Numerical simulation of surface reconstruction, inverse problems and differential equations in applications

我研究的总体目标是为涉及曲面重建、反问题和微分方程组(DES)的模型开发高效可靠的计算方法和软件。重点是约束DES和优化问题的方法，如3D数字几何、图像处理、虚拟现实仿真和分布式参数估计中出现的方法。我还对依赖于时间的确定性和随机DES的保守方法感兴趣。虽然我的研究重点是在应用领域之间转让知识和专门知识，但在这一框架内，重点将放在具体情况上。 1.涉及不连续的大规模分布参数估计问题的算法和软件：分段光滑曲面重构、界面处理、水平集方法和大型逆问题的求解。 2.面向图形和图像处理应用的科学计算：基于点云集的曲面重建、稀疏求解方法、布料模拟、时空重建、图像处理中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。 3.感觉运动应用中的数值方法：计算机肌图和眼跳运动中的控制和逆问题。 4.双曲型偏微分方程组的紧致守恒方法：不可积问题和数据同化。

The general objective of my research is to develop efficient and reliable computational methods and software for models involving surface reconstruction, inverse problems and differential equations (DEs). The focus is on methods for constrained DEs and optimization problems, such as those arising in 3D digital geometry, image processing, virtual reality simulations and distributed parameter estimation. I am also interested in conservative methods for time-dependent deterministic and stochastic DEs. While my research focuses on the transfer of knowledge and expertise among areas of application, emphasis will be placed within this framework on particular instances. 1. Algorithms and software for large scale distributed parameter estimation problems involving discontinuities: reconstructing piecewise smooth surfaces, handling interfaces, level set methods and solving large inverse problems. 2. Scientific computing for graphics and image processing applications: surface reconstruction from point cloud sets, sparse solution methods, cloth simulation, space-time reconstruction, discrete dynamics in image processing, and constrained flexible-body mechanical system simulations in robotics and virtual reality. 3. Numerical methods in sensorimotor applications: control and inverse problems in computed myography and saccadic eye movement. 4. Compact, conservative methods for hyperbolic systems of PDEs: non-integrable problems and data assimilation.