Hybrid Spectral Difference/Embedded Finite Volume Method for Conservation Laws

A novel hybrid spectral difference/embedded finite volume method is introduced in order to apply a discontinuous high-order method for large scale engineering applications involving discontinuities in the flows with complex geometries. In the proposed hybrid approach, the finite volume (FV) element, consisting of structured FV subcells, is embedded in the base hexahedral element containing discontinuity, and an FV based high-order shock-capturing scheme is employed to overcome the Gibbs phenomena. Thus, a discontinuity is captured at the resolution of FV subcells within an embedded FV element. In the smooth flow region, the SD element is used in the base hexahedral element. Then, the governing equations are solved by the SD method. The SD method is chosen for its low numerical dissipation and computational efficiency preserving high-order accurate solutions. The coupling between the SD element and the FV element is achieved by the globally conserved mortar method. In this paper, the 5th-order WENO scheme with the characteristic decomposition is employed as the shock-capturing scheme in the embedded FV element, and the 5th-order SD method is used in the smooth flow field. The order of accuracy study and various 1D and 2D test cases are carried out, which involve the discontinuities and vortex flows. Overall, it is shown that the proposed hybrid method results in comparable or better simulation results compared with the standalone WENO scheme when the same number of solution DOF is considered in both SD and FV elements.Comment: 27 pages, 17 figures, 2 tables, Accepted for publication in the Journal of Computational Physics, April 201

Zigzag Filamentary Theory of Broken Symmetry of Neutron and Infrared Vibronic Spectra of YBa2Cu3O(6+x)

Filamentary high-temperature superconductivity (HTSC) theory differs fundamentally from continuous HTSC theories because it emphasizes self-organized, discrete dopant networks and does not make the effective medium approximation (EMA). Analysis of neutron and infrared (especially with c-axis polarization) vibrational spectra, primarily for YBa2Cu3O(6+x), within the filamentary framework, shows that the observed vibronic anomalies near 400 cm-1 (50 meV) are associated with curvilinear filamentary paths. these paths pass through cuprate chains and planes, as well as resonant tunneling centers in the BaO layers. The analysis and the data confirm earlier filamentary structural models containing ferroelastic domains of 3-4 nm in the CuO2 planes; it is these nanodomains that are responsible for the discrete glassy nature of both electronic and vibronic properties. Chemical trends in vibronic energies and oscillator strengths, both for neutron and photon scattering, that were anomalous in continuum models, are readily explained by the filamentary model.Comment: 45 pages, 17 figures, PD

Perfect Regular Equilibrium

We propose a revised version of the perfect Bayesian equilibrium in general multi-period games with observed actions. In finite games, perfect Bayesian equilibria are weakly consistent and subgame perfect Nash equilibria. In general games that allow a continuum of types and strategies, however, perfect Bayesian equilibria might not satisfy these criteria of rational solution concepts. To solve this problem, we revise the definition of the perfect Bayesian equilibrium by replacing Bayes' rule with a regular conditional probability. We call this revised solution concept a perfect regular equilibrium. Perfect regular equilibria are always weakly consistent and subgame perfect Nash equilibria in general games. In addition, perfect regular equilibria are equivalent to simplified perfect Bayesian equilibria in finite games. Therefore, the perfect regular equilibrium is an extended and simple version of the perfect Bayesian equilibrium in general multi-period games with observed actions

PCA consistency in high dimension, low sample size context

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace consistency. We show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others, then the corresponding estimated PC directions are consistent or converge to the appropriate subspace (subspace consistency) and most other PC directions are strongly inconsistent. Broad sets of sufficient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations. In preparation for these results, we show that the geometric representation of HDLSS data holds under general conditions, which includes a $\rho$-mixing condition and a broad range of sphericity measures of the covariance matrix.Comment: Published in at http://dx.doi.org/10.1214/09-AOS709 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org