Computational inverse problems, optimization, differential equations and applications

点击翻译按钮获取中文摘要

The general objective of my research is to develop efficient and reliable computational algorithms for large-scale models involving surface reconstruction, inference problems and differential equations that arise in applications. The focus is on methods for randomization, optimization and constrained differential problems, such as those arising in 3D digital geometry and tracking, image processing, model calibration, virtual reality simulations and distributed parameter estimation.I am also interested in structure-preserving discretizations for time-dependent differential problems, and in finding convergence proofs for fast gradient descent methods.While my research focuses on the transfer of knowledge and expertise among different fields of science and engineering, emphasis will be placed within this framework on particular applications that arise in areas including computer graphics, sensorimotor computations, machine learning, geophysics and mathematical finance.Over the next five years I expect to work on the following topics:1. Data completion and manipulation. This includes (i) limitations on completion of "missing" data by approximation or interpolation; (ii) problems with uncertainty in data locations, not only data values; and (iii) problems where data completion appears to be necessary for obtaining plausible results.2. Algorithms and software for large-scale distributed parameter estimation problems involving discontinuities and many data sets. This includes reconstructing piece-wise smooth surfaces, randomized algorithms, heterogeneous interface problems, and solving large, sparse inverse problems.3. Scientific computing in computer graphics and image processing applications. This includes model calibration and simulation for various object ensembles, surface tracking and reconstruction from point cloud sets, sparse solution methods, soft body simulation, discrete dynamics with large and varying forces, and constrained flexible-body mechanical system simulations in robotics and virtual reality.4. Compact, structure-preserving methods for nonlinear hyperbolic and parabolic partial differential equations. Emphasis will be placed on (i) practical assessment of such methods; (ii) deriving new algorithms for complex problems (e.g., incorporating heterogeneous material); and (iii) application of such methods in computer graphics.5. Establishing convergence properties of faster gradient descent methods and investigating their occasionally very large steps.