Image-Based Visualization of Classifier Decision Boundaries

Understanding how a classifier partitions a high-dimensional input space and assigns labels to the parts is an important task in machine learning. Current methods for this task mainly use color-coded sample scatterplots, which do not explicitly show the actual decision boundaries or confusion zones. We propose an image-based technique to improve such visualizations. The method samples the 2D space of a dimensionality-reduction projection and color-code relevant classifier outputs, such as the majority class label, the confusion, and the sample density, to render a dense depiction of the high-dimensional decision boundaries. Our technique is simple to implement, handles any classifier, and has only two simple-to-control free parameters. We demonstrate our proposal on several real-world high-dimensional datasets, classifiers, and two different dimensionality reduction methods

Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics

Conservative numerical schemes for general relativistic magnetohydrodynamics (GRMHD) require a method for transforming between ``conserved'' variables such as momentum and energy density and ``primitive'' variables such as rest-mass density, internal energy, and components of the four-velocity. The forward transformation (primitive to conserved) has a closed-form solution, but the inverse transformation (conserved to primitive) requires the solution of a set of five nonlinear equations. Here we discuss the mathematical properties of the inverse transformation and present six numerical methods for performing the inversion. The first method solves the full set of five nonlinear equations directly using a Newton-Raphson scheme and a guess from the previous timestep. The other methods reduce the five nonlinear equations to either one or two nonlinear equations that are solved numerically. Comparisons between the methods are made using a survey over phase space, a two-dimensional explosion problem, and a general relativistic MHD accretion disk simulation. The run-time of the methods is also examined. Code implementing the schemes is available for download on the web.Comment: Accepted to ApJ, 33 pages, 8 figures (color and greyscale), 1 machine-readable table (tab2.txt), code available at http://rainman.astro.uiuc.edu/codelib, a high-resolution and full-color PDF version is located at http://rainman.astro.uiuc.edu/codelib/codes/pvs_grmhd/ms.pd

General-relativistic resistive magnetohydrodynamics in three dimensions: Formulation and tests

We present a new numerical implementation of the general-relativistic resistive magnetohydrodynamics (MHD) equations within the Whisky code. The numerical method adopted exploits the properties of implicit-explicit Runge-Kutta numerical schemes to treat the stiff terms that appear in the equations for large electrical conductivities. Using tests in one, two, and three dimensions, we show that our implementation is robust and recovers the ideal-MHD limit in regimes of very high conductivity. Moreover, the results illustrate that the code is capable of describing scenarios in a very wide range of conductivities. In addition to tests in flat spacetime, we report simulations of magnetized nonrotating relativistic stars, both in the Cowling approximation and in dynamical spacetimes. Finally, because of its astrophysical relevance and because it provides a severe testbed for general-relativistic codes with dynamical electromagnetic fields, we study the collapse of a nonrotating star to a black hole. We show that also in this case our results on the quasinormal mode frequencies of the excited electromagnetic fields in the Schwarzschild background agree with the perturbative studies within 0.7% and 5.6% for the real and the imaginary part of the l=1 mode eigenfrequency, respectively. Finally we provide an estimate of the electromagnetic efficiency of this process.Comment: 22 pages, 19 figure

A three-dimensional Galactic extinction model

A large-scale three-dimensional model of Galactic extinction is presented based on the Galactic dust distribution model of Drimmel and Spergel (2001). The extinction A_V to any point within the Galactic disk can be quickly deduced using a set of three-dimensional cartesian grids. Extinctions from the model are compared to empirical extinction measures, including lines-of-sight in and near the Galactic plane using optical and NIR extinction measures; in particular we show how extinction can be derived from NIR color-magnitude diagrams in the Galactic plane to a distance of 8 kiloparsec.Comment: 12 pages, to be published in A&