Correlations of peaks of Gaussian random fields

The high peaks of a Gaussian random field are studied. Asymptotic expansions, appropriate for high peak thresholds and large spatial separations, are developed for theN-point correlation functions of the number density of high peaks, in terms of the two-point correlation of the underlying Gaussian field. Similar expressions are derived for the correlations of points, not necessarily the positions of peaks, where the field exceeds a high threshold.Work supported in part by U.S. Department of Energy under contract DEAC03-81-ER40050.KFAS Graduate FellowAlfred P. Sloan Foundation Fellow and supported in part by U.S. Department of Energy Outstanding Junior Investigator Program under contract No. DE-FG03-84 ER40172     

Symmetric Euler and Navier-Stokes shocks in stationary barotropic flow on a bounded domain

We construct stationary solutions to the barotropic, compressible Euler and Navier-Stokes equations in several space dimensions with spherical or cylindrical symmetry. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in the exterior domain. On the other hand, stationary smooth solutions in the interior domain necessarily become sonic and cannot be continued beyond a critical inner radius. We then use these solutions to construct entropy-satisfying shocks for the Euler equations in the region between two concentric spheres or cylinders. Next we construct smooth Navier-Stokes solutions converging to the previously constructed Euler shocks in the small viscosity limit. In the process we introduce a new technique for constructing smooth solutions, which exhibit a fast transition in the interior, to a class of two-point boundary problems.     

Pattern search ranking and selection algorithms for mixed variable simulation-based optimization

The class of generalized pattern search (GPS) algorithms for mixed variable optimization is extended to problems with stochastic objective functions. Because random noise in the objective function makes it more difficult to compare trial points and ascertain which points are truly better than others, replications are needed to generate sufficient statistical power to draw conclusions. Rather than comparing pairs of points, the approach taken here augments pattern search with a ranking and selection (R&S) procedure, which allows for comparing many function values simultaneously. Asymptotic convergence for the algorithm is established, numerical issues are discussed, and performance of the algorithm is studied on a set of test problems.