Computational inverse problems, optimization, differential equations and applications

计算反问题、优化、微分方程和应用

• 批准号: RGPIN-2016-03855
• 负责人: Ascher, Uri
• 金额: $ 3.35万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614672
• 关键词: Computational; inverse; problems; optimization; differential

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.

本研究的总体目标是为大规模模型开发高效可靠的计算算法，这些模型包括曲面重建、推理问题和应用中出现的微分方程。我的研究重点是随机化、最优化和约束微分问题的方法，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方法。我还对依赖时间的微分问题的结构保持离散化和寻找快速梯度下降方法的收敛证明感兴趣。虽然我的研究集中在不同科学和工程领域之间的知识和专业知识的转移，但在这个框架内，重点将放在出现在计算机图形学、传感器计算、机器学习、地球物理和数学金融。在接下来的五年里，我希望在以下主题上工作：1。数据完成和处理。这包括：(1)通过近似或内插法补全“缺失”数据的局限性；(2)数据位置的不确定性问题，而不仅仅是数据值的不确定性问题；以及(3)数据补全似乎是获得可信结果所必需的问题。涉及不连续和许多数据集的大规模分布式参数估计问题的算法和软件。这包括分段光滑曲面的重建、随机化算法、异质界面问题以及大型稀疏逆问题的求解。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标定和仿真、从点云集合进行曲面跟踪和重建、稀疏求解方法、柔体仿真、大力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。求解非线性双曲型和抛物型偏微分方程解的紧致保结构方法。重点将放在：(1)对这种方法的实际评估；(2)为复杂问题(例如，结合不同种类的材料)得出新的算法；以及(3)这种方法在计算机图形学中的应用。建立了快速梯度下降法的收敛性质，并研究了它们偶尔的很大步长。