Topics in Geometric and Multiscale Numerical Methods

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There is an emerging interest in various disciplinesof science and engineering to develop parsimonious multiscalerepresentations of data that `lives' on a Riemannian manifold orLie group. Such application areas are growing by leaps and boundsand cognate application problems are arising all the time.Diffusion tensor imaging and collaborative motion modelling aresimple examples of new sensor types and deployments that give riseto massive volumes of data taking values in nonlinear manifolds.We believe that many more such methods are going to be seen in thefuture. The first proposed project allows a kind ofmultiscale representation of nonlinear data that does for suchdata what wavelets were able to do for images and signals. Theresulted multiscale representations are the key to datacompression, feature extraction, noise removal, fast search, andmany other important problems that arise in exploiting such data.There is also an increasing need to extend the current subdivisionmethods to handle also functions, vector fields, 1-forms, etc. on2-D and 3-D manifolds of arbitrary topology. The second proposed projectaddresses part of these needs. The holy grail is to designnumerical algorithms that, in an appropriate sense, respect thegeometric or topological characteristics of the underlyingproblem. The third proposed project is motivated by thevast interests in nanotechnologies. It is speculated thatcomputational nanoscience may gradually take the forefront ofscientific computing in the same way that computational fluiddynamics was at the forefront of scientific computing forseveral decades. In this project we study a central method inelectronic structure computation known as the Kohn-Shamfunctional minimization problem. A specific geometric structure is proposed to be studied. The ultimate goal is to take full advantage of the smooth manifold structure underlying the problem to come up with linear scaling methods that are more efficient, more robust and possess provable, well-understood convergence properties.Our immediate goal of analysis and synthesis of many new types of data, especially those taking values in nonlinear manifolds, as well as functions, vector fields, and differential forms on free-form manifolds, fits right into the broad and fundamental goal of finding efficient ways to organize and manipulate enormous and complex volumes of high-dimensional geometric data. The need of such methods is ubiquitous in science and engineering, so the potential impact of the project is even wider. We also believe that our focused effort here will eventually find their way into large scale scientific and engineering simulation problems, as the fields of computer-aided geometric design and computer-aided engineering are currently converging to each other. In a different direction, we combine rigorous geometric and numerical ideas to attack a centralproblem in electronic structure calculations. The broader impact of this project is evident from the the world-wide interests in material sciences and nanotechnologies. These projects also provide interdisciplinary research and training opportunities for graduate students, and stimulates collaboration among computational mathematicians, engineers and scientists. The publicly available software implementation of our research results further facilitates such training and collaborations.