Computational methods involving differential equations in computer graphics, machine learning and inference problems

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The general objective of my research is to develop efficient and reliable computational algorithms for large-scale models involving learning (neural networks), physics-based simulation in computer graphics and robotics, partial differential equations in applications, inference problems, and optimization. The focus is on methods for constrained differential problems in time and space, data driven learning and inference (inverse) problems, neural nets, nonlinear optimization algorithms, and data assimilation approaches. Such problems arise in physics-based simulation in computer graphics, image processing, finance, model calibration, and distributed parameter estimation. I have a lifelong interest in exploring the relationship between discrete and continuous processes, and I'm also interested in geometric integration, including conservative discretizations for time--dependent differential problems. While my research focuses on the transfer of knowledge and expertise amongst different areas of application, emphasis will be placed within this framework on particular applications that arise in areas such as computer graphics, vision, biology, and finance. Over the next five years I expect to work on the following topics: 1. Differential equation models for optimization algorithms and neural nets 2. Neural net methods for capturing structure and solution detail of constrained differential equations 3. Numerical integrators for physics-based simulations in computer animation 4. Handling contact and friction in physics--based animation 5. Learning deformable object properties and other inverse problems Almost all of my activities are collaborative and involve HQP.