Stony Brook AMS - Downloadable Preprints, 1998

We present spatial distributions for pore path length and coordination number, pore throat size and nodal pore volume obtained for a $1.5^3$ mm${}^3$ volume of 12.1\% porosity Fontainebleau sandstone. The sandstone was imaged using Synchrotron X-Ray computed microtomography at 6 micron resolution. The spatial distributions were computed based upon three dimensional medial axis analysis of the void space in the image. We also present vesicle size and vesicle-vesicle contact surface area distributions for a 1.36 mm length of a 6.36 mm diameter core of basalt from a vesiculated lava flow imaged at 20 micron resolution.

We consider the problem of segmenting a digitized 2D or 3D image consisting of two univariate populations. Assume a-priori knowledge allows incomplete assignment of voxels in the image, in the sense that a fraction of the voxels can be identified as belonging to population P₀, a second fraction to P₁, and the remaining fraction have no a-priori identification. Based upon estimates of the short length scale spatial covariance of the image, we develop a method utilizing indicator kriging to complete the image segmentation.

Front Tracking provides high resolution piecewise smooth numerical solutions through the active tracking of dynamically moving interfaces separating distinct fluids or materials. A major challenge to this computational method is the description and resolution of changes in the topology of the interface. In this paper, we describe two algorithms that have been implemented in the Front Tracking code \FronTier\ to solve the problem of robust computation of dynamic changes in the interface description. One algorithm is grid independent and the other one is grid based. The two methods can be used simultaneously to give a hybrid method. We show that these algorithms, in particular the hybrid method, can successfully address a dynamic change (bifurcation) in interface topology. We illustrate interface bifurcation in the simulation of Rayleigh-Taylor instability, which is a fluid interface instability driven by a steady acceleration directed across a fluid discontinuity interface. We validate our numerical results for 3-D \FronTier\ by comparison with analytic results for the single mode terminal velocity.

We report the results of testing the performance of a new, efficient, and highly general-purpose parallel optimization method based upon simulated annealing. This optimization algorithm was applied to analyze the network of interacting genes that control embryonic development and other fundamental biological processes. We found several sets of algorithmic parameters that lead to optimal parallel efficiency for up to 100 processors on distributed-memory MIMD architectures. Our strategy contains two major elements. First, we monitor and pool performance statistics obtained simultaneously on all processors. Second, we mix states at intervals to ensure a Boltzmann distribution of energies. The central scientific issue is the inverse problem, the determination of the parameters of a set of nonlinear ordinary differential equations by minimizing the total error between the model behavior and experimental observations.

Consider the problem of the conformal tetrahedralization of a three dimensional tensor product grid subject to the restrictions that i) no additional vertices beyond those required to specify the grid corners are introduced, ii) each tetrahedralization produced lie solely within some grid block, and iii) the resultant tetrahedralization contain a predefined set of conformal triangular faces. We present a necessary and sufficient condition for the achievement of such a conformal tetrahedralization.

This paper describes a simple and robust method for load balance and mesh smoothing in both 2d and 3d spaces. The method produces nearly perfect balance in 2d for three artificial load imbalance problems. We also explain the extension of the method to mesh smoothing, while pointing out the route to generalize it to higher dimensions.

Abstract not available.

This paper investigates the structure of two-dimensional Riemann problems for Hamilton-Jacobi equations. We show that it is possible for the viscosity solution to contain closed characteristic orbits, enclosing furthermore a periodic sonic structure, which in turn encloses a parabolic structure. The existence of such examples elucidates the difficulties encountered in designing construction methods for viscosity solutions to Riemann problems in dimension $\geq 2$. This investigation was prompted by the discovery of numerical evidence of examples displaying an even richer internal structure. Here we establishe the existence of Riemann problems with viscosity solutions of considerable complexity.

Solutions to Hamilton-Jacobi equations, which model many physical processes containing moving boundaries, generally develop discontinuities in gradient, and thus corners and edges in their graphs. Although unique viscosity solutions are guaranteed, construction of them is not simple, except in special cases. The evolution of singularities, determining local structure, can be studied by assuming self-similarity in solutions, and result in Riemann problems and front tracking schemes for numerical computation of solutions. Near infinity, self-similar solutions can be constructed by ODE methods, i.e. characteristics, and within a compact set they can be constructed by algebraic methods. A fundamental difficulty lies in where and how these partial solutions interact to form a unique global solution to the Riemann problem. We present an example with a periodic characteristic orbit, obstructing standard ODE continuation methods, and a numerically-based resolution of the non-trivial internal e of the Riemann solution which forms as a result.

New closures for two pressure two phase flow in the context of unstable fluid mixing have been proposed recently by the authors. Here we examine the physical basis for the models, the nature of the boundary conditions at the edges of the mixing layer, and an algorithm for the numerical solution of the two phase flow equations. Physically, the closures describe chunk mix, in which the flow is dominated by coherent structures of size comparable to the mixing zone thickness. The closed form solution previously introduced for the incompressible limit is reviewed and extended. Sufficient boundary conditions for the compressible equations are found from drag and buoyancy laws proposed by others, with coefficients fit to two sets of independent experiments. These laws complete the closure of the two phase flow equations. A postulate of stationary center of mass, previously introduced at a numerical level, is here related to a weak notion of self similarity and is solved analytically for the ratio of the growth rates of the two sides of the mixing zone in the self similar case.

Pattern recognition and data mining methods are developed for a large data set arising in the management of customers and a supplier network for a large organization. Development of quality control techniques to improve the data processing are based on multivariate regression and time series methods.

A new and simplified version of Front Tracking has been developed, as an aspect of the extension of this algorithm to three dimensions. Here we emphasize two main results: (1) a simplified description of the microtopology of the interface, based on interface crossings with cell block edges, and (2) an improved algorithm for the interaction of a tracked contact discontinuity with an untracked shock wave. For the latter question, we focus on the post interaction jump at the contact, which is a purely 1D issue. Comparisons to other methods, including the level set method, are included.

We consider transport properties for a class of periodic, Gaussian, stationary, divergence free, random velocity fields in R^2 with a finite number of spatial modes. Full asymptotics of the effective diffusivity is obtained in the case of short time correlations, on a fully rigorous basis. Existence and uniqueness for a hypoelliptic partial differential equation coupling velocity Fourier modes to physical space transport provides the main technical step in the justification of the asymptotic analysis.

We consider transport properties for Gaussian, stationary, divergence free, random velocity fields in R^2, which are Markov in time. We prove the existence of effective diffusivity. We also obtain its full asymptotics in the case of short time correlations, on a fully rigorous level. The main regularity assumption is that almost every realization of the random velocity field should be continuous in space and time, and Lipschitz continuous in space.

We study a model of turbulent transport described by the motion in a Gaussian random velocity field with Kolmogorov spectrum. The field is assumed to be divergence free, homogeneous in time and space, and Markovian in time.

The molecular viscosity defines the cutoff in the Fourier space, thus regularizing the generalized field of pure infinite Reynolds number Kolmogorov spectrum by ordinary vector fields. We provide an asymptotic bound on the effective diffusivity of the finite Reynolds number fields fields as R tends to infinity. Namely, with macroscopic parameters of the system fixed, and the viscosity tending to zero, the effective diffusivity is bounded above by a constant times the 2/3 power of the Reynolds number.

Prediction is based on the comparison of results from the statistical analysis of observational data and from the scientific modeling of the system being observed. Effective prediction imposes new as well as familiar requirements on observation and scientific modeling, as will be reviewed here. We emphasize issues specific to prediction in the context of technology. Recent results of the authors, colleagues, and others which address these requirements will be presented.