Construction of nonparametric Bayesian models from parametric Bayes equations

We consider the general problem of constructing nonparametric Bayesian models on infinite-dimensional random objects, such as functions, infinite graphs or infinite permutations. The problem has generated much interest in machine learning, where it is treated heuristically, but has not been studied in full generality in non-parametric Bayesian statistics, which tends to focus on models over probability distributions. Our approach applies a standard tool of stochastic process theory, the construction of stochastic processes from their finite-dimensional marginal distributions. The main contribution of the paper is a generalization of the classic Kolmogorov extension theorem to conditional probabilities. This extension allows a rigorous construction of nonparametric Bayesian models from systems of finite-dimensional, parametric Bayes equations. Using this approach, we show (i) how existence of a conjugate posterior for the nonparametric model can be guaranteed by choosing conjugate finite-dimensional models in the construction, (ii) how the mapping to the posterior parameters of the nonparametric model can be explicitly determined, and (iii) that the construction of conjugate models in essence requires the finite-dimensional models to be in the exponential family. As an application of our constructive framework, we derive a model on infinite permutations, the nonparametric Bayesian analogue of a model recently proposed for the analysis of rank data.

Shallow water equations: Magic and limitations

D Liang from Cambridge University explains the shallow water equations and their applications to the dam-break and other steep-fronted flow modeling. They assume that the horizontal scale of the flow is much greater than the vertical scale, which means the flow is restricted within a thin layer, thus the vertical momentum is insignificant and the pressure distribution is hydrostatic. The left hand sides of the two momentum equations represent the acceleration of the fluid particle in the horizontal plane. If the fluid acceleration is ignored, then the two momentum equations are simplified into the so-called diffusion wave equations. In contrast to the SWEs approach, it is much less convenient to model floods with the Navier-Stokes equations. In conventional computational fluid dynamics (CFD), cumbersome treatments are needed to accurately capture the shape of the free surface. The SWEs are derived using the assumptions of small vertical velocity component, smooth water surface, gradual variation and hydrostatic pressure distribution.

Energy stable and momentum conserving hybrid finite element method for the incompressible Navier-Stokes equations

A hybrid method for the incompressible Navier-Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equal-order interpolations can be used for the velocity and pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh. Different from our earlier investigations on the approach for the Navier-Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for an appropriate choice of finite element spaces, momentum conservation. Also, a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a pointwise solenoidal velocity field. Mass conservation, momentum conservation, and global energy stability are proved for the time-continuous case and for a fully discrete scheme. The presented analysis results are supported by a range of numerical simulations. © 2012 Society for Industrial and Applied Mathematics.

Development of realistic liquid equations of state for multi-phase CFD modelling

Several equations of state (EOS) have been incorporated into a novel algorithm to solve a system of multi-phase equations in which all phases are assumed to be compressible to varying degrees. The EOSs are used to both supply functional relationships to couple the conservative variables to the primitive variables and to calculate accurately thermodynamic quantities of interest, such as the speed of sound. Each EOS has a defined balance of accuracy, robustness and computational speed; selection of an appropriate EOS is generally problem-dependent. This work employs an AUSM+-up method for accurate discretisation of the convective flux terms with modified low-Mach number dissipation for added robustness of the solver. In this paper we show a newly-developed time-marching formulation for temporal discretisation of the governing equations with incorporated time-dependent source terms, as well as considering the system of eigenvalues that render the governing equations hyperbolic.

