Numerical methods for multiscale inverse problems

We consider the inverse problem of determining the highly oscillatory coefficient $a^\epsilon$ in partial differential equations of the form $-\nabla\cdot (a^\epsilon\nabla u^\epsilon)+bu^\epsilon = f$ from given measurements of the solutions. Here, $\epsilon$ indicates the smallest characteristic wavelength in the problem ($0<\epsilon\ll1$). In addition to the general difficulty of finding an inverse, the oscillatory nature of the forward problem creates an additional challenge of multiscale modeling, which is hard even for forward computations. The inverse problem in its full generality is typically ill-posed and one common approach is to replace the original problem with an effective parameter estimation problem. We will here include microscale features directly in the inverse problem and avoid ill-posedness by assuming that the microscale can be accurately represented by a low-dimensional parametrization. The basis for our inversion will be a coupling of the parametrization to analytic homogenization or a coupling to efficient multiscale numerical methods when analytic homogenization is not available. We will analyze the reduced problem, $b = 0$, by proving uniqueness of the inverse in certain problem classes and by numerical examples and also include numerical model examples for medical imaging, $b > 0$, and exploration seismology, $b < 0$

The Opacity of the Intergalactic Medium Measured Along Quasar Sightlines at z∼6z\sim 6

We publicly release a new sample of $34$ medium resolution quasar spectra at $5.77\leq z_{\rm em}\leq6.54$ observed with the Echellette Spectrograph and Imager (ESI) on the Keck telescope. This quasar sample represents an ideal laboratory to study the intergalactic medium (IGM) during the end stages of the epoch of reionization, and constrain the timing and morphology of the phase transition. For a subset of $23$ of our highest signal-to-noise ratio spectra (S/N$>7$, per $10\,{\rm km\,s^{-1}}$ pixel), we present a new measurement of the Lyman-$\alpha$ (Ly$\alpha$) forest opacity spanning the redshift range $4.8\lesssim z\lesssim6.3$. We carefully eliminate spectral regions that could be causing biases in our measurements due to additional transmitted flux in the proximity zone of the quasars, or extra absorption caused by strong intervening absorption systems along the line of sight. We compare the observed evolution of the IGM opacity with redshift to predictions from a hydrodynamical simulation with uniform ultraviolet background (UVB) radiation, as well as two semi-numerical patchy reionization models, one with a fluctuating UVB and another with a fluctuating temperature field. Our measurements show a steep rise in opacity at $z\gtrsim5.0$ and an increased scatter and thus support the picture of a spatially inhomogeneous reionization process, consistent with previous work. However, we measure significantly higher optical depths at $5.3\lesssim z\lesssim5.7$ than previous studies, which reduces the contrast between the highest opacity Gunn-Peterson troughs and the average opacity trend of the IGM, which may relieve some of the previously noted tension between these measurements and reionization models.Comment: accepted for publication at Ap

Physiological Effects of Binaural Beats and Meditative Musical Stimulation

The current study examined physiological effects of meditation music and binaural beats on humans, solo and in combination. A binaural beat is the presence of two separate auditory tones with equal amplitude and slightly differing frequencies (Goodin et al., 2012). Meditation music is often used with binaural beats to calm individuals (Chan et al., 2008). There is reason to believe binaural beats and meditative music impact human vital signs (Wahbeh et al., 2012). Heart rate, blood pressure, and oxygen saturation were recorded from 60 participants, tested individually, and randomly assigned to one of three listening groups: Beat + Music, Music Only, or Beat Only. Participants experienced their assigned auditory stimulation through headphones for 6 min. Physiological responses were recorded before and during auditory stimulation. A one- way ANOVA showed a significant difference in mean heart rate between listening groups (p = .046). Due to sample size limitations, a subsequent Tukey test (Abdi & Williams, 2010) could not identify the location of the significant difference. The largest difference in averages (at 9.05 bpm) existed between Beat Only and Music Only groups, therefore, indicating this as the location of the significant difference. No significant difference was found between listening groups in blood pressure (systolic: p = .937; diastolic: p = .954) or oxygen saturation (p = .752). It is recommended future studies in this domain incorporate larger sample sizes to ensure statistical sensitivity

The Discrete Logarithm Problem and Ternary Functional Graphs

Encryption is essential to the security of transactions and communications, but the algorithms on which they rely might not be as secure as we all assume. In this paper, we investigate the randomness of the discrete exponentiation function used frequently in encryption. We show how we used exponential generating functions to gain theoretical data for mapping statistics in ternary functional graphs. Then, we compare mapping statistics of discrete exponentiation functional graphs, for a range of primes, with mapping statistics of the respective ternary functional graphs

Applying Second Language Acquisition to Facilitate a Blended Learning of Programming Languages

This paper describes a recent NSF funded project under the Research Initiation Grant in Engineering Education (RIGEE) program. It correlates the programming language study to second language acquisition theory. The work begun in Fall 2014, and project materials are under development. This paper outlines the proposed work and the materials developed to support the implementation of the project in Fall 2015

Seeing through rock with help from optimal transport

Geophysicists and mathematicians work together to detect geological structures located deep within the earth by measuring and interpreting echoes from manmade earthquakes. This inverse problem naturally involves the mathematics of wave propagation, but we will see that a different mathematical theory – optimal transport – also turns out to be very useful