A Fully Discrete Framework for the Adaptive Solution of Inverse Problems

偏微分方程的参数估计、数据同化、最优工程设计和最优控制等反问题在大气科学和海洋学、航空航天工程、流体和结构力学等领域都具有重要的意义。并根据解的特征调整计算模式。自适应性是必要的，以控制由时间和空间离散化引入的数值误差，并保持解的定性特征（例如，避免伪摆动的形成）。相反，迄今为止，大多数反问题都是使用非自适应方法解决的（例如，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逆问题。该框架填补了国家的最先进的自适应方法（正）模拟和目前可用于thesolution的反问题的计算工具之间的差距。具体的研究目标是开发离散算法的反问题的模型，采用细化的空间离散化，自适应时间步进，以保证离散反演过程导致收敛的数值逼近，本文的研究成果是一些通用的算法和方法，它们将通过发展适应时间的能力来推进反问题的研究步长、网格大小和计算模式，例如控制逆解的质量和精度。 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、海洋学和环境科学中的数据同化;流的最优控制和最优工程设计。在这项研究中开发的算法和软件工具将通过专业期刊和会议广泛传播。这个项目为培养反问题和自适应计算领域的研究生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.