Divergence-free vector field interpolants properties are explored on uniform and scattered nodes, and also their application to fluid flow problems. These interpolants may be applied to physical problems that require the approximant to have zero divergence, such as the velocity field in the incompressible Navier-Stokes equations and the magnetic and electric fields in the Maxwell's equations. In addition, the methods studied here are meshfree, and are suitable for problems defined on complex domains, where mesh generation is computationally expensive or inaccurate, or for problems where the data is only available at scattered locations.

The contributions of this work include a detailed comparison between standard and divergence-free radial basis approximations, a study of the Lebesgue constants for divergence-free approximations and their dependence on node placement, and an investigation of the flat limit of divergence-free interpolants. Finally, numerical solvers for the incompressible Navier-Stokes equations in primitive variables are implemented using discretizations based on traditional and divergence-free kernels. The numerical results are compared to reference solutions obtained with a spectral

method.

Obtaining high-quality experimental designs to optimize statistical efficiency and data quality is quite challenging for functional magnetic resonance imaging (fMRI). The primary fMRI design issue is on the selection of the best sequence of stimuli based on a statistically meaningful optimality criterion. Some previous studies have provided some guidance and powerful computational tools for obtaining good fMRI designs. However, these results are mainly for basic experimental settings with simple statistical models. In this work, a type of modern fMRI experiments is considered, in which the design matrix of the statistical model depends not only on the selected design, but also on the experimental subject's probabilistic behavior during the experiment. The design matrix is thus uncertain at the design stage, making it diffcult to select good designs. By taking this uncertainty into account, a very efficient approach for obtaining high-quality fMRI designs is developed in this study. The proposed approach is built upon an analytical result, and an efficient computer algorithm. It is shown through case studies that the proposed approach can outperform an existing method in terms of computing time, and the quality of the obtained designs.

Urbanization and infrastructure development often brings dramatic changes in the surface and groundwater regimes. These changes in moisture content may be particularly problematic when subsurface soils are moisture sensitive such as expansive soils. Residential foundations such as slab-on ground may be built on unsaturated expansive soils and therefore have to resist the deformations associated with change in moisture content (matric suction) in the soil. The problem is more pronounced in arid and semi arid regions with drying periods followed by wet season resulting in large changes in soil suction. Moisture content change causes volume change in expansive soil which causes serious damage to the structures. In order to mitigate these ill effects various mitigation are adopted. The most commonly adopted method in the US is the removal and replacement of upper soils in the profile. The remove and replace method, although heavily used, is not well understood with regard to its impact on the depth of soil wetting or near-surface differential soil movements. In this study the effectiveness of the remove and replace method is studied. A parametric study is done with various removal and replacement materials used and analyzed to obtain the optimal replacement depths and best material. The depth of wetting and heave caused in expansive soil profile under climatic conditions and common irrigation scenarios are studied for arid regions. Soil suction changes and associated soil deformations are analyzed using finite element codes for unsaturated flow and stress/deformation, SVFlux and SVSolid, respectively. The effectiveness and fundamental mechanisms at play in mitigation of expansive soils for remove and replace methods are studied, and include (1) its role in reducing the depth and degree of wetting, and (2) its effect in reducing the overall heave potential, and (3) the effectiveness of this method in pushing the seat of movement deeper within the soil profile to reduce differential soil surface movements. Various non-expansive replacement layers and different surface flux boundary conditions are analyzed, and the concept of optimal depth and soil is introduced. General observations are made concerning the efficacy of remove and replace as a mitigation method.

In many classication problems data samples cannot be collected easily, example in drug trials, biological experiments and study on cancer patients. In many situations the data set size is small and there are many outliers. When classifying such data, example cancer vs normal patients the consequences of mis-classication are probably more important than any other data type, because the data point could be a cancer patient or the classication decision could help determine what gene might be over expressed and perhaps a cause of cancer. These mis-classications are typically higher in the presence of outlier data points. The aim of this thesis is to develop a maximum margin classier that is suited to address the lack of robustness of discriminant based classiers (like the Support Vector Machine (SVM)) to noise and outliers. The underlying notion is to adopt and develop a natural loss function that is more robust to outliers and more representative of the true loss function of the data. It is demonstrated experimentally that SVM's are indeed susceptible to outliers and that the new classier developed, here coined as Robust-SVM (RSVM), is superior to all studied classier on the synthetic datasets. It is superior to the SVM in both the synthetic and experimental data from biomedical studies and is competent to a classier derived on similar lines when real life data examples are considered.