Stony Brook AMS - Downloadable Preprints, 2001

The structure of neuronal dendrites and their spines underlie the connectivity of neural networks. Dendrites, spines, and their dynamics are shaped by genetic programs as well as sensory experience. Dendritic structures and dynamics may therefore be important predictors of the function of neural networks.

Based on new imaging approaches and increases in the speed of computation it has become possible to acquire large sets of high resolution optical micrographs of neuron structure at length scales small enough to resolve spines. This advance in data acquisition has not been accompanied by comparable advances in data analysis techniques; the analysis of dendritic and spine morphology is still accomplished largely manually. In addition to being extremely time intensive, manual analysis also introduces systematic and hard to characterize biases. We present a geometric approach for automatically detecting and quantifying the three dimensional structure of dendritic spines from stacks of image data acquired using laser scanning microscopy.

We present results on the measurement of dendritic spine length, volume, density, and shape classification for both static and time-lapse images of dendrites of hippocampal pyramidal neurons. For spine length and density, the automated measurements in static images are compared with manual measurements. Comparisons are also made between automated and manual spine length measurements for a time-series data set. The algorithm performs well compared to a human analyzer, especially on time-series data.

Automated analysis of dendritic spine morphology will enable objective analysis of large morphological data sets. The approaches presented here are generalizable to other aspects of neuronal morphology.

A front tracking method for inviscid gas dynamics is presented. The key constructions and algorithms used in the code are described and the interrelations between shock capturing, interface dynamics, computational geometry, grid construction, and parallelism are discussed for the code as a whole. Validation is carried out by comparing the single mode bubble velocity for Rayleigh-Taylor instability with theoretical models and experimental results. The calculations are validated by mesh refinement studies and by the comparison of the asymptotic limit of the minimum radius r_min as it tends to infinity to a pure planar computation in two dimensions.

In this paper, we present a numerical study of the axisymmetric Richtmyer-Meshkov instability in converging spherical geometry by the front tracking method for the first time. The front tracking method has been successfully used in solving fluid instability problems in both rectangular and curved geometry. The central issue for axisymmetric flows is the absence of the rotational symmetry in the $(r, z)$ plane, although the perturbed shape of the initial contact interface appears to have it. The cause of the asymmetry is somewhat obvious. The sinusoidal perturbations appear symmetric only in the cross-sectional view; in actuality they are not symmetric because they represent rings around the z-axis and hence the perturbed mass at the equator, for example, is different from the perturbed mass at the pole.

The first purpose of this paper is to quantify the effect of this inherited asymmetry on the growth of the spherical mixing. We find this asymmetry drives the original structure to some degree so that the mixing radius at the north pole is noticeably larger than at the equator during the evolution of chaotic mixing. We also study quantitatively the azimuthal dependence of the mixing statistics, such as the mixing edges, the growth rate and volume fraction.

Richtmyer-Meshkov (RM) instabilities in spherical geometry have been a challenge due to the inherent difficulty of theiraccurate simulation. Our second purpose is to demonstrate that our Front Tracking method can describe the Richtmyer-Meshkov instability growth in a complex flow involving multiple reshocks. We have successfully displayed the converging geometry, reshock process, asymmetry phenomenon through the density and pressure color plots. The quantitative growth rate analysis is also presented.

In this paper we formulate a model for the merger of bubbles at the edge of an unstable acceleration driven (Rayliegh-Taylor) mixing layer. Steady acceleration defines a self similar mixing process, with a time dependent inverse cascade of structures of increasing size. The time evolution is itself a renormalization group evolution, and so the large time asymptotics define an RNG fixed point. We solve the model introduced here at this fixed point. The model predicts the growth rate of a Rayleigh-Taylor chaotic fluid mixing layer. The model has three main components: the velocity of a single bubble in this unstable flow regime, an envelope velocity, which describes collective excitations in the mixing region, and a merger process, which drives an inverse cascade, with a steady increase of bubble size. The present model differs from an earlier 2D merger model in several important ways. Beyond the extension of the model to 3D, the present model contains one phenomenological parameter, the variance of the bubble radii at fixed time. The model also predicts several experimental numbers: the bubble mixing rate, $\alpha_b = h_b/Agt^2 \approx 0.05 - 0.06$, the mean bubble radius, and the bubble height separation at the time of merger. From these we also obtain the bubble height to radius aspect ratio. Using experiment results of Smeeton and Youngs \cite{SmeYou87} to fix a value for the radius variance, we determine $\alpha_b$ within the range of experimental uncertainty. We also obtain the experimental values for the bubble height to width aspect ratio in agreement with experimental values.

