A Fully Discrete Framework for the Adaptive Solution of Inverse Problems

逆问题自适应求解的完全离散框架

• 批准号: 1218454
• 负责人: Adrian Sandu
• 金额: $ 25万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1218454&HistoricalAwards=false
• 关键词: Fully; Discrete; Framework; Adaptive; Solution

Inverse problems like parameter estimation, data assimilation, optimalengineering design, and optimal control of large scale systems governed bypartial differential equations, are of considerable importance in manyfields including atmospheric science and oceanography, aerospaceengineering, and fluid and structural mechanics.State-of-the-art solvers for large scale partial differential equationsadaptively refine the time step and the mesh, and adjust the computationalpattern according to the features of the solution. Adaptivity is necessaryto control the numerical errors introduced by temporal and spatialdiscretizations and to preserve the qualitative features of the solution(e.g., avoid the formation of spurious wiggles). In contrast, most inverseproblems to date have been solved using non-adaptive methods (e.g., fixedgrids and timesteps).This project develops a fully discrete framework for solving inverseproblems in the context of adaptive models. The framework fills the gapbetween the state-of-the-art adaptive methods used in (forward)simulations and the computational tools currently available for thesolution of inverse problems. The specific research objectives are todevelop discrete algorithms for inverse problems with models that employrefinement of the spatial discretization, and adaptive time stepping, toguarantee that the discrete inversion process leads to convergentnumerical approximations, and to control the accuracy of the inversesolution.The results of this work are general algorithms and methodologies thatwill advance the field of inverse problems by developing the capability toadapt time steps, grid sizes, and computational patterns, such as tocontrol the quality and accuracy of the inverse solutions. These resultshave the potential to impact any maturefield which relies on adaptive simulations, such as data assimilation inatmospheric sciences, oceanography, and environmental sciences; optimalcontrol of flows, and optimal engineering design.The algorithmic and software tools developed during this research will belargely disseminated through specialized journals and conferences. Thisproject provides an excellent opportunity for training graduate students inthe areas of inverse problems and adaptive computations.

诸如参数估计，数据同化，最佳启动设计以及对大规模系统控制的副系统的最佳控制等逆问题在许多赛场中都非常重要解决方案。适应性是必要的，以控制由时间和水上性散布引入的数值误差，并保留溶液的定性特征（例如，避免形成虚假的摇摆）。相比之下，迄今为止，大多数逆势都使用非自适应方法（例如，固定格栅和时间段）来解决。该项目在自适应模型的背景下开发了一个完全离散的框架。该框架填充了（正向）模拟中使用的最新自适应方法与当前可用于反问题的解决方案之间的最新自适应方法。具体的研究目标是用于与模型的反问题的开发离散算法，这些算法的雇佣会导致空间离散化，并且自适应时间阶梯阶段，即离散反演过程可导致收敛性的近似值，并导致contressition the eversive the eversive the Argyoly corressive aologiese and Ondoologies and Ondoologies and Ondoologiese the Ofdermiss and Modecties to and and of gyolodies to and of gyolodies to and ofdyolices the and ofdyolices。 Toadapt时间步骤，网格大小和计算模式，例如逆向解决方案的质量和准确性。 这些结果可能会影响任何依赖于自适应模拟的天真田野，例如数据同化Inatmospheric科学，海洋学和环境科学；最佳流量和最佳工程设计。在这项研究中开发的算法和软件工具将通过专门的期刊和会议来暗中传播。 ThisProject为培训逆问题和适应性计算领域的研究生提供了绝佳的机会。