Efficient Higher Order Derivatives of Objective Functions Composed of Matrix Operations

This paper is concerned with the efficient evaluation of higher-order derivatives of functions $f$ that are composed of matrix operations. I.e., we want to compute the $D$-th derivative tensor $\nabla^D f(X) \in \mathbb R^{N^D}$, where $f:\mathbb R^{N} \to \mathbb R$ is given as an algorithm that consists of many matrix operations. We propose a method that is a combination of two well-known techniques from Algorithmic Differentiation (AD): univariate Taylor propagation on scalars (UTPS) and first-order forward and reverse on matrices. The combination leads to a technique that we would like to call univariate Taylor propagation on matrices (UTPM). The method inherits many desirable properties: It is easy to implement, it is very efficient and it returns not only $\nabla^D f$ but yields in the process also the derivatives $\nabla^d f$ for $d \leq D$. As performance test we compute the gradient $\nabla f(X)$ % and the Hessian $\nabla_A^2 f(A)$ by a combination of forward and reverse mode of f(X) = \trace (X^{-1}) in the reverse mode of AD for $X \in \mathbb R^{n \times n}$. We observe a speedup of about 100 compared to UTPS. Due to the nature of the method, the memory footprint is also small and therefore can be used to differentiate functions that are not accessible by standard methods due to limited physical memory

Dust reverberation mapping in the era of big optical surveys and its cosmological application

The time lag between optical and near-infrared (IR) flux variability can be taken as a means to determine the sublimation radius of the dusty "torus" around supermassive black holes in active galactic nuclei (AGN). I will show that data from big optical survey telescopes, e.g. the Large Synoptic Survey Telescope (LSST), can be used to measure dust sublimation radii as well. The method makes use of the fact that the Wien tail of the hot dust emission reaches into the optical and can be reliably recovered with high-quality photometry. Simulations show that dust sublimation radii for a large sample of AGN can be reliably established out to redshift z ~ 0.1-0.2 with the LSST. Owing to the ubiquitous presence of AGN up to high redshifts, they have been studies as cosmological probes. Here, I discuss how optically-determined dust time lags fit into the suggestion of using the dust sublimation radius as a "standard candle" and propose and extension of the dust time lags as "standard rulers" in combination with IR interferometry.Comment: 6 pages, 4 figures, 1 table; accepted by ApJL (Feb 18, 2014

Non-Singular Bounces Catalysed by Dark Energy

We investigate classically non-singular bounces caused by dark energy. In the presence of positive spatial curvature, vacuum energy, either in the form of a cosmological constant or a scalar field potential, allows for an open set of initial conditions leading to non-singular bounces, without any violation of the null energy condition. We study anisotropic Bianchi IX cosmologies, and demonstrate that they can even have multiple bounces, accompanied by intricate evolutions of the anisotropies that provide a non-singular analogue of mixmaster crunches. The relation of these solutions to more complete cosmological models, as well as to the recently proposed swampland criteria, are briefly discussed.Comment: 30 pages, 25 figure files; v2: references adde

Sum-frequency ionic Raman scattering

In a recent report sum-frequency excitation of a Raman-active phonon was experimentally demonstrated for the first time. This mechanism is the sibling of impulsive stimulated Raman scattering, in which difference-frequency components of a light field excite a Raman-active mode. Here we propose that ionic Raman scattering analogously has a sum-frequency counterpart. We compare the four Raman mechanisms, photonic and ionic difference- and sum-frequency excitation, for three different example materials using a generalized oscillator model for which we calculate the parameters with density functional theory. Sum-frequency ionic Raman scattering completes the toolkit for controlling materials properties by means of selective excitation of lattice vibrations

The dust sublimation radius as an outer envelope to the bulk of the narrow Fe Kalpha line emission in Type 1 AGN

The Fe Kalpha emission line is the most ubiquitous feature in the X-ray spectra of active galactic nuclei (AGN), but the origin of its narrow core remains uncertain. Here, we investigate the connection between the sizes of the Fe core emission regions and the measured sizes of the dusty tori in 13 local Type 1 AGN. The observed Fe K emission radii (R_fe) are determined from spectrally resolved line widths in X-ray grating spectra, and the dust sublimation radii (R_dust) are measured either from optical/near-infrared reverberation time lags or from resolved near-infrared interferometric data. This direct comparison shows, on an object-by-object basis, that the dust sublimation radius forms an outer envelope to the bulk of the Fe K emission. R_fe matches R_dust well in the AGN with the best constrained line widths currently. In a significant fraction of objects without a clear narrow line core, R_fe is similar to, or smaller than the radius of the optical broad line region. These facts place important constraints on the torus geometries for our sample. Extended tori in which the solid angle of fluorescing gas peaks at well beyond the dust sublimation radius can be ruled out. We also test for luminosity scalings of R_fe, finding that Eddington ratio is not a prime driver in determining the line location in our sample. We discuss in detail potential caveats due to data analysis and instrumental limitations, simplistic line modeling, uncertain black hole masses, as well as sample selection, showing that none of these is likely to bias our core result. The calorimeter on board Astro-H will soon vastly increase the parameter space over which line measurements can be made, overcoming many of these limitations.Comment: ApJ in press. Community comments greatly appreciated. 13 pages, 4 figures and 2 tables including an appendi

Constraining Generalized Non-local Cosmology from Noether Symmetries

We study a generalized nonlocal theory of gravity which, in specific limits, can become either the curvature non-local or teleparallel non-local theory. Using the Noether Symmetry Approach, we find that the coupling functions coming from the non-local terms are constrained to be either exponential or linear in form. It is well known that in some non-local theories, a certain kind of exponential non-local couplings are needed in order to achieve a renormalizable theory. In this paper, we explicitly show that this kind of coupling does not need to by introduced by hand, instead, it appears naturally from the symmetries of the Lagrangian in flat Friedmann-Robertson-Walker cosmology. Finally, we find de-Sitter and power law cosmological solutions for different nonlocal theories. The symmetries for the generalized non-local theory is also found and some cosmological solutions are also achieved under the full theory.Comment: 15 pages, to be published in Eur.Phys.J.