Parallel Computation of Finite Element Navier-Stokes codes using MUMPS Solver

The study deals with the parallelization of 2D and 3D finite element based Navier-Stokes codes using direct solvers. Development of sparse direct solvers using multifrontal solvers has significantly reduced the computational time of direct solution methods. Although limited by its stringent memory requirements, multifrontal solvers can be computationally efficient. First the performance of MUltifrontal Massively Parallel Solver (MUMPS) is evaluated for both 2D and 3D codes in terms of memory requirements and CPU times. The scalability of both Newton and modified Newton algorithms is tested

A domain decomposing parallel sparse linear system solver

The solution of large sparse linear systems is often the most time-consuming part of many science and engineering applications. Computational fluid dynamics, circuit simulation, power network analysis, and material science are just a few examples of the application areas in which large sparse linear systems need to be solved effectively. In this paper we introduce a new parallel hybrid sparse linear system solver for distributed memory architectures that contains both direct and iterative components. We show that by using our solver one can alleviate the drawbacks of direct and iterative solvers, achieving better scalability than with direct solvers and more robustness than with classical preconditioned iterative solvers. Comparisons to well-known direct and iterative solvers on a parallel architecture are provided.Comment: To appear in Journal of Computational and Applied Mathematic

On large-scale diagonalization techniques for the Anderson model of localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi–Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude

Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers

In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration

Sympiler: Transforming Sparse Matrix Codes by Decoupling Symbolic Analysis

Sympiler is a domain-specific code generator that optimizes sparse matrix computations by decoupling the symbolic analysis phase from the numerical manipulation stage in sparse codes. The computation patterns in sparse numerical methods are guided by the input sparsity structure and the sparse algorithm itself. In many real-world simulations, the sparsity pattern changes little or not at all. Sympiler takes advantage of these properties to symbolically analyze sparse codes at compile-time and to apply inspector-guided transformations that enable applying low-level transformations to sparse codes. As a result, the Sympiler-generated code outperforms highly-optimized matrix factorization codes from commonly-used specialized libraries, obtaining average speedups over Eigen and CHOLMOD of 3.8X and 1.5X respectively.Comment: 12 page

An efficient null space inexact Newton method for hydraulic simulation of water distribution networks

Null space Newton algorithms are efficient in solving the nonlinear equations arising in hydraulic analysis of water distribution networks. In this article, we propose and evaluate an inexact Newton method that relies on partial updates of the network pipes' frictional headloss computations to solve the linear systems more efficiently and with numerical reliability. The update set parameters are studied to propose appropriate values. Different null space basis generation schemes are analysed to choose methods for sparse and well-conditioned null space bases resulting in a smaller update set. The Newton steps are computed in the null space by solving sparse, symmetric positive definite systems with sparse Cholesky factorizations. By using the constant structure of the null space system matrices, a single symbolic factorization in the Cholesky decomposition is used multiple times, reducing the computational cost of linear solves. The algorithms and analyses are validated using medium to large-scale water network models.Comment: 15 pages, 9 figures, Preprint extension of Abraham and Stoianov, 2015 (https://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0001089), September 2015. Includes extended exposition, additional case studies and new simulations and analysi

SparseM: A Sparse Matrix Package for R *

SparseM provides some basic R functionality for linear algebra with sparse matrices. Use of the package is illustrated by a family of linear model fitting functions that implement least squares methods for problems with sparse design matrices. Significant performance improvements in memory utilization and computational speed are possible for applications involving large sparse matrices.