Computational inverse problems, optimization, differential equations and applications

我的研究的总体目标是开发高效和可靠的计算算法的大型模型，涉及表面重建，推理问题和微分方程中出现的应用程序。重点是随机化，优化和约束微分问题的方法，例如3D数字几何和跟踪，图像处理，模型校准，虚拟现实模拟和分布参数估计中出现的问题。我还对时间相关微分问题的结构保持离散化感兴趣，以及寻找快速梯度下降法的收敛性证明。虽然我的研究集中在科学和工程的不同领域之间的知识和专业知识的转移，在这个框架内，重点将放在计算机图形学、感觉运动计算、机器学习、物理学和数学金融等领域的特定应用上。数据完成和操作。这包括：（一）通过近似或插值法完成“缺失”数据的局限性;（二）数据位置不确定的问题，而不仅仅是数据值的不确定性;（三）数据完成似乎是获得合理结果所必需的问题。涉及不连续性和许多数据集的大规模分布参数估计问题的算法和软件。这包括重建分段光滑表面，随机算法，异构界面问题，以及解决大型稀疏逆问题。计算机图形学和图像处理应用中的科学计算。这包括各种对象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，具有大的和变化的力的离散动力学，以及机器人和虚拟现实中的约束柔性体机械系统仿真。4.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）这些方法的实际评估;（ii）为复杂问题推导新的算法（例如，结合异质材料）;和（iii）这些方法在计算机图形学中的应用。建立更快的梯度下降法的收敛特性，并研究它们偶尔非常大的步骤。

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.