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In this thesis, we present a method for the automated generation of numerical evaluation routines for bivariate functions via tensor product series and develop a toolkit to assist with the generation of the approximations. The final approximations can be evaluated in user-defined precision or in hardware floating point precision by default. The evaluation routines can also be compiled into a C library (or a library in some other language) for more efficient evaluations. The toolkit can be used for various mathematical functions of two variables, such as Bessel functions or user-defined functions, at any given precision. The method of tensor product series expansion reduces the bivariate approximation problem to a sequence of univariate approximation problems. In order to control the degrees of the approximating functions so that evaluation will be accurate and efficient, we recursively divide the bivariate intervals into subintervals until both the number of terms in the tensor product series and the degrees of the univariate approximations are less than specified bounds. We then generate in each subinterval rational approximations using Chebyshev-Padé approximants or polynomial approximations using Chebyshev series according to the user's specification. Finally we show the experimental results for a variety of bivariate functions, which achieve a significant speedup over the original Maple functions for evaluation in hardware floating point precision. We also compare the results of choosing polynomial approximations versus rational approximations for the univariate subproblems.

Over one million simulations were conducted using the Hec-Ras4b (US Army Corps of Engineers, 2004) model to evaluate the sensitivity of model predictions to field data accuracy, density and estimation techniques and provide guidance towards balancing human resource allocation with model accuracy. Notable differences were identified in model accuracy if a project is concerned with river processes occurring within the limits of the bankfull channel versus floodplain regions. Increased cross section discretization, bankfull channel detail and main channel roughness were of greatest field survey and measurement importance when processes relevant to the bankfull channel are of concern (i.e. geomorphic processes or sediment transport). Conversely, where flood conditions are of highest consideration, estimates of floodplain roughness dominate the accuracy of the results of computed water surface elevations. Results for this case study also demonstrate that higher orders of total station field surveys provide little additional accuracy in final predicted water surface elevations, relative to proper estimates of in-channel and floodplain roughness. As long as drift in field surveys has been accounted for during or subsequent to total station surveys, survey techniques such as hangers can be readily employed with very little increase in final model prediction error, while improving field data acquisition efficiency.

In this work, the peroxide-initiated degradation of polypropylene (PP) in co-rotating intermeshing twin-screw extruders (COITSEs) is analyzed by means of numerical simulations. This reactive extrusion (REX) operation is simulated by implementing (i) a one-dimensional and (ii) a three-dimensional (3D) modeling approach. In the case of the 1D modeling, a REX mathematical model previously developed and implemented as a computer code is used for the evaluation of two scale-up rules for COITSEs of various sizes. The first scale-up rule which is proposed in this work is based on the concept of thermal time introduced by Nauman (1977), and the second one is based on specific energy consumption (SEC) requirements. The processing parameters used in testing the previously referred to scale-up approaches are the mass throughput, the screw rotating speed, and the peroxide concentration, whereas the extruder screw configuration and the barrel temperature profiles are kept constant. The results for the simulated operating conditions show that when the REX operation is scaled-up under constant thermal time, very good agreement is obtained between the weight-average molecular weight (Mw) and poly-dispersity index (PDI) from the larger extruders and the values of these parameters corresponding to the reference extruder. For the constant SEC approach, on the other hand, more significant variations are observed for both of the aforementioned parameters. In the case of the implemented constant thermal time procedure, a further analysis of the effect of the mass throughput and screw speed of the reference device on the scaled-up operation is performed. It is observed that when the lower mass throughput is implemented for the smaller extruder keeping a constant screw speed, the predicted residence times of extrusion for the larger extruders are lower, in general terms, than those corresponding to the reference device, and a converse situation occurs for the higher implemented value of the mass throughput. Also, in general terms, the higher increase of the reaction temperature on the scaled-up operation corresponds to the lower mass throughputs and higher screw speeds specified for the reference extruder. For the 3D modeling approach, two different case studies are analyzed by means of a commercial FEM software package. The REX simulations are performed under the assumption of steady-state conditions using the concept of a moving relative system (MRS). To complement the information obtained from the MRS calculations, simulations for selected conditions (for non-reactive cases) are performed considering the more realistic transient-state (TS) flow conditions. The TS flow conditions are associated to the time periodicity of the flow field inside the conveying elements of COITSEs. In the first case study, the peroxide-initiated degradation of PP is simulated in fully-filled screw elements of two different size COITSEs in order to evaluate scale-up implications of the REX operation. In the second case, the reacting flow is simulated for a conventional conveying screw element and a conveying screw element having a special design and corresponding to the same extruder size. For both of the analyzed cases, the effects of the initial peroxide concentration and mass throughput on the final Mw and PDI of the degraded resin are studied. The effect of the processing conditions is discussed in terms of the residence time distribution (RTD), the temperature of reaction, and the distributive mixing capabilities of the REX system. When analyzing the scale-up case, it is found that for the implemented processing conditions, the final Mws and PDIs are very close to each other in both of the analyzed flow geometries when the specified flow is close to that corresponding to the maximum conveying capabilities of the screw elements. For more restrictive flow conditions, the final Mws and PDIs are lower in the case of the screw element of the larger extruder. It is found that the distributive mixing ability of the reactive flow is mainly related to the specified mass throughput and almost independent of the specified peroxide concentration for a particular extruder size. For the analyzed screw elements, the conveying element corresponding to the small size extruder shows a slightly better distributive mixing performance. For this same case study, a further evaluation of the proposed scale-up criterion under constant thermal time confirms the trend of the results observed for the 1D simulations. In the second case study, the special type of screw element consists of screws rotating at different speeds which have different cross sections. In this case, the outer and inner diameters of both the special and the conventional type of screw elements are specified to be the same. As in the previous case study, the distributive mixing capabilities appear to be independent of the specified peroxide concentrations but dependent on the mass flow rate. It is speculated from the simulation results, from both the transient- as well as the steady-state flow conditions, that the screw element with the special design would yield lower final values of the PDI and Mw. Also, this screw element appears to have improved distributive mixing capabilities as well as a wider RTD.

