510 research outputs found
From stripe to checkerboard order on the square lattice in the presence of quenched disorder
We discuss the effects of quenched disorder on a model of charge density wave
(CDW) ordering on the square lattice. Our model may be applicable to the
cuprate superconductors, where a random electrostatic potential exists in the
CuO2 planes as a result of the presence of charged dopants. We argue that the
presence of a random potential can affect the unidirectionality of the CDW
order, characterized by an Ising order parameter. Coupling to a unidirectional
CDW, the random potential can lead to the formation of domains with 90 degree
relative orientation, thus tending to restore the rotational symmetry of the
underlying lattice. We find that the correlation length of the Ising order can
be significantly larger than the CDW correlation length. For a checkerboard CDW
on the other hand, disorder generates spatial anisotropies on short length
scales and thus some degree of unidirectionality. We quantify these disorder
effects and suggest new techniques for analyzing the local density of states
(LDOS) data measured in scanning tunneling microscopy experiments.Comment: 10 pages, 11 figures; added referenc
Load-Balancing for Parallel Delaunay Triangulations
Computing the Delaunay triangulation (DT) of a given point set in
is one of the fundamental operations in computational geometry.
Recently, Funke and Sanders (2017) presented a divide-and-conquer DT algorithm
that merges two partial triangulations by re-triangulating a small subset of
their vertices - the border vertices - and combining the three triangulations
efficiently via parallel hash table lookups. The input point division should
therefore yield roughly equal-sized partitions for good load-balancing and also
result in a small number of border vertices for fast merging. In this paper, we
present a novel divide-step based on partitioning the triangulation of a small
sample of the input points. In experiments on synthetic and real-world data
sets, we achieve nearly perfectly balanced partitions and small border
triangulations. This almost cuts running time in half compared to
non-data-sensitive division schemes on inputs exhibiting an exploitable
underlying structure.Comment: Short version submitted to EuroPar 201
Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes
We consider an algorithm called FEMWARP for warping triangular and
tetrahedral finite element meshes that computes the warping using the finite
element method itself. The algorithm takes as input a two- or three-dimensional
domain defined by a boundary mesh (segments in one dimension or triangles in
two dimensions) that has a volume mesh (triangles in two dimensions or
tetrahedra in three dimensions) in its interior. It also takes as input a
prescribed movement of the boundary mesh. It computes as output updated
positions of the vertices of the volume mesh. The first step of the algorithm
is to determine from the initial mesh a set of local weights for each interior
vertex that describes each interior vertex in terms of the positions of its
neighbors. These weights are computed using a finite element stiffness matrix.
After a boundary transformation is applied, a linear system of equations based
upon the weights is solved to determine the final positions of the interior
vertices. The FEMWARP algorithm has been considered in the previous literature
(e.g., in a 2001 paper by Baker). FEMWARP has been succesful in computing
deformed meshes for certain applications. However, sometimes FEMWARP reverses
elements; this is our main concern in this paper. We analyze the causes for
this undesirable behavior and propose several techniques to make the method
more robust against reversals. The most successful of the proposed methods
includes combining FEMWARP with an optimization-based untangler.Comment: Revision of earlier version of paper. Submitted for publication in
BIT Numerical Mathematics on 27 April 2010. Accepted for publication on 7
September 2010. Published online on 9 October 2010. The final publication is
available at http://www.springerlink.co
On the stability of viscous free-surface flow supported by a rotating cylinder
Using an adaptive finite-element (FE) scheme developed recently by the authors, we shed new light on the long-standing fundamental problem of the unsteady free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field. For supportable loads, we observe that the steady-state is more readily attained for near-maximal fluid loads on the cylinder than for significantly sub-maximal loads. For the latter, we investigate large-time dynamics by means of a finite-difference approximation to the thin-film equations, which is also used to validate the adaptive FE simulations (applied to the full Stokes equations) for these significantly sub-maximal loads.
