2,566 research outputs found
An Efficient Dual Approach to Distance Metric Learning
Distance metric learning is of fundamental interest in machine learning
because the distance metric employed can significantly affect the performance
of many learning methods. Quadratic Mahalanobis metric learning is a popular
approach to the problem, but typically requires solving a semidefinite
programming (SDP) problem, which is computationally expensive. Standard
interior-point SDP solvers typically have a complexity of (with
the dimension of input data), and can thus only practically solve problems
exhibiting less than a few thousand variables. Since the number of variables is
, this implies a limit upon the size of problem that can
practically be solved of around a few hundred dimensions. The complexity of the
popular quadratic Mahalanobis metric learning approach thus limits the size of
problem to which metric learning can be applied. Here we propose a
significantly more efficient approach to the metric learning problem based on
the Lagrange dual formulation of the problem. The proposed formulation is much
simpler to implement, and therefore allows much larger Mahalanobis metric
learning problems to be solved. The time complexity of the proposed method is
, which is significantly lower than that of the SDP approach.
Experiments on a variety of datasets demonstrate that the proposed method
achieves an accuracy comparable to the state-of-the-art, but is applicable to
significantly larger problems. We also show that the proposed method can be
applied to solve more general Frobenius-norm regularized SDP problems
approximately
Very High-Order A-stable Stiffly Accurate Diagonally Implicit Runge-Kutta Methods with Error Estimators
A numerical search approach is used to design high-order diagonally implicit
Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of
which have identical diagonal elements (SDIRK) and explicit first stage
(ESDIRK). In each of these classes, we present new A-stable schemes of order
six (the highest order of previously known A-stable DIRK-type schemes) up to
order eight. For each order, we include one scheme that is only A-stable as
well as schemes that are L-stable, stiffly accurate, and/or have stage order
two. The latter types require more stages, but give better convergence rates
for differential-algebraic equations (DAEs), and those which have stage order
two give better accuracy for moderately stiff problems. The development of the
eighth-order schemes requires, in addition to imposing A-stability, finding
highly accurate numerical solutions for a system of 200 equations in over 100
variables, which is accomplished via a combination of global and local
optimization strategies. The accuracy, stability, and adaptive stepsize control
of the schemes are demonstrated on diverse problems
SpECTRE: A Task-based Discontinuous Galerkin Code for Relativistic Astrophysics
We introduce a new relativistic astrophysics code, SpECTRE, that combines a
discontinuous Galerkin method with a task-based parallelism model. SpECTRE's
goal is to achieve more accurate solutions for challenging relativistic
astrophysics problems such as core-collapse supernovae and binary neutron star
mergers. The robustness of the discontinuous Galerkin method allows for the use
of high-resolution shock capturing methods in regions where (relativistic)
shocks are found, while exploiting high-order accuracy in smooth regions. A
task-based parallelism model allows efficient use of the largest supercomputers
for problems with a heterogeneous workload over disparate spatial and temporal
scales. We argue that the locality and algorithmic structure of discontinuous
Galerkin methods will exhibit good scalability within a task-based parallelism
framework. We demonstrate the code on a wide variety of challenging benchmark
problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the
code's scalability including its strong scaling on the NCSA Blue Waters
supercomputer up to the machine's full capacity of 22,380 nodes using 671,400
threads.Comment: 41 pages, 13 figures, and 7 tables. Ancillary data contains
simulation input file
An Unsplit, Cell-Centered Godunov Method for Ideal MHD
We present a second-order Godunov algorithm for multidimensional, ideal MHD.
