Solver For Dense Linear Systems Research Articles

Performance impact of precision reduction in sparse linear systems solvers.

It is well established that reduced precision arithmetic can be exploited to accelerate the solution of dense linear systems. Typical examples are mixed precision algorithms that reduce the execution time and the energy consumption of parallel solvers for dense linear systems by factorizing a matrix at a precision lower than the working precision. Much less is known about the efficiency of reduced precision in parallel solvers for sparse linear systems, and existing work focuses on single core experiments. We evaluate the benefits of using single precision arithmetic in solving a double precision sparse linear system using multiple cores. We consider both direct methods and iterative methods and we focus on using single precision for the key components of LU factorization and matrix–vector products. Our results show that the anticipated speedup of 2 over a double precision LU factorization is obtained only for the very largest of our test problems. We point out two key factors underlying the poor speedup. First, we find that single precision sparse LU factorization is prone to a severe loss of performance due to the intrusion of subnormal numbers. We identify a mechanism that allows cascading fill-ins to generate subnormal numbers and show that automatically flushing subnormals to zero avoids the performance penalties. The second factor is the lack of parallelism in the analysis and reordering phases of the solvers and the absence of floating-point arithmetic in these phases. For iterative solvers, we find that for the majority of the matrices computing or applying incomplete factorization preconditioners in single precision provides at best modest performance benefits compared with the use of double precision. We also find that using single precision for the matrix–vector product kernels provides an average speedup of 1.5 over double precision kernels. In both cases some form of refinement is needed to raise the single precision results to double precision accuracy, which will reduce performance gains.

Parallelization of the inverse fast multipole method with an application to boundary element method

We present an algorithm to parallelize the inverse fast multipole method (IFMM), which is an approximate direct solver for dense linear systems. The parallel scheme is based on a greedy coloring algorithm, where two nodes in the hierarchy with the same color are separated by at least σ nodes. We proved that when σ≥6, the workload associated with one color is embarrassingly parallel. However, the number of nodes in a group (color) may be small when σ=6. Therefore, we also explored σ=3, where a small fraction of the algorithm needs to be serialized, and the overall parallel efficiency was improved. We implemented the parallel IFMM using OpenMP for shared-memory machines. Successively, we applied it to a fast-multipole accelerated boundary element method (FMBEM) as a preconditioner, and compared its efficiency with (a) the original IFMM parallelized by linking a multi-threaded linear algebra library and (b) the commonly used parallel block-diagonal preconditioner. Our results showed that our parallel IFMM achieved at most 4× and 11× speedups over the reference method (a) and (b), respectively, in realistic examples involving more than one million variables.

Efficient Scalable Parallel Higher Order Direct MoM-SIE Method With Hierarchically Semiseparable Structures for 3-D Scattering

A novel fast scalable parallel algorithm is proposed for the solution of large 3-D scattering problems based on: 1) the double (geometrical and current-approximation) higher order (DHO) method of moments (MoM) in the surface integral equation (SIE) formulation and 2) a direct solver for dense linear systems utilizing hierarchically semiseparable (HSS) structures. Namely, an HSS matrix representation is used for compression, factorization, and solution of the system matrix. In addition, a rank-revealing QR decomposition for memory compression is used, with a stopping criterion in terms of the relative rank tolerance value. A method for geometrical preprocessing of the scatterers based on the cobblestone distance sorting technique is employed in order to enhance the HSS algorithm accuracy and parallelization. Numerical examples show how the accuracy of the DHO HSS-MoM-SIE method is easily controllable by using the relative tolerance for the matrix compression. Moreover, the examples demonstrate low memory consumption, as well as much faster simulation time, when compared to the direct LU decomposition. The method enables dramatically faster monostatic scattering computations than iterative solvers and reduced number of unknowns when compared to low-order discretizations. Finally, great scalability of the algorithm is demonstrated on more than one thousand processes.

Exploring Capabilities within ForTrilinos by Solving the 3D Burgers Equation

We present the first three-dimensional, partial differential equation solver to be built atop the recently released, open-source ForTrilinos package (http://trilinos.sandia.gov/packages/fortrilinos). ForTrilinos currently provides portable, object-oriented Fortran 2003 interfaces to the C++ packages Epetra, AztecOO and Pliris in the Trilinos library and framework [ACM Trans. Math. Softw.31(3) (2005), 397–423]. Epetra provides distributed matrix and vector storage and basic linear algebra calculations. Pliris provides direct solvers for dense linear systems. AztecOO provides iterative sparse linear solvers. We demonstrate how to build a parallel application that encapsulates the Message Passing Interface (MPI) without requiring the user to make direct calls to MPI except for startup and shutdown. The presented example demonstrates the level of effort required to set up a high-order, finite-difference solution on a Cartesian grid. The example employs an abstract data type (ADT) calculus [Sci. Program.16(4) (2008), 329–339] that empowers programmers to write serial code that lower-level abstractions resolve into distributed-memory, parallel implementations. The ADT calculus uses compilable Fortran constructs that resemble the mathematical formulation of the partial differential equation of interest.

Dense Linear System: A Parallel Self-verified Solver

This article presents a parallel self-verified solver for dense linear systems of equations. This kind of solver is commonly used in many different kinds of real applications which deal with large matrices. Nevertheless, two key problems appear to limit the use of linear system solvers to a more extensive range of real applications: solution correctness and high computational cost. In order to solve the first one, verified computing would be an interesting choice. An algorithm that uses this concept is able to find a highly accurate and automatically verified result providing more reliability. However, the performance of these algorithms quickly becomes a drawback. Aiming at a better performance, parallel computing techniques were employed. Two main parts of this method were parallelized: the computation of the approximate inverse of matrix A and the preconditioning step. The results obtained show that these optimizations increase significantly the overall performance.

Algebraic preconditioning versus direct solvers for dense linear systems as arising in crack propagation problems

AbstractPreconditioned iterative solution methods are compared with the direct Gaussian elimination method to solve dense linear systems Ax=b which originate from problems, discretized by boundary element method (BEM) techniques. Numerical experiments are presented and compared with the direct solution method available in a commercial BEM package, which show that the preconditioned iterative schemes are highly competitive with respect to both arithmetic operations required and memory demands. Copyright © 2004 John Wiley & Sons, Ltd.