Stony Brook AMS - Downloadable Preprints, 2002

In this paper, we study systems of two conservation laws with homogeneous quadratic flux functions. We use the viscous profile criterion for shock admissibility. This criterion leads to the occurrence of non-classical transitional shock waves, which are sensitively dependent on the form of the viscosity matrix. The goal of this paper is to lay a foundation for investigating how the structure of solutions of the Riemann problem is affected by the choice of viscosity matrix.

Working in the framework of the fundamental wave manifold, we derive a necessary and sufficient condition on the model parameters for the presence of transitional shock waves. Using this condition, we are able to identify the regions in the wave manifold that correspond to transitional shock waves. Also, we determine the boundaries in the space of model parameters that separate models with differing numbers of transitional regions.

X-ray computed microtomography was used to investigate why gels reduce permeability to water more than to oil in strongly water-wet Berea sandstone and in an oil-wet porous polyethylene core. Although the two porous media had very different porosities (22% versus 40%), the distributions of pore sizes and aspect ratios were similar. A Cr(III)-acetate-HPAM gel caused comparable oil and water permeability reductions in both porous media. In both cores, the gel reduced permeability to water by a factor of 80 to 90 times more than to oil. However, the distributions of water an oil saturations (versus pore size) were substantially difference before, during, and after gel placement.

The disproportionate permeability reduction appeared to occur by different mechanisms in the two porous media. In Berea, gel caused disproportionate permeability reduction by trapping substantial volumes of oil that remained immobile during water flooding. With this high trapped oil saturation, water was forced to flow through narrow films, through the smallest pores, and through the gel itself. In contrast, during oil flooding, oil pathways remained relatively free from constriction by the gel.

In the polyethylene core, oil trapping did not contribute significantly to the disproportionate permeability reduction. Instead, oil films and a relatively small number of pore pathways provided conduits for the oil. For reasons yet to be understood, the small pore pathways appeared larely unavailable for water flow.

X-ray computed microtomography was used to investigate why gels reduce permeability to water more than to oil in strongly water-wet Berea sandstone. We studied a Cr(III)-acetate-HPAM gel that reduced permeability to water by a factor of 80 to 90 times more than to oil. In Berea, the gel caused disproportionate permeability reduction by trapping substantial volumes of oil that remained immobile during water flooding. With this high trapped oil saturation, water was forced to flow through narrow films, through the smallest pores, and through the gel itself. In contrast, during oil flooding, oil pathways remained relatively free from constriction by the gel.

The functional significance of dendritic spines and their plasticity to a wide spectrum of developmental and pathological conditions has led to extensive studies based on spine morphology. The advances in image acquisition techniques and the associated generation of large 3D data sets of optical micrographs have not been accompanied by comparable advances in data analysis techniques. We present an automated 3D spine detection and quantification procedure suitable for images obtained by laser scanning microscopy. The image is first processed by deconvolution and the dendritic phase consisting of the neuronal cytoplasm is extracted by segmentation. Spines are detected as geometrical protrusions relative to the dendritic backbone. As very thin necks may not be imaged, some spine `heads' may be detached from the dendrite and are detected as detached components. These detected heads are merged with spine `bases' where appropriate. Morphological characterizations on spine length, volume, density and shape classifications are obtained. For time-lapse data, images are registered and individual spines are traced through the image sequence. Successful comparison results on spine lengths and densities with manual analysis are obtained. This method is highly automatic and allows detailed and objective quantification of the structure and dynamics of dendritic spines, which can be important predictors for the function of neural networks.

Pre-clinical animal carcinogenicity studies are usually concerned with testing the statistical significance of a dose-response relationship. When the response consists of a rare event such as the development of a certain type of tumor, exact statistical methods are often employed. The exact randomization trend test based on the multivariate hypergeometric distribution is less powerful in the presence of treatment-related risks other than the specified response. Particularly, the loss of power becomes more pronounced when competing risks cause progressively higher mortality rates with increasing dose, which is usual in practice. An age-adjusted form of the randomization test is proposed to adjust for this effect. Permutational distribution for Peto's cause-of-death (COD) test is also explored and compared to its asymptotic counterpart by simulation. The use of COD information has been a controversial issue due to the subjectivity in the pathologists' determinations as well as economic reasons. The proposed age-adjusted exact test does not require COD, and it is shown to compare favorably to the COD tests via an extensive Monte Carlo simulation. Applications of the methods to two real data sets are included.

