Efficient algorithms for evolving continuum processes on curved surfaces

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Continuum processes on surfaces are essential components to a remarkable variety of modern applications.  These include the physics-based modelling of computer-animated objects, the understanding and characterization of shape, the application and enhancement of texture on objects, and the mapping of cortical change in Alzheimer's disease.  Partial differential equations (PDEs) are the fundamental tools for formulating mathematical algorithms for continuum processes on flat spaces and curved surfaces.  However, working with such equations is much more complicated when the processes occur on curved surfaces rather than on standard Cartesian coordinate spaces.  As a consequence, the algorithms and software needed to solve the underlying equations are often poorly understood, inefficient, or simply unavailable. My long term vision is the development of efficient algorithms and software for general continuum models (involving standard as well as degenerate differential operators,  constraints, integrals, etc.) on general geometries (static or moving, open or closed, piecewise smooth or point cloud, and of arbitrary co-dimension within some general embedding space).  Consistent with this, we have introduced and developed closest point methods.  Such methods have the advantage of dramatically simplifying complex problems into the two standard problems of interpolation and continuum evolution.  To date, most work on closest point methods has focused on the numerical approximation of PDEs on certain smooth, moving surfaces and on the practical application of the method by end-users.  In the proposed research, we (i) conduct the first detailed analysis of the original explicit closest point method, (ii) analyze and develop parallel algorithms and software for the CPM, (iii) derive efficient time-evolution strategies, (iv) extend methods to the approximation of new flows of practical interest, and (v) construct maps between surfaces thereby enabling new, efficient methods for the processing of surfaces. The program of research develops algorithms and software that are accurate and efficient, yet are simple in the sense that they compute solutions to different continuum models as uniformly as possible while leveraging the use of existing standard algorithms and software in 3D. It improves the efficiency of methods in current use, conducts analysis for the improved understanding of existing and new methods, and enables the numerical approximation of surface processes that cannot presently be computed. It also develops the first domain decomposition software for the parallel computing of solutions to some of the most frequently occurring problems on moving surfaces.  As a consequence, the methods and software developed under this grant will enable researchers and end-users to numerically investigate new and realistic models of general continuum processes on complex moving surfaces with high accuracy.