One-Dimensional Navier-Stokes Finite Element Flow Model

This technical report documents the theoretical, computational, and practical aspects of the one-dimensional Navier-Stokes finite element flow model. The document is particularly useful to those who are interested in implementing, validating and utilizing this relatively-simple and widely-used model.Comment: 46 pages, 1 tabl

Consistently Simulating a Wide Range of Atmospheric Scenarios for K2-18b with a Flexible Radiative Transfer Module

The atmospheres of small, potentially rocky exoplanets are expected to cover a diverse range in composition and mass. Studying such objects therefore requires flexible and wide-ranging modeling capabilities. We present in this work the essential development steps that lead to our flexible radiative transfer module, REDFOX, and validate REDFOX for the Solar system planets Earth, Venus and Mars, as well as for steam atmospheres. REDFOX is a k-distribution model using the correlated-k approach with random overlap method for the calculation of opacities used in the $\delta$-two-stream approximation for radiative transfer. Opacity contributions from Rayleigh scattering, UV / visible cross sections and continua can be added selectively. With the improved capabilities of our new model, we calculate various atmospheric scenarios for K2-18b, a super-Earth / sub-Neptune with $\sim$8 M$_\oplus$ orbiting in the temperate zone around an M-star, with recently observed H$_2$O spectral features in the infrared. We model Earth-like, Venus-like, as well as H$_2$-He primary atmospheres of different Solar metallicity and show resulting climates and spectral characteristics, compared to observed data. Our results suggest that K2-18b has an H$_2$-He atmosphere with limited amounts of H$_2$O and CH$_4$. Results do not support the possibility of K2-18b having a water reservoir directly exposed to the atmosphere, which would reduce atmospheric scale heights, hence too the amplitudes of spectral features inconsistent with the observations. We also performed tests for H$_2$-He atmospheres up to 50 times Solar metallicity, all compatible with the observations.Comment: 28 pages, 13 figures, accepted for publication in Ap

Calculation of Generalized Polynomial-Chaos Basis Functions and Gauss Quadrature Rules in Hierarchical Uncertainty Quantification

Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a set of orthonormal polynomials and a proper numerical quadrature rule. The former are used as the basis functions in a generalized polynomial chaos expansion. The latter is used to compute the integrals involved in stochastic spectral methods. Obtaining such information requires knowing the density function of the random input {\it a-priori}. However, individual system components are often described by surrogate models rather than density functions. In order to apply stochastic spectral methods in hierarchical uncertainty quantification, we first propose to construct physically consistent closed-form density functions by two monotone interpolation schemes. Then, by exploiting the special forms of the obtained density functions, we determine the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator. The effectiveness of our proposed algorithm is verified by both synthetic and practical circuit examples.Comment: Published by IEEE Trans CAD in May 201

Variational Inference for Generalized Linear Mixed Models Using Partially Noncentered Parametrizations

The effects of different parametrizations on the convergence of Bayesian computational algorithms for hierarchical models are well explored. Techniques such as centering, noncentering and partial noncentering can be used to accelerate convergence in MCMC and EM algorithms but are still not well studied for variational Bayes (VB) methods. As a fast deterministic approach to posterior approximation, VB is attracting increasing interest due to its suitability for large high-dimensional data. Use of different parametrizations for VB has not only computational but also statistical implications, as different parametrizations are associated with different factorized posterior approximations. We examine the use of partially noncentered parametrizations in VB for generalized linear mixed models (GLMMs). Our paper makes four contributions. First, we show how to implement an algorithm called nonconjugate variational message passing for GLMMs. Second, we show that the partially noncentered parametrization can adapt to the quantity of information in the data and determine a parametrization close to optimal. Third, we show that partial noncentering can accelerate convergence and produce more accurate posterior approximations than centering or noncentering. Finally, we demonstrate how the variational lower bound, produced as part of the computation, can be useful for model selection.Comment: Published in at http://dx.doi.org/10.1214/13-STS418 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org