We propose a fully conservative Front Tracking algorithm. In 1D, the algorithm is second order accurate in L$_1$ for piecewise smooth solutions all of whose discontinuities are tracked, except for interactions of tracked waves. The local truncation errors in L$_\infty$ are third order in the interior (away from tracked fronts) and away from local maxima, second order in L$_\infty$ near isolated tracked fronts, and ${\mathcal O}(1)$ near the finite number of space time points at which tracked fronts collide. 1D Numerical tests confirming these theoretical estimates are presented. Extension of the algorithm to higher dimensions is proposed. The higher dimensional algorithm is also fully conservative and has a first order local truncation error for cells near the tracked discontinuity. The algorithm improves by one order of accuracy over most finite difference schemes, which have ${\mathcal O}(1)$ local truncation errors near discontinuities.

The leading X-ray computed microtomographic imaging facilities can now provide $1024^3$ voxel images of rock and other porous media samples at a voxel resolution of under 5 microns. Such data sets are extremely rich in information, and overwhelming in size; a $1024^3$ data set corresponds to a gigabyte of character data. Automated computer analysis is necessary in order to extract quantitative information from such images. In this paper we discuss automated extraction of geometrical features using computerized image analysis. Typical algorithms required include segmentation to identify the material type of each voxel in the image; medial axis reduction of objects in the image to provide a skeleton enabling efficient searching and geometrical characterization as well as a network for the application of graph theoretic tools; feature extraction; measurement of length, cross sectional area and volume; and stochastic characterization of measured properties. With current memory limitations in desktop workstations, data sets beyond $512^3$ voxels in size require parallelization of the algorithms.

Dynamic behavior of mixing fronts plays a crucial role in multifluid turbulent mixing. In this paper, we derive an analytic solution for the entire dynamic evolution of the mixing fronts driven by steady acceleration Rayleigh-Taylor (RT) and impulsive acceleration Richtmyer-Meshkov (RM) instabilities, from a simple physics description as a pair of ordinary differential equations. Our solutions agree with available experimental data.

A general discussion of the quantification of uncertainty in numerical simulations is presented. A principal conclusion is that the distribution of solution errorsis the leading term in the assessment of the validity of a simulation and its associated uncertainty in the Bayesian framework. Key issues that arise in uncertainty quantification are discussed for two examples drawn from shock wave physics and modeling of petroleum reservoirs. Solution error models, confidence intervals and Gaussian error statistics based on simulation studies are presented.

Hyperbolic conservation laws are foundational for many branches of continuum physics. Discontinuities in the solutions of these partial differential equations are widely recognized as a primary difficulty for numerical simulation, especially for thermal and shear discontinuities and fluid-fluid internal boundaries. We propose numerical algorithms which will (a) track these discontinuities as sharp internal boundaries, (b) fully conserve the conserved quantities at a discrete level, even at the discontinuities, and (c) display one order of numerical accuracy higher globally (at the discontinuity) than algorithms in common use. A significant improvement in simulation capabilities is anticipated through use of the proposed algorithms.

We present a numerical method which provides the ability to analyze digitized microscope images of retinal explants and quantify neurite outgrowth. Few parameters are required as input and limited user interaction is necessary to process an entire experiment of images. This eliminates fatigue related errors and user-related bias common to manual analysis. The method does not rely on stained images and handles images of variable quality. We present comparisons with results obtained from manual analysis of in vitro neurotoxic effects of 1 GeV/nucleon iron particles in retinal explants.

We present the solutions for a two dimensional Riemann problem for a 2 x 2 hyperbolic nonlinear system based upon the Keyfitz-Kranzer-Isaacson-Temple model. This model arises in the polymer flooding of an oil reservoir. For isotropic media, we have constructed the solution for the single and four quadrant Riemann problems.

We construct the solutions to a two dimensional Riemann problem for a 2 x 2 hyperbolic nonlinear system which models polymer flooding in an anisotropic oil reservoir. The construction demonstrates the importance of the shock, rarefaction and contact `base points' and `base curves' in the determination of the solutions for two dimensional Riemann problems. In particular, we establish some new relations between these. While specific details of the base points and curves are applicable only to this model, the existence of the curves and the existence of relationships between these curves are general features to be exploited for any hyperbolic system.

We study the motion of the slightly compressible multi-phase flow model proposed by Chen, Glimm, Sharp and Zhang. The interface velocity and constitutive law are analyzed by derivation of the exact quantity. Using singular perturbation theory, a formal asymptotic expansion is derived for the solution of the compressible equations. An asymptotic analysis in the incompressible limit, for slightly compressible flows supplies important new information to resolve nonuniqueness of the pressure difference between the two fluid species of the incompressible flow equations.

