Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions

GMC Collisions As Triggers of Star Formation. IV. The Role of Ambipolar Diffusion

We investigate the role of ambipolar diffusion (AD) in collisions between magnetized giant molecular clouds (GMCs), which may be an important mechanism for triggering star cluster formation. Three dimensional simulations of GMC collisions are performed using a version of the Enzo magnetohydrodynamics code that has been extended to include AD. The resistivities are calculated using the 31-species chemical model of Wu et al. (2015). We find that in the weak-field, $10\:{\rm \mu G}$ case, AD has only a modest effect on the dynamical evolution during the collision. However, for the stronger-field, $30\:{\rm \mu G}$ case involving near-critical clouds, AD results in formation of dense cores in regions where collapse is otherwise inhibited. The overall efficiency of formation of cores with $n_{\rm H}\geq10^{6}\:{\rm cm}^{-3}$ in these simulations is increases from about 0.2% to 2% once AD is included, comparable to observed values in star-forming GMCs. The gas around these cores typically has relatively slow infall at speeds that are a modest fraction of the free-fall speed.Comment: 15 pages, 15 figures, Accepted to Ap

GMC Collisions As Triggers of Star Formation. IV. The Role of Ambipolar Diffusion

We investigate the role of ambipolar diffusion (AD) in collisions between magnetized giant molecular clouds (GMCs), which may be an important mechanism for triggering star cluster formation. Three dimensional simulations of GMC collisions are performed using a version of the Enzo magnetohydrodynamics code that has been extended to include AD. The resistivities are calculated using the 31-species chemical model of Wu et al. (2015). We find that in the weak-field, $10\:{\rm \mu G}$ case, AD has only a modest effect on the dynamical evolution during the collision. However, for the stronger-field, $30\:{\rm \mu G}$ case involving near-critical clouds, AD results in formation of dense cores in regions where collapse is otherwise inhibited. The overall efficiency of formation of cores with $n_{\rm H}\geq10^{6}\:{\rm cm}^{-3}$ in these simulations is increases from about 0.2% to 2% once AD is included, comparable to observed values in star-forming GMCs. The gas around these cores typically has relatively slow infall at speeds that are a modest fraction of the free-fall speed.Comment: 15 pages, 15 figures, Accepted to Ap

An average-case depth hierarchy theorem for Boolean circuits

We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of $\mathsf{AND}$, $\mathsf{OR}$, and $\mathsf{NOT}$ gates. Our hierarchy theorem says that for every $d \geq 2$, there is an explicit $n$-variable Boolean function $f$, computed by a linear-size depth-$d$ formula, which is such that any depth-$(d-1)$ circuit that agrees with $f$ on $(1/2 + o_n(1))$ fraction of all inputs must have size $\exp({n^{\Omega(1/d)}}).$ This answers an open question posed by H{\aa}stad in his Ph.D. thesis. Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus answering a question posed by O'Donnell, Kalai, and Hatami. A key ingredient in our proof is a notion of \emph{random projections} which generalize random restrictions

Computational Modelling of Wing Downwash Profile with Reynolds-Averaged and Delayed Detached-Eddy Simulations

This paper describes the computational model to predict downwash for a conventional fixed wing configuration at flight scales (ReMAC = 2.26 × 107 ). The lack of resolution in the downwash wake region resulted in an over-dissipation of the turbulent behaviour of airflow in the wing’s wake. This artificially inflates the effectiveness of the horizontal stabilizer where an over-prediction of pitch stiffness was observed. To resolve this over-dissipation, both the Reynolds-Averaged and Delayed Detached-Eddy Simulation methodology were adopted to accurately capture the downwash profile leaving the wing. Comparisons between the estimation of wall shear stresses and viscous wall unit against a ‘first-cut’ simulation are made and discussed. Fundamental features of the downwash profile including the spatial and temporal scales used for the mesh are also presented and detailed in this paper

