1,643 research outputs found
Three-Level Parallel J-Jacobi Algorithms for Hermitian Matrices
The paper describes several efficient parallel implementations of the
one-sided hyperbolic Jacobi-type algorithm for computing eigenvalues and
eigenvectors of Hermitian matrices. By appropriate blocking of the algorithms
an almost ideal load balancing between all available processors/cores is
obtained. A similar blocking technique can be used to exploit local cache
memory of each processor to further speed up the process. Due to diversity of
modern computer architectures, each of the algorithms described here may be the
method of choice for a particular hardware and a given matrix size. All
proposed block algorithms compute the eigenvalues with relative accuracy
similar to the original non-blocked Jacobi algorithm.Comment: Submitted for publicatio
A GPU-based hyperbolic SVD algorithm
A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm,
using a massively parallel graphics processing unit (GPU), is developed. The
algorithm also serves as the final stage of solving a symmetric indefinite
eigenvalue problem. Numerical testing demonstrates the gains in speed and
accuracy over sequential and MPI-parallelized variants of similar Jacobi-type
HSVD algorithms. Finally, possibilities of hybrid CPU--GPU parallelism are
discussed.Comment: Accepted for publication in BIT Numerical Mathematic
Novel Modifications of Parallel Jacobi Algorithms
We describe two main classes of one-sided trigonometric and hyperbolic
Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian
matrices. These types of algorithms exhibit significant advantages over many
other eigenvalue algorithms. If the matrices permit, both types of algorithms
compute the eigenvalues and eigenvectors with high relative accuracy.
We present novel parallelization techniques for both trigonometric and
hyperbolic classes of algorithms, as well as some new ideas on how pivoting in
each cycle of the algorithm can improve the speed of the parallel one-sided
algorithms. These parallelization approaches are applicable to both
distributed-memory and shared-memory machines.
The numerical testing performed indicates that the hyperbolic algorithms may
be superior to the trigonometric ones, although, in theory, the latter seem
more natural.Comment: Accepted for publication in Numerical Algorithm
A hierarchically blocked Jacobi SVD algorithm for single and multiple graphics processing units
We present a hierarchically blocked one-sided Jacobi algorithm for the
singular value decomposition (SVD), targeting both single and multiple graphics
processing units (GPUs). The blocking structure reflects the levels of GPU's
memory hierarchy. The algorithm may outperform MAGMA's dgesvd, while retaining
high relative accuracy. To this end, we developed a family of parallel pivot
strategies on GPU's shared address space, but applicable also to inter-GPU
communication. Unlike common hybrid approaches, our algorithm in a single GPU
setting needs a CPU for the controlling purposes only, while utilizing GPU's
resources to the fullest extent permitted by the hardware. When required by the
problem size, the algorithm, in principle, scales to an arbitrary number of GPU
nodes. The scalability is demonstrated by more than twofold speedup for
sufficiently large matrices on a Tesla S2050 system with four GPUs vs. a single
Fermi card.Comment: Accepted for publication in SIAM Journal on Scientific Computin
Structure Preserving Parallel Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem
The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue
problem arising from discretized Bethe-Salpeter equation in the context of
computing exciton energies and states. A computational challenge is that at
least half of the eigenvalues and the associated eigenvectors are desired in
practice. We establish the equivalence between Bethe-Salpeter eigenvalue
problems and real Hamiltonian eigenvalue problems. Based on theoretical
analysis, structure preserving algorithms for a class of Bethe-Salpeter
eigenvalue problems are proposed. We also show that for this class of problems
all eigenvalues obtained from the Tamm-Dancoff approximation are overestimated.
In order to solve large scale problems of practical interest, we discuss
parallel implementations of our algorithms targeting distributed memory
systems. Several numerical examples are presented to demonstrate the efficiency
and accuracy of our algorithms
Approximate matrix and tensor diagonalization by unitary transformations: convergence of Jacobi-type algorithms
We propose a gradient-based Jacobi algorithm for a class of maximization
problems on the unitary group, with a focus on approximate diagonalization of
complex matrices and tensors by unitary transformations. We provide weak
convergence results, and prove local linear convergence of this algorithm.The
convergence results also apply to the case of real-valued tensors
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