833 research outputs found
A Class of Parallel Tiled Linear Algebra Algorithms for Multicore Architectures
As multicore systems continue to gain ground in the High Performance
Computing world, linear algebra algorithms have to be reformulated or new
algorithms have to be developed in order to take advantage of the architectural
features on these new processors. Fine grain parallelism becomes a major
requirement and introduces the necessity of loose synchronization in the
parallel execution of an operation. This paper presents an algorithm for the
Cholesky, LU and QR factorization where the operations can be represented as a
sequence of small tasks that operate on square blocks of data. These tasks can
be dynamically scheduled for execution based on the dependencies among them and
on the availability of computational resources. This may result in an out of
order execution of the tasks which will completely hide the presence of
intrinsically sequential tasks in the factorization. Performance comparisons
are presented with the LAPACK algorithms where parallelism can only be
exploited at the level of the BLAS operations and vendor implementations
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Alternative methods for representing the inverse of linear programming basis matrices
Methods for representing the inverse of Linear Programming (LP) basis matrices are closely related to techniques for solving a system of sparse unsymmetric linear equations by direct methods. It is now well accepted that for these problems the static process of reordering the matrix in the lower block triangular (LBT) form constitutes the initial step. We introduce a combined static and dynamic factorisation of a basis matrix and derive its inverse which we call the partial elimination form of the inverse (PEFI). This factorization takes advantage of the LBT structure and produces a sparser representation of the inverse than the elimination form of the inverse (EFI). In this we make use of the original columns (of the constraint matrix) which are in the basis. To represent the factored inverse it is, however, necessary to introduce special data structures which are used in the forward and the backward transformations (the two major algorithmic steps) of the simplex method. These correspond to solving a system of equations and solving a system of equations with the transposed matrix respectively. In this paper we compare the nonzero build up of PEFI with that of EFI. We have also investigated alternative methods for updating the basis inverse in the PEFI representation. The results of our experimental investigation are presented in this pape
Fast Parallel Randomized QR with Column Pivoting Algorithms for Reliable Low-rank Matrix Approximations
Factorizing large matrices by QR with column pivoting (QRCP) is substantially
more expensive than QR without pivoting, owing to communication costs required
for pivoting decisions. In contrast, randomized QRCP (RQRCP) algorithms have
proven themselves empirically to be highly competitive with high-performance
implementations of QR in processing time, on uniprocessor and shared memory
machines, and as reliable as QRCP in pivot quality.
We show that RQRCP algorithms can be as reliable as QRCP with failure
probabilities exponentially decaying in oversampling size. We also analyze
efficiency differences among different RQRCP algorithms. More importantly, we
develop distributed memory implementations of RQRCP that are significantly
better than QRCP implementations in ScaLAPACK.
As a further development, we introduce the concept of and develop algorithms
for computing spectrum-revealing QR factorizations for low-rank matrix
approximations, and demonstrate their effectiveness against leading low-rank
approximation methods in both theoretical and numerical reliability and
efficiency.Comment: 11 pages, 14 figures, accepted by 2017 IEEE 24th International
Conference on High Performance Computing (HiPC), awarded the best paper priz
Domain Decomposition Based High Performance Parallel Computing\ud
The study deals with the parallelization of finite element based Navier-Stokes codes using domain decomposition and state-ofart sparse direct solvers. There has been significant improvement in the performance of sparse direct solvers. Parallel sparse direct solvers are not found to exhibit good scalability. Hence, the parallelization of sparse direct solvers is done using domain decomposition techniques. A highly efficient sparse direct solver PARDISO is used in this study. The scalability of both Newton and modified Newton algorithms are tested
Hybrid static/dynamic scheduling for already optimized dense matrix factorization
We present the use of a hybrid static/dynamic scheduling strategy of the task
dependency graph for direct methods used in dense numerical linear algebra.
This strategy provides a balance of data locality, load balance, and low
dequeue overhead. We show that the usage of this scheduling in communication
avoiding dense factorization leads to significant performance gains. On a 48
core AMD Opteron NUMA machine, our experiments show that we can achieve up to
64% improvement over a version of CALU that uses fully dynamic scheduling, and
up to 30% improvement over the version of CALU that uses fully static
scheduling. On a 16-core Intel Xeon machine, our hybrid static/dynamic
scheduling approach is up to 8% faster than the version of CALU that uses a
fully static scheduling or fully dynamic scheduling. Our algorithm leads to
speedups over the corresponding routines for computing LU factorization in well
known libraries. On the 48 core AMD NUMA machine, our best implementation is up
to 110% faster than MKL, while on the 16 core Intel Xeon machine, it is up to
82% faster than MKL. Our approach also shows significant speedups compared with
PLASMA on both of these systems
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