The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations

Simple inequalities for the Riemann problem for a Hamilton-Jacobi equation in N space dimension when neither the initial data nor the Hamiltonian need be convex (or concave) are presented. The initial data is globally continuous, affine in each orthant, with a possible jump in normal derivative across each coordinate plane, x sub i = 0. The inequalities become equalities wherever a maxmin equals a minmax and thus an exact closed form solution to this problem is then obtained

The discrete one-sided Lipschitz condition for convex scalar conservation laws

Physical solutions to convex scalar conservation laws satisfy a one-sided Lipschitz condition (OSLC) that enforces both the entropy condition and their variation boundedness. Consistency with this condition is therefore desirable for a numerical scheme and was proved for both the Godunov and the Lax-Friedrichs scheme--also, in a weakened version, for the Roe scheme, all of them being only first order accurate. A new, fully second order scheme is introduced here, which is consistent with the OSLC. The modified equation is considered and shows interesting features. Another second order scheme is then considered and numerical results are discussed

Essentially non-oscillatory shock capturing methods applied to turbulence amplification in shock wave calculations

ENO (essentially non-oscillatory) schemes can provide uniformly high order accuracy right up to discontinuities while keeping sharp, essentially non-oscillatory shock transitions. Recently, an efficient implementation of ENO schemes was obtained based on fluxes and TVD Runge-Kutta time discretizations. The resulting code is very simple to program for multi-dimensions. ENO schemes are especially suitable for computing problems with both discontinuities and fine structures in smooth regions, such as shock interaction with turbulence, for which results for 1-D and 2-D Euler equations are presented. Much better resolution is observed by using third order ENO schemes than by using second order TVD schemes for such problems

Numerical solution of the unsteady Navier-Stokes equation

The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws are discussed. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper a uniformly second-order approximation is constructed, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell

Optimal Data Collection For Informative Rankings Expose Well-Connected Graphs

Given a graph where vertices represent alternatives and arcs represent pairwise comparison data, the statistical ranking problem is to find a potential function, defined on the vertices, such that the gradient of the potential function agrees with the pairwise comparisons. Our goal in this paper is to develop a method for collecting data for which the least squares estimator for the ranking problem has maximal Fisher information. Our approach, based on experimental design, is to view data collection as a bi-level optimization problem where the inner problem is the ranking problem and the outer problem is to identify data which maximizes the informativeness of the ranking. Under certain assumptions, the data collection problem decouples, reducing to a problem of finding multigraphs with large algebraic connectivity. This reduction of the data collection problem to graph-theoretic questions is one of the primary contributions of this work. As an application, we study the Yahoo! Movie user rating dataset and demonstrate that the addition of a small number of well-chosen pairwise comparisons can significantly increase the Fisher informativeness of the ranking. As another application, we study the 2011-12 NCAA football schedule and propose schedules with the same number of games which are significantly more informative. Using spectral clustering methods to identify highly-connected communities within the division, we argue that the NCAA could improve its notoriously poor rankings by simply scheduling more out-of-conference games.Comment: 31 pages, 10 figures, 3 table

Fast wavelet based algorithms for linear evolution equations

A class was devised of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin which they applied to general Calderon-Zygmund type integral operators. A modification of their idea is applied to linear hyperbolic and parabolic equations, with spatially varying coefficients. A significant speedup over standard methods is obtained when applied to hyperbolic equations in one space dimension and parabolic equations in multidimensions