50,840 research outputs found
Applications and accuracy of the parallel diagonal dominant algorithm
The Parallel Diagonal Dominant (PDD) algorithm is a highly efficient, ideally scalable tridiagonal solver. In this paper, a detailed study of the PDD algorithm is given. First the PDD algorithm is introduced. Then the algorithm is extended to solve periodic tridiagonal systems. A variant, the reduced PDD algorithm, is also proposed. Accuracy analysis is provided for a class of tridiagonal systems, the symmetric, and anti-symmetric Toeplitz tridiagonal systems. Implementation results show that the analysis gives a good bound on the relative error, and the algorithm is a good candidate for the emerging massively parallel machines
Efficient Parallel Kernel Solvers for Computational Fluid Dynamics Applications
Distributed-memory parallel computers dominate today's parallel computing arena. These machines, such as Intel Paragon, IBM SP2, and Cray Origin2OO, have successfully delivered high performance computing power for solving some of the so-called "grand-challenge" problems. Despite initial success, parallel machines have not been widely accepted in production engineering environments due to the complexity of parallel programming. On a parallel computing system, a task has to be partitioned and distributed appropriately among processors to reduce communication cost and to attain load balance. More importantly, even with careful partitioning and mapping, the performance of an algorithm may still be unsatisfactory, since conventional sequential algorithms may be serial in nature and may not be implemented efficiently on parallel machines. In many cases, new algorithms have to be introduced to increase parallel performance. In order to achieve optimal performance, in addition to partitioning and mapping, a careful performance study should be conducted for a given application to find a good algorithm-machine combination. This process, however, is usually painful and elusive. The goal of this project is to design and develop efficient parallel algorithms for highly accurate Computational Fluid Dynamics (CFD) simulations and other engineering applications. The work plan is 1) developing highly accurate parallel numerical algorithms, 2) conduct preliminary testing to verify the effectiveness and potential of these algorithms, 3) incorporate newly developed algorithms into actual simulation packages. The work plan has well achieved. Two highly accurate, efficient Poisson solvers have been developed and tested based on two different approaches: (1) Adopting a mathematical geometry which has a better capacity to describe the fluid, (2) Using compact scheme to gain high order accuracy in numerical discretization. The previously developed Parallel Diagonal Dominant (PDD) algorithm and Reduced Parallel Diagonal Dominant (RPDD) algorithm have been carefully studied on different parallel platforms for different applications, and a NASA simulation code developed by Man M. Rai and his colleagues has been parallelized and implemented based on data dependency analysis. These achievements are addressed in detail in the paper
An Efficient Parallel Solver for SDD Linear Systems
We present the first parallel algorithm for solving systems of linear
equations in symmetric, diagonally dominant (SDD) matrices that runs in
polylogarithmic time and nearly-linear work. The heart of our algorithm is a
construction of a sparse approximate inverse chain for the input matrix: a
sequence of sparse matrices whose product approximates its inverse. Whereas
other fast algorithms for solving systems of equations in SDD matrices exploit
low-stretch spanning trees, our algorithm only requires spectral graph
sparsifiers
Fast, Accurate Second Order Methods for Network Optimization
Dual descent methods are commonly used to solve network flow optimization
problems, since their implementation can be distributed over the network. These
algorithms, however, often exhibit slow convergence rates. Approximate Newton
methods which compute descent directions locally have been proposed as
alternatives to accelerate the convergence rates of conventional dual descent.
The effectiveness of these methods, is limited by the accuracy of such
approximations. In this paper, we propose an efficient and accurate distributed
second order method for network flow problems. The proposed approach utilizes
the sparsity pattern of the dual Hessian to approximate the the Newton
direction using a novel distributed solver for symmetric diagonally dominant
linear equations. Our solver is based on a distributed implementation of a
recent parallel solver of Spielman and Peng (2014). We analyze the properties
of the proposed algorithm and show that, similar to conventional Newton
methods, superlinear convergence within a neighbor- hood of the optimal value
is attained. We finally demonstrate the effectiveness of the approach in a set
of experiments on randomly generated networks.Comment: arXiv admin note: text overlap with arXiv:1502.0315
- …