2,994 research outputs found
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
An O(log sup 2 N) parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix
An O(log sup 2 N) parallel algorithm is presented for computing the eigenvalues of a symmetric tridiagonal matrix using a parallel algorithm for computing the zeros of the characteristic polynomial. The method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level. Intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros. The exact behavior of the polynomials at the interval endpoints is used to eliminate the usual problems induced by finite precision arithmetic
Lattice density functional theory at finite temperature with strongly density-dependent exchange-correlation potentials
The derivative discontinuity of the exchange-correlation (xc) energy at
integer particle number is a property of the exact, unknown xc functional of
density functional theory (DFT) which is absent in many popular local and
semilocal approximations. In lattice DFT, approximations exist which exhibit a
discontinuity in the xc potential at half filling. However, due to convergence
problems of the Kohn-Sham (KS) self-consistency cycle, the use of these
functionals is mostly restricted to situations where the local density is away
from half filling. Here a numerical scheme for the self-consistent solution of
the lattice KS Hamiltonian with a local xc potential with rapid (or
quasi-discontinuous) density dependence is suggested. The problem is formulated
in terms of finite-temperature DFT where the discontinuity in the xc potential
emerges naturally in the limit of zero temperature. A simple parametrization is
suggested for the xc potential of the uniform 1D Hubbard model at finite
temperature which is obtained from the solution of the thermodynamic Bethe
ansatz. The feasibility of the numerical scheme is demonstrated by application
to a model of fermionic atoms in a harmonic trap. The corresponding density
profile exhibits a plateau of integer occupation at low temperatures which
melts away for higher temperatures.Comment: 14 pages, 11 figure
Projected Power Iteration for Network Alignment
The network alignment problem asks for the best correspondence between two
given graphs, so that the largest possible number of edges are matched. This
problem appears in many scientific problems (like the study of protein-protein
interactions) and it is very closely related to the quadratic assignment
problem which has graph isomorphism, traveling salesman and minimum bisection
problems as particular cases. The graph matching problem is NP-hard in general.
However, under some restrictive models for the graphs, algorithms can
approximate the alignment efficiently. In that spirit the recent work by Feizi
and collaborators introduce EigenAlign, a fast spectral method with convergence
guarantees for Erd\H{o}s-Reny\'i graphs. In this work we propose the algorithm
Projected Power Alignment, which is a projected power iteration version of
EigenAlign. We numerically show it improves the recovery rates of EigenAlign
and we describe the theory that may be used to provide performance guarantees
for Projected Power Alignment.Comment: 8 page
Branch-and-Prune Search Strategies for Numerical Constraint Solving
When solving numerical constraints such as nonlinear equations and
inequalities, solvers often exploit pruning techniques, which remove redundant
value combinations from the domains of variables, at pruning steps. To find the
complete solution set, most of these solvers alternate the pruning steps with
branching steps, which split each problem into subproblems. This forms the
so-called branch-and-prune framework, well known among the approaches for
solving numerical constraints. The basic branch-and-prune search strategy that
uses domain bisections in place of the branching steps is called the bisection
search. In general, the bisection search works well in case (i) the solutions
are isolated, but it can be improved further in case (ii) there are continuums
of solutions (this often occurs when inequalities are involved). In this paper,
we propose a new branch-and-prune search strategy along with several variants,
which not only allow yielding better branching decisions in the latter case,
but also work as well as the bisection search does in the former case. These
new search algorithms enable us to employ various pruning techniques in the
construction of inner and outer approximations of the solution set. Our
experiments show that these algorithms speed up the solving process often by
one order of magnitude or more when solving problems with continuums of
solutions, while keeping the same performance as the bisection search when the
solutions are isolated.Comment: 43 pages, 11 figure
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