Our algorithm is based on the unsplit formulation of Colella (J. Comput. Phys.
vol. 87, 1990), with all of the primary dependent variables centered at the
same location. To properly represent the divergence-free condition of the
magnetic fields, we apply a discrete projection to the intermediate values of
the field at cell faces, and apply a filter to the primary dependent variables
at the end of each time step. We test the method against a suite of linear and
nonlinear tests to ascertain accuracy and stability of the scheme under a
variety of conditions. The test suite includes rotated planar linear waves, MHD
shock tube problems, low-beta flux tubes, and a magnetized rotor problem. For
all of these cases, we observe that the algorithm is second-order accurate for
smooth solutions, converges to the correct weak solution for problems involving
shocks, and exhibits no evidence of instability or loss of accuracy due to the
possible presence of non-solenoidal fields.Comment: 37 Pages, 9 Figures, submitted to Journal of Computational Physic
A global residualâbased stabilization for equalâorder finite element approximations of incompressible flows
Due to simplicity in implementation and data structure, elements with equal-order interpolation of velocity and pressure are very popular in finite-element-based flow simulations. Although such pairs are inf-sup unstable, various stabilization techniques exist to circumvent that and yield accurate approximations. The most popular one is the pressure-stabilized PetrovâGalerkin (PSPG) method, which consists of relaxing the incompressibility constraint with a weighted residual of the momentum equation. Yet, PSPG can perform poorly for low-order elements in diffusion-dominated flows, since first-order polynomial spaces are unable to approximate the second-order derivatives required for evaluating the viscous part of the stabilization term. Alternative techniques normally require additional projections or unconventional data structures. In this context, we present a novel technique that rewrites the second-order viscous term as a first-order boundary term, thereby allowing the complete computation of the residual even for lowest-order elements. Our method has a similar structure to standard residual-based formulations, but the stabilization term is computed globally instead of only in element interiors. This results in a scheme that does not relax incompressibility, thereby leading to improved approximations. The new method is simple to implement and accurate for a wide range of stabilization parameters, which is confirmed by various numerical examples
The Configurable SAT Solver Challenge (CSSC)
It is well known that different solution strategies work well for different
types of instances of hard combinatorial problems. As a consequence, most
solvers for the propositional satisfiability problem (SAT) expose parameters
that allow them to be customized to a particular family of instances. In the
international SAT competition series, these parameters are ignored: solvers are
run using a single default parameter setting (supplied by the authors) for all
benchmark instances in a given track. While this competition format rewards
solvers with robust default settings, it does not reflect the situation faced
by a practitioner who only cares about performance on one particular
application and can invest some time into tuning solver parameters for this
application. The new Configurable SAT Solver Competition (CSSC) compares
solvers in this latter setting, scoring each solver by the performance it
achieved after a fully automated configuration step. This article describes the
CSSC in more detail, and reports the results obtained in its two instantiations
so far, CSSC 2013 and 2014
Three Dimensional Modeling of Hot Jupiter Atmospheric Flows
We present a three dimensional hot Jupiter model, extending from 200 bar to 1
mbar, using the Intermediate General Circulation Model from the University of
Reading. Our horizontal spectral resolution is T31 (equivalent to a grid of
48x96), with 33 logarithmically spaced vertical levels. A simplified
(Newtonian) scheme is employed for the radiative forcing. We adopt a physical
set up nearly identical to the model of HD 209458b by Cooper & Showman
(2005,2006) to facilitate a direct model inter-comparison. Our results are
broadly consistent with theirs but significant differences also emerge. The
atmospheric flow is characterized by a super-rotating equatorial jet, transonic
wind speeds, and eastward advection of heat away from the dayside. We identify
a dynamically-induced temperature inversion ("stratosphere") on the planetary
dayside and find that temperatures at the planetary limb differ systematically
from local radiative equilibrium values, a potential source of bias for transit
spectroscopic interpretations. While our model atmosphere is quasi-identical to
that of Cooper & Showman (2005,2006) and we solve the same meteorological
equations, we use different algorithmic methods, spectral-implicit vs.
grid-explicit, which are known to yield fully consistent results in the Earth
modeling context. The model discrepancies identified here indicate that one or
both numerical methods do not faithfully capture all of the atmospheric
dynamics at work in the hot Jupiter context. We highlight the emergence of a
shock-like feature in our model, much like that reported recently by Showman et
al. (2009), and suggest that improved representations of energy conservation
may be needed in hot Jupiter atmospheric models, as emphasized by Goodman
(2009).Comment: 25 pages, 6 figures, minor revisions, ApJ accepted, version with
hi-res figures:
http://www.astro.columbia.edu/~kristen/Hires/hotjup.3d.deep.ps.g
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