569 research outputs found
High-Performance Solvers for Dense Hermitian Eigenproblems
We introduce a new collection of solvers - subsequently called EleMRRR - for
large-scale dense Hermitian eigenproblems. EleMRRR solves various types of
problems: generalized, standard, and tridiagonal eigenproblems. Among these,
the last is of particular importance as it is a solver on its own right, as
well as the computational kernel for the first two; we present a fast and
scalable tridiagonal solver based on the Algorithm of Multiple Relatively
Robust Representations - referred to as PMRRR. Like the other EleMRRR solvers,
PMRRR is part of the freely available Elemental library, and is designed to
fully support both message-passing (MPI) and multithreading parallelism (SMP).
As a result, the solvers can equally be used in pure MPI or in hybrid MPI-SMP
fashion. We conducted a thorough performance study of EleMRRR and ScaLAPACK's
solvers on two supercomputers. Such a study, performed with up to 8,192 cores,
provides precise guidelines to assemble the fastest solver within the ScaLAPACK
framework; it also indicates that EleMRRR outperforms even the fastest solvers
built from ScaLAPACK's components
An Optimized and Scalable Eigensolver for Sequences of Eigenvalue Problems
In many scientific applications the solution of non-linear differential
equations are obtained through the set-up and solution of a number of
successive eigenproblems. These eigenproblems can be regarded as a sequence
whenever the solution of one problem fosters the initialization of the next. In
addition, in some eigenproblem sequences there is a connection between the
solutions of adjacent eigenproblems. Whenever it is possible to unravel the
existence of such a connection, the eigenproblem sequence is said to be
correlated. When facing with a sequence of correlated eigenproblems the current
strategy amounts to solving each eigenproblem in isolation. We propose a
alternative approach which exploits such correlation through the use of an
eigensolver based on subspace iteration and accelerated with Chebyshev
polynomials (ChFSI). The resulting eigensolver is optimized by minimizing the
number of matrix-vector multiplications and parallelized using the Elemental
library framework. Numerical results show that ChFSI achieves excellent
scalability and is competitive with current dense linear algebra parallel
eigensolvers.Comment: 23 Pages, 6 figures. First revision of an invited submission to
special issue of Concurrency and Computation: Practice and Experienc
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
Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach
The real symmetric tridiagonal eigenproblem is of outstanding importance in
numerical computations; it arises frequently as part of eigensolvers for
standard and generalized dense Hermitian eigenproblems that are based on a
reduction to tridiagonal form. For its solution, the algorithm of Multiple
Relatively Robust Representations (MRRR) is among the fastest methods. Although
fast, the solvers based on MRRR do not deliver the same accuracy as competing
methods like Divide & Conquer or the QR algorithm. In this paper, we
demonstrate that the use of mixed precisions leads to improved accuracy of
MRRR-based eigensolvers with limited or no performance penalty. As a result, we
obtain eigensolvers that are not only equally or more accurate than the best
available methods, but also -in most circumstances- faster and more scalable
than the competition
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