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.

我的研究的一般目标是为涉及表面重建，推理问题和在应用中出现的微分方程的大规模模型开发有效且可靠的计算算法。重点是用于随机化，优化和约束差异问题的方法，例如在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式参数估计中产生的方法。我也有兴趣结构性地提供时间相关的差异性问题的结构，以及在各个领域的转化和渐进率的研究中的转化。工程学，重点将在此框架内放置在包括计算机图形，感觉运动计算，机器学习，地球物理和数学金融等领域中出现的特定应用程序中。数据完成和操纵。这包括（i）通过近似或插值完成“丢失”数据的限制； （ii）数据位置不确定性的问题，不仅是数据值； （iii）数据完成似乎是获得合理结果所必需的问题。2。用于大规模分布式参数估计问题的算法和软件，涉及不连续性和许多数据集。这包括重建零件平滑的表面，随机算法，异质界面问题以及解决较大的稀疏逆问题。3。计算机图形和图像处理应用程序中的科学计算。这包括用于各种对象集合，点云集的表面跟踪和重建的模型校准和仿真，稀疏解决方案方法，软体模拟，具有大而变化的力的离散动力学以及在机器人和虚拟现实中受约束的柔性机械系统仿真。4。非线性双曲线和抛物线部分微分方程的紧凑，结构保护方法。重点将放在（i）对此类方法的实际评估上； （ii）得出复杂问题的新算法（例如，合并异质材料）； （iii）在计算机图形中应用此类方法5。建立更快的梯度下降方法的收敛特性，并研究其偶尔的步骤。