Network flow models have evolved into an important tool for linking pore scale structure and bulk fluid properties at the scale of rock core samples by providing parametric relationships that are required in continuum-scale descriptions of multiphase flow. To accurately model flow in real rock, network flow models have three challenges to resolve: the role of thin wetting films and amount of trapping of the wetting phase; identification of size distributions for pore bodies, throats and coordination numbers; and the relative importance of pairwise and spatial correlations of these variables. This paper address the last two of these three points. We summarize computer analysis of a suite of X-ray computed microtomographic images of Fontainebleau core samples which provides accurate measurement for these marginal and correlated distributions. We then discuss initial results of network flow model computations which demonstrate the importance of correctly capturing these distribution in a network model.

The temporal development of a single mode Rayleigh-Taylor instability consists of three stages: the linear, free fall and terminal velocity regimes. The purpose of this paper is to report on new phenomena observed in the approach to terminal velocity. Our numerical study shows an unexpected nonuniform approach to terminal velocity. The nonuniformity applies especially to the spikes, which are fingers of heavy fluid falling into the light fluid, but it also applies to the rising bubbles of light fluid. For spikes especially, our results call into question the meaningfulness of a terminal velocity for moderate values of the Atwood velocity $A$. After a short time period of pseudo--terminal plateau, the spike velocity increases to a significantly higher maximum, followed by a decrease. This phenomena appears to be due to a slow evolution in the shape of the spike and bubble. We find a relation between the spike (bubble) acceleration and the tip curvature. In correlation with an increase in the spike velocity, the main body of the spike becomes narrower and the tip curvature increases Our numerical results are by the Front Tracking method. The very late time simulations considered here required stabilization by a small value for the viscosity, so that the compressible Navier-Stokes equations govern the dynamics.

We propose a fully conservative Front Tracking algorithm in 1D and examine the close connection between the tracked front position error and the local truncation error near the front. The algorithm is second order accurate in L_1 for piecewise smooth solutions all of whose discontinuities are tracked, except for interactions of tracked waves. The local truncation errors in L_{infinity} are third order in the interior (away from tracked fronts) and away from local maxima, second order in L_{infinity} near isolated tracked fronts, and O(1) near the finite number of space time points at which tracked fronts collide. 1D Numerical tests confirming these theoretical estimates are presented.

We propose a fully conservative Front Tracking algorithm in two space dimension. The algorithm first uses the point shifted algorithm on two adjacent time levels and then constructs space time hexahedra as computational units. We develope and prove a successful geometric construction under certain interface requirement. This algorithm has a first order local truncation error for cells near the tracked discontinuity, which is an improvement by one order of accuracy over most finite difference schemes, which have O(1) local truncation errors near discontinuities.

We present a Rayleigh-Taylor mixing rate simulation with an acceleration rate falling within the range of experiments. The simulation uses front tracking to prevent interfacial mass diffusion. We present evidence to support the assertion that the lower acceleration rate found in untracked simulations is caused, at least to a large extent, by a reduced buoyancy force due to numerical interfacial mass diffusion. Quantitative evidence includes results from a time dependent Atwood number analysis of the diffusive simulation, which yields a renormalized mixing rate coefficient for the diffusive simulation in agreement with experiment.

We present a simple model for simulation errors for upscaled flow in porous media. The error model is Gaussian with a very reduced number of degrees of freedom. We show robustness of the model by applying it to six different geology models (five different correlation lengths and a mixture model combining all five correlation lengths) with no change in the model parameterization. Compared to previous models of the authors, the present one yields similar confidence intervals (giving prediction with approximately equal uncertainty). The authors are not aware of comparable studies of scale up simulation and modeling error by other groups.

We study conic programming in which the right-hand side is a linear function of a scalar parameter. We extend the idea of optimal partition from linear programming (LP) and semidefinite programming (SDP) to general convex cones and propose two approaches to find the range of the parameter. We define a weaker notion of nondegeneracy and show that both of the approaches reduce to a minimum ratio test under appropriate nondegeneracy assumptions. We also study properties of the optimal value as a function of the parameter and extend some of the results from LP and convex quadratic programming to general conic programming. We present an example illustrating the extent to which other results can be generalized.

Utilizing a reservoir model consisting of purely stochastic, small scale geological information we study ensemble-based (multiple realization) reservoir prediction. We address the following questions: what inherent prediction accuracy is achievable given purely stochastic knowledge; how sensitive is the accuracy to the governing stochastic parameters; how many realizations are necessary in order to approach the inherent accuracy to within a specified tolerance; how much can history matching reduce the inherent uncertainty; and how does uncertainty change as a function of time? Related to this last question we show that the `predictability lifetime' of a single realization depends on the outcome being observed. In the model studied, we provide evidence that the prediction lifetime for the oil cut is governed by a Markov process, whereas cumulative oil production is not. Thus a single realization, even one that is history matched, is expected to provide accurate prediction of oil cut over a much shorter period of time then it will provide accurate cumulative oil production prediction.