solution of a series of augmented least squares problems, which are generated by adding information to an original least squares model repetitively. Instead of solving the least squares problems directly, we transform them into a batch of saddle point linear systems and subsequently solve the linear systems using restrictively preconditioned conjugate gradient (RPCG) methods. Approximation of the new Schur complement is generated effectively based on a previously approximated Schur complement. Owing to the variations of the preconditioned conjugate gradient method, the proposed methods generate convergence results similar to the conjugate gradient method and achieve a very fast convergent iterative sequence when the coefficient matrix is well preconditioned. Numerical tests show that the new methods are more effective than some standard Krylov subspace methods. Updated RPCG methods meet the requirement of real-time computing successfully for multifactor models.