New Developments in Geometric and Multiscale Numerical Methods

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This research project will study representation and approximation of complex data structures involving geometrical configurations and shapes. The methods developed in this project will shed light on questions such as: Why do human red blood cells have a bi-concave donut shape instead of, say, a peanut shape? It is known that the lipid bilayer is the most elementary and indispensable structural component of biological membranes that form the boundary of all cells. To understand the amazing variety of different shapes they exhibit, biophysicists strive to find an accurate mathematical model for such bio-membranes. With the mathematical and computational techniques under development in this project, the solution of such models can be obtained accurately and efficiently. Similar mathematical techniques can also be applied to unravel white matter structure from diffusion tensor magnetic resonance images (DT-MRI) of the brain and to predict, in real-time, a life-threatening phenomenon called aerodynamic flutter in air transportation. These will help diagnose psychiatric disorders and save lives of pilots and passengers. The project studies multi-scale representation and approximation of manifold-valued data, reduced order modeling, as well as design of numerical algorithms that respect the geometric or topological characteristics of the underlying problem. In particular, this research addresses the following: I. Approximation of manifold-valued data (scattered manifold-valued data and reduced order modeling, and theory of nonlinear subdivision algorithms); II. New multi-scale geometric modeling tools (numerical simulation of lipid bilayers, subdivision differential forms for genus 0 topology, and 3-D subdivision methods). The research will combine techniques from mathematical analysis, geometry, numerical analysis, optimization theory, and computing to advance the understanding of applied geometry problems.