We present an overview of the technical objectives of the Terascale Simulation Tools and Technologies center. The primary goal of this multi-institution collaboration is to develop technologies that enable application scientists to easily use multiple mesh and discretization strategies within a single simulation on terascale computers. The discussion focuses on our efforts to create interoperable mesh generation tools, high-order discretization techniques, and adaptive meshing strategies.

We study the motion of a coherent structure of bubbles and spikes in the Rayleigh-Taylor instability for fluids with a finite density contrast. The theoretical and numerical solutions for the system of conservation laws are found, and the dynamics of the bubble shape and velocity in the nonlinear regime is described. Good agreement between theory and simulations is achieved. A comparison to earlier models is performed.

Acceleration driven fluid mixing is studied here from a theoretical point of view. Considerable progress has been achieved in the understanding of mix. Theories of the authors are reviewed which allow prediction of the edge of the mixing zone, in agreement with experimental data. Theories which describe the distribution of masses within the mixing region are alos reviewed. The theory we present describes a chunk mix regime, in which two phases are mixed at a chunk level, but for which there is no atomic mixing. Thus the two phases are segregated into disjoint regions of space.

A new statistical testing approach is developed for rodent tumorigencity assays that have a single terminal sacrifice or occasionally interim sacrifices but not cause-of-death data. For experiments that lack cause-of-death data, statistically imputed numbers of fatal tumors and incidental tumors are used to modify Peto's cause-of-death test which is usually implemented using pathologist-assigned cause-of-death information. The numbers of fatal tumors are estimated using a constrained nonparametric maximum likelihood estimation method. A new Newton-based approach under inequality constraints is proposed for finding the global maximum likelihood estimates. In this study, the proposed method is concentrated on data with a single sacrifice experiment without implementing further assumptions. The new testing approach may be more reliable than Peto's test because of the potential for a misclassification of cause-of-death by pathologists. A Monte Carlo simulation study for the proposed test is conducted to assess size and power of the test. Asymptotic normality for the statistic of the proposed test is also investigated. The proposed testing approach is illustrated using a real data set.

We describe a parallelized, ``point-shifted'', tetrahedral grid scheme for finite element applications. The scheme was developed for front-tracking based methods that track moving discontinuity surfaces, across which solution variables and governing parameters may have large jump discontinuities. This requires conformity between the tracked surface and the finite element grid. The point-shifted technique is a hybrid structured-unstructured grid scheme, providing local unstructuring within the confines of a rectangularly indexed global structure.

We report the theoretical and numerical solutions for the system of conservation laws, describing the nonlinear dynamics of the Rayleigh-Taylor bubbles for fluids with a finite density contrast. Good agreement between theory and simulations is achieved. Comparison to earlier models is performed.

We present a numerical model of two fluid mixing based on hyperbolic equations having complete state variables (velocity, pressure, temperature) for each fluid. The model is designed for the study of acceleration driven mixing layers in a chunk mix regime dominated by large scale coherent mixing structures. The numerical solution of the model is validated by comparison to the incompressible limit. For the purpose of this comparison, we present a newly obtained analytic solution of the pressure equation for this model and an analytic constraint derived from the asymptotic limit of the compressible pressures, which determines uniquely the incompressible pressure solution. The numerical solution is also validated by a mesh convergence study.

We propose an algorithm for sequentially constructing non-isomorphic orthogonal designs (including both regular and non-regular orthogonal designs). An essential element of the algorithm is using minimal column base to reduce the computations for determining isomorphism between any two designs. By using this algorithm, we obtain the complete catalogs of $n \times k$ ($k=2, \ldots, n-1$) two-level orthogonal designs for $n=12,$ 16, and 20. Based on this result, we study the statistical properties of the designs using the extended word length pattern (EWLP) criterion. The minimum-aberration designs according to this criterion are obtained and provided for practical use. For 20-run designs, we discover two minimum-aberration designs that are not the projections of Hadamard matrices. These designs were not given in the literature previously.

We consider a model for immiscible three-phase (e.g., water, oil, and gas) flow in a porous medium. We allow the relative permeability of the gas phase to exhibit hysteresis, in that it varies irreversibly along two extreme paths (the imbibition and drainage curves) that bound a region foliated by reversible paths (scanning curves). By numerically solving one-dimensional flow problems involving simultaneous and alternating injection of water and gas into a rock core, we demonstrate the effects of hysteresis.