Conversely, by comparing results of the two methods, we assess the validity of the thin-film approximation as either the load is increased or the rotation rate of the cylinder is decreased. Results are presented on the independent effects of gravity, surface tension and initial film thickness on the decay to steady-state. Finally, new numerical simulations of load shedding are presented
Universal Scaling of Optimal Current Distribution in Transportation Networks
Transportation networks are inevitably selected with reference to their
global cost which depends on the strengths and the distribution of the embedded
currents. We prove that optimal current distributions for a uniformly injected
d-dimensional network exhibit robust scale-invariance properties, independently
of the particular cost function considered, as long as it is convex. We find
that, in the limit of large currents, the distribution decays as a power law
with an exponent equal to (2d-1)/(d-1). The current distribution can be exactly
calculated in d=2 for all values of the current. Numerical simulations further
suggest that the scaling properties remain unchanged for both random injections
and by randomizing the convex cost functions.Comment: 5 pages, 5 figure
Minimizing the stabbing number of matchings, trees, and triangulations
The (axis-parallel) stabbing number of a given set of line segments is the
maximum number of segments that can be intersected by any one (axis-parallel)
line. This paper deals with finding perfect matchings, spanning trees, or
triangulations of minimum stabbing number for a given set of points. The
complexity of these problems has been a long-standing open question; in fact,
it is one of the original 30 outstanding open problems in computational
geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide
is negative for a number of minimum stabbing problems by showing them NP-hard
by means of a general proof technique. It implies non-trivial lower bounds on
the approximability. On the positive side we propose a cut-based integer
programming formulation for minimizing the stabbing number of matchings and
spanning trees. We obtain lower bounds (in polynomial time) from the
corresponding linear programming relaxations, and show that an optimal
fractional solution always contains an edge of at least constant weight. This
result constitutes a crucial step towards a constant-factor approximation via
an iterated rounding scheme. In computational experiments we demonstrate that
our approach allows for actually solving problems with up to several hundred
points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational
Geometry". Previous version (extended abstract) appears in SODA 2004, pp.
430-43
Evaluating Elevated Convection with the Downdraft Convective Inhibition
A method for evaluating the penetration of a stable layer by an elevated convective downdraft is discussed. Some controversy exists on the community’s ability to define truly elevated convection from surface-based convection. By comparing the downdraft convective inhibition (DCIN) to the downdraft convective available potential energy (DCAPE), we determine that downdraft penetration potential is progressively enabled as the DCIN is progressively smaller than the DCAPE; inversely as DCIN increases over DCAPE, so does the likelihood of purely elevated convection. Serial vertical soundings and accompanying analyses are provided to support this finding
Exploiting asynchrony from exact forward recovery for DUE in iterative solvers
This paper presents a method to protect iterative solvers from Detected and Uncorrected Errors (DUE) relying on error detection techniques already available in commodity hardware. Detection operates at the memory page level, which enables the use of simple algorithmic redundancies to correct errors. Such redundancies would be inapplicable under coarse grain error detection, but become very powerful when the hardware is able to precisely detect errors.
Relations straightforwardly extracted from the solver allow to recover lost data exactly. This method is free of the overheads of backwards recoveries like checkpointing, and does not compromise mathematical convergence properties of the solver as restarting would do. We apply this recovery to three widely used Krylov subspace methods, CG, GMRES and BiCGStab, and their preconditioned versions.
We implement our resilience techniques on CG considering scenarios from small (8 cores) to large (1024 cores) scales, and demonstrate very low overheads compared to state-of-the-art solutions. We deploy our recovery techniques either by overlapping them with algorithmic computations or by forcing them to be in the critical path of the application. A trade-off exists between both approaches depending on the error rate the solver is suffering. Under realistic error rates, overlapping decreases overheads from 5.37% down to 3.59% for a non-preconditioned CG on 8 cores.This work has been partially supported by the European Research Council under the European Union's 7th FP, ERC Advanced Grant 321253, and by the Spanish Ministry of Science and Innovation under grant TIN2012-34557. L. Jaulmes has been partially supported by the Spanish Ministry of Education, Culture and Sports under grant FPU2013/06982.
M. Moreto has been partially supported by the Spanish Ministry of Economy and Competitiveness under Juan de la
Cierva postdoctoral fellowship JCI-2012-15047. M. Casas
has been partially supported by the Secretary for Universities and Research of the Ministry of Economy and Knowledge of the Government of Catalonia and the Co-fund programme of the Marie Curie Actions of the European Union's 7th FP (contract 2013 BP
B 00243).Peer ReviewedPostprint (author's final draft
Incoherent dynamics in the toric code subject to disorder
We numerically study the effects of two forms of quenched disorder on the
anyons of the toric code. Firstly, a new class of codes based on random
lattices of stabilizer operators is presented, and shown to be superior to the
standard square lattice toric code for certain forms of biased noise. It is
further argued that these codes are close to optimal, in that they tightly
reach the upper bound of error thresholds beyond which no correctable CSS codes
can exist. Additionally, we study the classical motion of anyons in toric codes
with randomly distributed onsite potentials. In the presence of repulsive
long-range interaction between the anyons, a surprising increase with disorder
strength of the lifetime of encoded states is reported and explained by an
entirely incoherent mechanism. Finally, the coherent transport of the anyons in
the presence of both forms of disorder is investigated, and a significant
suppression of the anyon motion is found.Comment: 13 pages, 12 figure
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