Heart disease, cancer, diabetes and other complex diseases account for more than half of human mortality in the United States. Other diseases such as AIDS, asthma, Parkinson’s disease, Alzheimer’s disease and cerebrovascular ailments such as stroke not only augment this mortality but also severely deteriorate the quality of human life experience. In spite of enormous financial support and global scientific effort over an extended period of time to combat the challenges posed by these ailments, we find ourselves short of sighting a cure or vaccine. It is widely believed that a major reason for this failure is the traditional reductionist approach adopted by the scientific community in the past. In recent times, however, the systems biology based research paradigm has gained significant favor in the research community especially in the field of complex diseases. One of the critical components of such a paradigm is computational systems biology which is largely driven by mathematical modeling and simulation of biochemical systems. The most common methods for simulating a biochemical system are either: a) continuous deterministic methods or b) discrete event stochastic methods. Although highly popular, none of them are suitable for simulating multi-scale models of biological systems that are ubiquitous in systems biology based research. In this work a novel method for simulating biochemical systems based on a deterministic solution is presented with a modification that also permits the incorporation of stochastic effects. This new method, through extensive validation, has been proven to possess the efficiency of a deterministic framework combined with the accuracy of a stochastic method. The new crossover method can not only handle the concentration and spatial gradients of multi-scale modeling but it does so in a computationally efficient manner. The development of such a method will undoubtedly aid the systems biology researchers by providing them with a tool to simulate multi-scale models of complex diseases.

The Mid Ocean Ridge system with its volcanism and related hydrothermal activity has been a subject for many studies since the discovery of high temperature hydrothermal vents at the ridge surfaces in the 1970´s. This thesis focuses on deep sea hydrothermal activity on a superfast spreading ridge, the SouthernEast Pacific Rise (SEPR).The ridge is located in the South Pacific, off the coast of South America, and separates the Nazca Plate and the Pacific Plate. A fossil high temperature hydrothermal zone hosted by a fault was sampled 80 m below the lava/dike transition zone in the Pito Deep (a tectonic window intothe SEPR). Geochemical data from the fault zone indicates that cold (<150°C)and hot (<390°) fluids coexisted at the same time whilst the hydrothermal system was active. A numerical model (HYDROTHERM) developed by the USGS was used to recreate the geological settings in the SEPR in order to try to model the hydrothermal activity and fluid flow. The model solves two governingpartial differential equations numerically, the water component flow equation(Darcy law for flow in porous media) and the thermal energy transport equation(conservation of enthalpy for the water component and the porous media). The result of the modeling indicates that cold seawater can penetrate from the relatively permeable volcanic material into a highly permeable fault zone in the sheeted dike unit. The cooler seawater fluid flows down the fault zone,reheats and flows up again in a narrow upflow zone at the edge of the fracture/sheeted dike boundary. The result is a horizontal temperature gradient created in the fractured zone supporting the theory that hot and cold fluids can coexist in a fault hosted hydrothermal zone.

This thesis presents a new preconditioner for elliptic PDE problems on unstructured meshes. Using ideas from second generation wavelets, a multi-resolution basis is constructed to effectively compress the inverse of the matrix, resolving the sparsity vs. quality problem of standard approximate inverses. This finally allows the approximate inverse approach to scale well, giving fast convergence for Krylov subspace accelerators on a wide variety of large unstructured problems. Implementation details are discussed, including ordering and construction of factored approximate inverses, discretization and basis construction in one and two dimensions, and possibilities for parallelism. The numerical experiments in one and two dimensions confirm the capabilities of the scheme. Along the way I highlight many new avenues for research, including the connections to multigrid and other multi-resolution schemes.

In this thesis, we present a method for the automated generation of numerical evaluation routines for bivariate functions via tensor product series and develop a toolkit to assist with the generation of the approximations. The final approximations can be evaluated in user-defined precision or in hardware floating point precision by default. The evaluation routines can also be compiled into a C library (or a library in some other language) for more efficient evaluations. The toolkit can be used for various mathematical functions of two variables, such as Bessel functions or user-defined functions, at any given precision. The method of tensor product series expansion reduces the bivariate approximation problem to a sequence of univariate approximation problems. In order to control the degrees of the approximating functions so that evaluation will be accurate and efficient, we recursively divide the bivariate intervals into subintervals until both the number of terms in the tensor product series and the degrees of the univariate approximations are less than specified bounds. We then generate in each subinterval rational approximations using Chebyshev-Padé approximants or polynomial approximations using Chebyshev series according to the user's specification. Finally we show the experimental results for a variety of bivariate functions, which achieve a significant speedup over the original Maple functions for evaluation in hardware floating point precision. We also compare the results of choosing polynomial approximations versus rational approximations for the univariate subproblems.