Minimal Carbon Requirements for Potential Colonizers of Other Planets

The NASA Office of Planetary Protection regulates the safe scientific exploration of other planets. Specifically, the office enacts rules to discourage interplanetary mission practices that would lead to the contamination of Earth-originating microbial life on other planets. Interplanetary contamination jeopardizes the potential to obtain reliable scientific evidence for extraterrestrial life. In coordination with this office, the biosignatures of potential colonizers of other celestials bodies are studied. Several organisms of Earth qualify as potential colonizers of other planets. This experiment focused on the environment of Mars in particular. Two organisms were tested: Desulfovibrio arcticus and Desulfotalea psychrophila. Both are psychrotolerant or psychrophilic anaerobic bacteria isolated from the arctic circle that metabolize by reducing sulfates to sulfide. Due to these characteristics, they are prime candidates to survive on the Martian surface, where sulfates and carbon sources are present. Four carbon sources were tested in different percentages: 0.5%, 1%, 2%, 3% and 4% of dextrose, glycerol, sodium acetate, and isobutyric acid. The carbon sources on Mars are not yet known, but these represent four different lengths of carbon sources that could be found on Mars. Based on the results of the experiment, the evidence supports the notion that D. arcticus was not able to metabolize any of the four carbon sources presented in any of the five different percentages. This could be due to suboptimal growth conditions or the complexity of the carbon sources themselves being incompatible with the growth of D. arcticus. However, the evidence indicates that D. psychrophila was able to metabolize some percentages of all carbon sources except for isobutyric acid. It was able to metabolize higher percentages of dextrose and lower percentages of glycerol and sodium acetate. The next project should be the investigation of other Martian conditions, such as temperature and pressure, in order to test the viability of the organisms in the Martian environment

Minimal Carbon Requirements for Potential Colonizers of Other Planets

The NASA Office of Planetary Protection regulates the safe scientific exploration of other planets. Specifically, the office enacts rules to discourage interplanetary mission practices that would lead to the contamination of Earth-originating microbial life on other planets. Interplanetary contamination jeopardizes the potential to obtain reliable scientific evidence for extraterrestrial life. In coordination with this office, the biosignatures of potential colonizers of other celestials bodies are studied. Several organisms of Earth qualify as potential colonizers of other planets. This experiment focused on the environment of Mars in particular. Two organisms were tested: Desulfovibrio arcticus and Desulfotalea psychrophila. Both are psychrotolerant or psychrophilic anaerobic bacteria isolated from the arctic circle that metabolize by reducing sulfates to sulfide. Due to these characteristics, they are prime candidates to survive on the Martian surface, where sulfates and carbon sources are present. Four carbon sources were tested in different percentages: 0.5%, 1%, 2%, 3% and 4% of dextrose, glycerol, sodium acetate, and isobutyric acid. The carbon sources on Mars are not yet known, but these represent four different lengths of carbon sources that could be found on Mars. Based on the results of the experiment, the evidence supports the notion that D. arcticus was not able to metabolize any of the four carbon sources presented in any of the five different percentages. This could be due to suboptimal growth conditions or the complexity of the carbon sources themselves being incompatible with the growth of D. arcticus. However, the evidence indicates that D. psychrophila was able to metabolize some percentages of all carbon sources except for isobutyric acid. It was able to metabolize higher percentages of dextrose and lower percentages of glycerol and sodium acetate. The next project should be the investigation of other Martian conditions, such as temperature and pressure, in order to test the viability of the organisms in the Martian environment

Where and Who? Automatic Semantic-Aware Person Composition

Image compositing is a method used to generate realistic yet fake imagery by inserting contents from one image to another. Previous work in compositing has focused on improving appearance compatibility of a user selected foreground segment and a background image (i.e. color and illumination consistency). In this work, we instead develop a fully automated compositing model that additionally learns to select and transform compatible foreground segments from a large collection given only an input image background. To simplify the task, we restrict our problem by focusing on human instance composition, because human segments exhibit strong correlations with their background and because of the availability of large annotated data. We develop a novel branching Convolutional Neural Network (CNN) that jointly predicts candidate person locations given a background image. We then use pre-trained deep feature representations to retrieve person instances from a large segment database. Experimental results show that our model can generate composite images that look visually convincing. We also develop a user interface to demonstrate the potential application of our method.Comment: 10 pages, 9 figure