A Fully Discrete Framework for the Adaptive Solution of Inverse Problems

反问题，如参数估计、数据同化、优化工程设计和由偏微分方程控制的大型系统的最优控制，在许多领域都具有相当重要的意义，包括大气科学和海洋学、航空航天工程、流体和结构力学。目前最先进的大规模偏微分方程求解器自适应地细化时间步长和网格，并根据解的特征调整计算模式。自适应是控制由时间和空间离散化引入的数值误差和保持解的定性特征(例如。，避免形成虚假的摆动)。相比之下，迄今为止大多数逆问题都是使用非自适应方法解决的（例如，固定网格和时间步长）。该项目开发了一个完全离散的框架，用于解决自适应模型背景下的反问题。该框架填补了（正向）模拟中使用的最先进的自适应方法与当前可用于求解逆问题的计算工具之间的空白。具体的研究目标是利用空间离散化的改进和自适应时间步进，开发求解模型反问题的离散算法，以保证离散反演过程导致收敛的数值逼近，并控制反解的精度。这项工作的结果是通用算法和方法，将通过发展适应时间步长，网格大小和计算模式的能力来推进反问题领域，例如控制反解的质量和准确性。这些结果有可能影响任何依赖自适应模拟的成熟领域，如大气科学、海洋学和环境科学的数据同化；流动的最优控制和最优工程设计。在这项研究期间开发的算法和软件工具将主要通过专门期刊和会议传播。本项目为研究生在反问题和自适应计算方面的训练提供了一个极好的机会。

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.