A generalization of the concepts of extrinsic curvature and Weingarten endomorphism is introduced to study a class of nonlinear maps over embedded matrix manifolds. These (nonlinear) oblique projections, generalize (nonlinear) orthogonal projections, i.e. applications mapping a point to its closest neighbor on a matrix manifold. Examples of such maps include the truncated SVD, the polar decomposition, and functions mapping symmetric and non-symmetric matrices to their linear eigenprojectors. This paper specifically investigates how oblique projections provide their image manifolds with a canonical extrinsic differential structure, over which a generalization of the Weingarten identity is available. By diagonalization of the corresponding Weingarten endomorphism, the manifold principal curvatures are explicitly characterized, which then enables us to (i) derive explicit formulas for the differential of oblique projections and (ii) study the global stability of a governing generic Ordinary Differential Equation (ODE) computing their values. This methodology, exploited for the truncated SVD in (Feppon 2018), is generalized to non-Euclidean settings, and applied to the four other maps mentioned above and their image manifolds: respectively, the Stiefel, the isospectral, the Grassmann manifolds, and the manifold of fixed rank (non-orthogonal) linear projectors. In all cases studied, the oblique projection of a target matrix is surprisingly the unique stable equilibrium point of the above gradient flow. Three numerical applications concerned with ODEs tracking dominant eigenspaces involving possibly multiple eigenvalues finally showcase the results.

Comprehensive spatiotemporal modeling and forecasting systems for ocean dynamics necessitate robust and efficient data delivery and visualization techniques. The multidisciplinary simulation, estimation, and assimilation systems group at MIT (MSEAS) focuses on capturing and predicting diverse ocean dynamics, including physics, acoustics, and biology on varied scales, thereby developing new methods for multi-resolution ocean prediction and analysis, including data generation and assimilation. The group has primarily used non-interactive ocean plots to visualize its simulated and measured data. Although these maps and sections allow for analysis of ocean physics and the underlying numerical schemes, more interactive maps provide more user control over depicted data, allowing easier study and pattern identification on multiple scales. Integrating static and geospatial data in dynamic visualization creates a heightened viewpoint for analysis, enhances ocean monitoring and prediction, and contributes to building scientific knowledge. This thesis focuses on explaining the motivation behind and the methodologies applied in designing these interactive maps.

We combine decision theory with fundamental stochastic time-optimal path planning to develop partial-differential-equations-based schemes for risk-optimal path planning in uncertain, strong and dynamic flows. The path planning proceeds in three steps: (i) predict the probability distribution of environmental flows, (ii) compute the distribution of exact time-optimal paths for the above flow distribution by solving stochastic dynamically orthogonal level set equations, and (iii) compute the risk of being suboptimal given the uncertain time-optimal path predictions and determine the plan that minimizes the risk. We showcase our theory and schemes by planning risk-optimal paths of unmanned and/or autonomous vehicles in illustrative idealized canonical flow scenarios commonly encountered in the coastal oceans and urban environments. The step-by-step procedure for computing the risk-optimal paths is presented and the key properties of the risk-optimal paths are analyzed.