Rayleigh-Taylor mixing rate is studied through the front tracking simulation; the result shows that the acceleration rate falls within the range of experiments. The front tracking method prevent interfacial mass diffusion. An analysis is presented 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 mass diffusion across the interface. 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. The Richtmyer-Meshkov instability is also studied through the front tracking method in spherical geometry. The growth rate of fingers at an unstable shell driven by an imploding spherical shock is studied through simulations; a qualitative understanding of this system has been achieved.

Uncertainty quantification requires a robust model for solution errors. We present a statistical model of solution errors for this purpose. We illustrate these ideas through application to the simulation and prediction of petroleum reservoir production with confidence intervals for estimates of future production. Prediction is based on use of past production data with the inverse or history matching problem solved stochastically within the uncertainty quantification framework. We have developed a simple model for solution errors for upscaled flow in porous media. A suitable choice of the error functional and its non-dimensionalization yield an error model with a small number of degrees of freedom and only moderate sensitivity to the various parameters (explanatory variables) of the model. The authors are not aware of comparable studies of scale up solution errors by other groups.

Dose response experiments are crucial in biomedical studies. There are usually multiple objectives in such experiments and the most common goals are the estimation of several percentiles on the dose response curve. Here we present the first nonparametric adaptive design approach to estimate several percentiles simultaneously via generalized P\'{o}lya urns. Theoretical properties of these designs were investigated and their performance was gauged by the locally compound optimal designs. As an example, we reinvestigated a psychophysical experiment where the goals were to estimate the three quartiles (Rosenberger and Grill, 1997). We show that these multiple-objective adaptive designs are more efficient than the original single-objective adaptive design targeting the median only.

Compound optimal design is one of two equivalent types of multiple-objective optimal designs, the other being the constrained optimal design. This paper intends to find the structure and study the properties of the locally compound optimal designs for the logit model. These designs can be adopted directly or serve as the `gold standard' in designing multiple-objective dose response experiments. By far we have shown that the locally compound optimal design for estimating the two model parameters (location and scale) with possibly unequal interests will be two-point symmetrical around the location parameter. The design support points can be obtained analytically via Elfving's Theorem on trace optimal design (Elfving, 1952). Furthermore, we have extended our results to another most commonly used dose response model -- the probit model.

In dose response studies, the dose range is often restricted due to concerns over drug toxicity and/or efficacy. We present the restricted interval compound optimal designs for estimating the underlying dose response curve. These curves have a common canonical form (Wu, 1988) and include the most common binary response models -- the logit and the probit. As an example, the results were applied towards the re-designing of a dose ranging trial conducted at the Merck Research Laboratories (Zeng and Zhu, 1997).

The breakup of injected fuel into spray is of key interest to the design of a fuel efficient, nonpolluting diesel engine. We report preliminary progress on the numerical simulation of diesel fuel injection spray with the front tracking code {\it FronTier}. Our simulation design is set to match experiments at ANL, and our present agreement is semi-quantitative. Future efforts will include mesh refinement studies, which will better model the turbulent flow.

We present numerical computations for single phase flow through 3D digitized rock fractures under varied simulated confining pressures appropriate to midcrustal depths. The computations are performed using a finite difference, lattice Boltzmann method and thus simulate Navier-Stokes flow. The digitized fracture data sets come from profiled elevations taken on tensile induced fractures in Harcourt granite. Numerical predictions of fracture permeability are compared with laboratory measurements performed on the same fractures. Use of the finite difference lattice Boltzmann method allows computation on nonuniform grid spacing, enabling accurate resolution across the aperture width without extensive refinement in the other two directions.

We generate the network model equivalents of four samples of Fontainebleau sandstone obtained from the analysis of microtomographic images. We present the measured distributions of flow-relevant geometric and topological properties of the pore space. We generate via bond dilution from a regular lattice, stochastic network models with identical geometric (pore-size, throat-size) and topological (coordination number distribution) properties. We then simulate the two-phase flow properties directly on the network model and the stochastic equivalent for each sample. The simulations on the stochastic networks are found to provide a poor representation of the results on the direct network equivalents. We find that the description of the network topology is particularly crucial in accurately predicting the residual phase saturations. We also find it necessary to introduce into the stochastic network geometry both extended pore-pore correlations and local pore-throat conrrelations to obtain good agreement with flow properties on the rock-equivalent network.

Appears in Transport in Porous Media 46: 345-372, 2002.
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