Convergence of the restricted Nelder-Mead algorithm in two dimensions

The Nelder-Mead algorithm, a longstanding direct search method for unconstrained optimization published in 1965, is designed to minimize a scalar-valued function f of n real variables using only function values, without any derivative information. Each Nelder-Mead iteration is associated with a nondegenerate simplex defined by n+1 vertices and their function values; a typical iteration produces a new simplex by replacing the worst vertex by a new point. Despite the method's widespread use, theoretical results have been limited: for strictly convex objective functions of one variable with bounded level sets, the algorithm always converges to the minimizer; for such functions of two variables, the diameter of the simplex converges to zero, but examples constructed by McKinnon show that the algorithm may converge to a nonminimizing point. This paper considers the restricted Nelder-Mead algorithm, a variant that does not allow expansion steps. In two dimensions we show that, for any nondegenerate starting simplex and any twice-continuously differentiable function with positive definite Hessian and bounded level sets, the algorithm always converges to the minimizer. The proof is based on treating the method as a discrete dynamical system, and relies on several techniques that are non-standard in convergence proofs for unconstrained optimization.Comment: 27 page

Using visualization for visualization : an ecological interface design approach to inputting data

Visualization is experiencing growing use by a diverse community, with continuing improvements in the availability and usability of systems. In spite of these developments the problem of how first to get the data in has received scant attention: the established approach of pre-defined readers and programming aids has changed little in the last two decades. This paper proposes a novel way of inputting data for scientific visualization that employs rapid interaction and visual feedback in order to understand how the data is stored. The approach draws on ideas from the discipline of ecological interface design to extract and control important parameters describing the data, at the same time harnessing our innate human ability to recognize patterns. Crucially, the emphasis is on file format discovery rather than file format description, so the method can therefore still work when nothing is known initially of how the file was originally written, as is often the case with legacy binary data. © 2013 Elsevier Ltd

The Rotation Measure and 3.5mm Polarization of Sgr A*

We report the detection of variable linear polarization from Sgr A* at a wavelength of 3.5mm, the longest wavelength yet at which a detection has been made. The mean polarization is 2.1 +/- 0.1% at a position angle of 16 +/- 2 deg with rms scatters of 0.4% and 9 deg over the five epochs. We also detect polarization variability on a timescale of days. Combined with previous detections over the range 150-400GHz (750-2000 microns), the average polarization position angles are all found to be consistent with a rotation measure of -4.4 +/- 0.3 x 10^5 rad/m^2. This implies that the Faraday rotation occurs external to the polarized source at all wavelengths. This implies an accretion rate ~0.2 - 4 x 10^-8 Msun/yr for the accretion density profiles expected of ADAF, jet and CDAF models and assuming that the region at which electrons in the accretion flow become relativistic is within 10 R_S. The inferred accretion rate is inconsistent with ADAF/Bondi accretion. The stability of the mean polarization position angle between disparate polarization observations over the frequency range limits fluctuations in the accretion rate to less than 5%. The flat frequency dependence of the inter-day polarization position angle variations also makes them difficult to attribute to rotation measure fluctuations, and suggests that both the magnitude and position angle variations are intrinsic to the emission.Comment: Ap.J.Lett. accepte

A comparison of the excess mass around CFHTLenS galaxy-pairs to predictions from a semi-analytic model using galaxy-galaxy-galaxy lensing

The matter environment of galaxies is connected to the physics of galaxy formation and evolution. Utilising galaxy-galaxy-galaxy lensing as a direct probe, we map out the distribution of correlated surface mass-density around galaxy pairs for different lens separations in the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS). We compare, for the first time, these so-called excess mass maps to predictions provided by a recent semi-analytic model, which is implanted within the dark-matter Millennium Simulation. We analyse galaxies with stellar masses between $10^9-10^{11}\,{\rm M}_\odot$ in two photometric redshift bins, for lens redshifts $z\lesssim0.6$, focusing on pairs inside groups and clusters. To allow us a better interpretation of the maps, we discuss the impact of chance pairs, i.e., galaxy pairs that appear close to each other in projection only. Our tests with synthetic data demonstrate that the patterns observed in the maps are essentially produced by correlated pairs that are close in redshift ($\Delta z\lesssim5\times10^{-3}$). We also verify the excellent accuracy of the map estimators. In an application to the galaxy samples in the CFHTLenS, we obtain a $3\sigma-6\sigma$ significant detection of the excess mass and an overall good agreement with the galaxy model predictions. There are, however, a few localised spots in the maps where the observational data disagrees with the model predictions on a $\approx3.5\sigma$ confidence level. Although we have no strong indications for systematic errors in the maps, this disagreement may be related to the residual B-mode pattern observed in the average of all maps. Alternatively, misaligned galaxy pairs inside dark matter halos or lensing by a misaligned distribution of the intra-cluster gas might also cause the unanticipated bulge in the distribution of the excess mass between lens pairs.Comment: 21 pages, 12 figures; abridged abstract; revised version for A&A after addressing all comments by the refere

Variable Linear Polarization from Sagittarius A*: Evidence for a Hot Turbulent Accretion Flow

We report the discovery of variability in the linear polarization from the Galactic Center black hole source, Sagittarius A*. New polarimetry obtained with the Berkeley-Illinois-Maryland Association array at a wavelength of 1.3 mm shows a position angle that differs by 28 +/- 5 degrees from observations 6 months prior and then remains stable for 15 months. This difference may be due to a change in the source emission region on a scale of 10 Schwarzschild radii or due to a change of 3 x 10^5 rad m^-2 in the rotation measure. We consider a change in the source physics unlikely, however, since we see no corresponding change in the total intensity or polarized intensity fraction. On the other hand, turbulence in the accretion region at a radius ~ 10 to 1000 R_s could readily account for the magnitude and time scale of the position angle change.Comment: accepted for publication in ApJ