34,033 research outputs found
Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics
We study a probabilistic numerical method for the solution of both boundary
and initial value problems that returns a joint Gaussian process posterior over
the solution. Such methods have concrete value in the statistics on Riemannian
manifolds, where non-analytic ordinary differential equations are involved in
virtually all computations. The probabilistic formulation permits marginalising
the uncertainty of the numerical solution such that statistics are less
sensitive to inaccuracies. This leads to new Riemannian algorithms for mean
value computations and principal geodesic analysis. Marginalisation also means
results can be less precise than point estimates, enabling a noticeable
speed-up over the state of the art. Our approach is an argument for a wider
point that uncertainty caused by numerical calculations should be tracked
throughout the pipeline of machine learning algorithms.Comment: 11 page (9 page conference paper, plus supplements
Faster SDP hierarchy solvers for local rounding algorithms
Convex relaxations based on different hierarchies of linear/semi-definite
programs have been used recently to devise approximation algorithms for various
optimization problems. The approximation guarantee of these algorithms improves
with the number of {\em rounds} in the hierarchy, though the complexity of
solving (or even writing down the solution for) the 'th level program grows
as where is the input size.
In this work, we observe that many of these algorithms are based on {\em
local} rounding procedures that only use a small part of the SDP solution (of
size instead of ). We give an algorithm to
find the requisite portion in time polynomial in its size. The challenge in
achieving this is that the required portion of the solution is not fixed a
priori but depends on other parts of the solution, sometimes in a complicated
iterative manner.
Our solver leads to time algorithms to obtain the same
guarantees in many cases as the earlier time algorithms based on
rounds of the Lasserre hierarchy. In particular, guarantees based on rounds can be realized in polynomial time.
We develop and describe our algorithm in a fairly general abstract framework.
The main technical tool in our work, which might be of independent interest in
convex optimization, is an efficient ellipsoid algorithm based separation
oracle for convex programs that can output a {\em certificate of infeasibility
with restricted support}. This is used in a recursive manner to find a sequence
of consistent points in nested convex bodies that "fools" local rounding
algorithms.Comment: 30 pages, 8 figure
Fully-Coupled Simulation of Cosmic Reionization. I: Numerical Methods and Tests
We describe an extension of the Enzo code to enable fully-coupled radiation
hydrodynamical simulation of inhomogeneous reionization in large cosmological volumes with thousands to millions of point sources. We
solve all dynamical, radiative transfer, thermal, and ionization processes
self-consistently on the same mesh, as opposed to a postprocessing approach
which coarse-grains the radiative transfer. We do, however, employ a simple
subgrid model for star formation which we calibrate to observations. Radiation
transport is done in the grey flux-limited diffusion (FLD) approximation, which
is solved by implicit time integration split off from the gas energy and
ionization equations, which are solved separately. This results in a faster and
more robust scheme for cosmological applications compared to the earlier
method. The FLD equation is solved using the hypre optimally scalable geometric
multigrid solver from LLNL. By treating the ionizing radiation as a grid field
as opposed to rays, our method is scalable with respect to the number of
ionizing sources, limited only by the parallel scaling properties of the
radiation solver. We test the speed and accuracy of our approach on a number of
standard verification and validation tests. We show by direct comparison with
Enzo's adaptive ray tracing method Moray that the well-known inability of FLD
to cast a shadow behind opaque clouds has a minor effect on the evolution of
ionized volume and mass fractions in a reionization simulation validation test.
We illustrate an application of our method to the problem of inhomogeneous
reionization in a 80 Mpc comoving box resolved with Eulerian grid
cells and dark matter particles.Comment: 32 pages, 23 figures. ApJ Supp accepted. New title and substantial
revisions re. v
Corner contribution to the entanglement entropy of strongly-interacting O(2) quantum critical systems in 2+1 dimensions
In a D=2+1 quantum critical system, the entanglement entropy across a
boundary with a corner contains a subleading logarithmic scaling term with a
universal coefficient. It has been conjectured that this coefficient is, to
leading order, proportional to the number of field components N in the
associated O(N) continuum field theory. Using density matrix
renormalization group calculations combined with the powerful numerical linked
cluster expansion technique, we confirm this scenario for the O(2)
Wilson-Fisher fixed point in a striking way, through direct calculation at the
quantum critical points of two very different microscopic models. The value of
this corner coefficient is, to within our numerical precision, twice the
coefficient of the Ising fixed point. Our results add to the growing body of
evidence that this universal term in the R\'enyi entanglement entropy reflects
the number of low-energy degrees of freedom in a system, even for strongly
interacting theories.Comment: 6 pages, 6 figure
The Astrophysical Multipurpose Software Environment
We present the open source Astrophysical Multi-purpose Software Environment
(AMUSE, www.amusecode.org), a component library for performing astrophysical
simulations involving different physical domains and scales. It couples
existing codes within a Python framework based on a communication layer using
MPI. The interfaces are standardized for each domain and their implementation
based on MPI guarantees that the whole framework is well-suited for distributed
computation. It includes facilities for unit handling and data storage.
Currently it includes codes for gravitational dynamics, stellar evolution,
hydrodynamics and radiative transfer. Within each domain the interfaces to the
codes are as similar as possible. We describe the design and implementation of
AMUSE, as well as the main components and community codes currently supported
and we discuss the code interactions facilitated by the framework.
Additionally, we demonstrate how AMUSE can be used to resolve complex
astrophysical problems by presenting example applications.Comment: 23 pages, 25 figures, accepted for A&
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