Computational inverse problems, optimization, differential equations and applications

我研究的总体目标是为涉及表面重建、推理问题和应用中出现的微分方程的大规模模型开发高效可靠的计算算法。重点是随机化、优化和约束微分问题的方法，例如在三维数字几何和跟踪、图像处理、模型校准、虚拟现实仿真和分布参数估计中出现的问题。我也对时间相关微分问题的结构保持离散化，以及快速梯度下降方法的收敛证明感兴趣。虽然我的研究重点是在不同科学和工程领域之间的知识和专业知识的转移，但重点将放在这个框架内的特定应用领域，包括计算机图形学，感觉运动计算，机器学习，地球物理学和数学金融。在接下来的五年里，我希望在以下几个方面进行工作：数据完成和操作。这包括(i)通过近似或插值补全“缺失”数据的限制；（ii）数据位置不确定的问题，而不仅仅是数据值；（iii）数据补全似乎是获得可信结果所必需的问题。涉及不连续和多数据集的大规模分布参数估计问题的算法和软件。这包括重建逐块光滑表面、随机算法、异构界面问题以及解决大型稀疏逆问题。科学计算在计算机图形学和图像处理中的应用。这包括各种目标集合的模型校准和仿真，点云集的表面跟踪和重建，稀疏解方法，软体仿真，具有大和变化力的离散动力学，以及机器人和虚拟现实中的约束柔性体机械系统仿真。非线性双曲型和抛物型偏微分方程的紧致、保结构方法。重点将放在(i)对这些方法的实际评价；（ii）推导复杂问题的新算法（例如，结合异质材料）；（三）这些方法在计算机图形学中的应用。建立更快的梯度下降方法的收敛性，并研究它们偶尔非常大的步长。

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.