Development of accurate, robust liquid equations of state for multi-phase CFD simulations with a modified AUSM+-up scheme

Eight equations of state (EOS) have been evaluated for the simulation of compressible liquid water properties, based on empirical correlations, the principle of corresponding states and thermodynamic relations. The IAPWS-IF97 EOS for water was employed as the reference case. These EOSs were coupled to a modified AUSM+-up convective flux solver to determine flow profiles for three test cases of differing flow conditions. The impact of the non-viscous interaction term discretisation scheme, interfacial pressure method and selection of low-Mach number diffusion were also compared. It was shown that a consistent discretisation scheme using the AUSM+-up solver for both the convective flux and the non-viscous interfacial term demonstrated both robustness and accuracy whilst facilitating a computationally cheaper solution than discretisation of the interfacial term independently by a central scheme. The simple empirical correlations gave excellent results in comparison to the reference IAPWS-IF97 EOS and were recommended for developmental work involving water as a cheaper and more accurate EOS than the more commonly used stiffened-gas model. The correlations based on the principles of corresponding-states and the modified Peng-Robinson cubic EOS also demonstrated a high degree of accuracy, which is promising for future work with generic fluids. Further work will encompass extension of the solver to multiple dimensions and to account for other source terms such as surface tension, along with the incorporation of phase changes. © 2013.

Multi scale shape index for 3D object recognition

We present Multi Scale Shape Index (MSSI), a novel feature for 3D object recognition. Inspired by the scale space filtering theory and Shape Index measure proposed by Koenderink & Van Doorn [6], this feature associates different forms of shape, such as umbilics, saddle regions, parabolic regions to a real valued index. This association is useful for representing an object based on its constituent shape forms. We derive closed form scale space equations which computes a characteristic scale at each 3D point in a point cloud without an explicit mesh structure. This characteristic scale is then used to estimate the Shape Index. We quantitatively evaluate the robustness and repeatability of the MSSI feature for varying object scales and changing point cloud density. We also quantify the performance of MSSI for object category recognition on a publicly available dataset. © 2013 Springer-Verlag.

Cabaret finite-difference schemes for the one-dimensional Euler equations

In the present paper we consider second order compact upwind schemes with a space split time derivative (CABARET) applied to one-dimensional compressible gas flows. As opposed to the conventional approach associated with incorporating adjacent space cells we use information from adjacent time layer to improve the solution accuracy. Taking the first order Roe scheme as the basis we develop a few higher (i.e. second within regions of smooth solutions) order accurate difference schemes. One of them (CABARET3) is formulated in a two-time-layer form, which makes it most simple and robust. Supersonic and subsonic shock-tube tests are used to compare the new schemes with several well-known second-order TVD schemes. In particular, it is shown that CABARET3 is notably more accurate than the standard second-order Roe scheme with MUSCL flux splitting.

Parabolic tapered photonic crystal cavity in silicon

We demonstrate the design, fabrication, transmission and nearfield characterization of a novel parabolic tapered 1D photonic crystal cavity in silicon. The design allows repeatable device fabrication, high quality factor and small modal volume. © 2012 OSA.

Parabolic tapered photonic crystal cavity in silicon

We demonstrate the design, fabrication, transmission and nearfield characterization of a novel parabolic tapered 1D photonic crystal cavity in silicon. The design allows repeatable device fabrication, high quality factor and small modal volume. © OSA 2012.

Parabolic tapered photonic crystal cavity in silicon

We demonstrate the design, fabrication, transmission and nearfield characterization of a novel parabolic tapered 1D photonic crystal cavity in silicon. The design allows repeatable device fabrication, high quality factor and small modal volume. © 2012 OSA.

Parabolic tapered photonic crystal cavity in silicon

We demonstrate the design, fabrication, transmission spectrum measurement, and near-field characterization of a parabolic tapered one-dimensional photonic crystal cavity in silicon. The results shows a relatively high quality factor (∼43 000), together with a small modal volume of ∼ 1. 1 (λ/n) 3. Moreover, the design allows repeatable device fabrication, as evident by the similar characteristics obtained for several tens of devices that were fabricated and tested. These demonstrated 1D PhC cavities may be used as a building block in integrated photonic circuits for optical on-chip interconnects and sensing applications. © 2012 American Institute of Physics.

Parabolic tapered photonic crystal cavity in silicon

We demonstrate the design, fabrication, transmission and nearfield characterization of a novel parabolic tapered 1D photonic crystal cavity in silicon. The design allows repeatable device fabrication, high quality factor and small modal volume. © OSA 2012.