Collaborative Research: Solution of Inverse Problems with Adaptive Models

参数估计、数据同化和由偏微分方程（PDEs）控制的大规模系统的最优控制等逆问题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、最优流量控制和国土安全。最先进的大规模偏微分方程求解器自适应地改进时间步长和网格，并调整计算模式，以控制数值误差并保持解的定性特征。相比之下，迄今为止，大多数逆问题都是使用非自适应方法来解决的，因为获取自适应模拟的梯度存在相当大的挑战。该项目的目标是推进在自适应模型背景下解决逆问题所需的基本算法。这项工作的智力价值得到以下研究要素的证实：(1)理解使用自适应网格细化，自适应时间步进和自适应计算模式（例如，上绕或通量限制）的方法的离散伴随；(2)开发技术以最小化离散伴随格式与（连续）伴随PDE之间的不一致性；(3)建立先验伴随误差估计；(4)评估不同伴随方法对几种数值优化方案性能的影响；(5)将新算法应用于实际的海洋学数据同化问题。利用自适应正演模型求解逆问题的通用算法和方法有望在许多领域产生广泛的影响，包括大气科学、海洋学、环境科学、流量最优控制、结构力学、国土安全等。

Inverse problems like parameter estimation, data assimilation, and optimalcontrol for large scale systems governed by partial differential equations(PDEs) are of considerable importance in many fields including atmosphericscience and oceanography, optimal flow control, and homeland security.State-of-the-art solvers for large scale PDEs adaptively refine the timestep and the mesh, and adjust the computational pattern in order tocontrol the numerical errors and to preserve the qualitative features ofthe solution. In contrast, most inverse problems to date have been solvedusing non-adaptive methods due to the considerable challenges associated with obtaining gradients for adaptive simulations. The goal of the project is to advance the fundamental algorithms neededfor solving inverse problems in the context of adaptive models. Theintellectual merit of this work is substantiated by the following researchelements: (1) understand the discrete adjoints for methods that use adaptive mesh refinement, adaptive time stepping, and adaptive computational patterns (e.g., upwinding or flux limiting);(2) develop techniques to minimize the inconsistencies between the discrete adjoint scheme and the (continuous) adjoint PDE;(3) develop apriori adjoint error estimates;(4) assess the impact of different adjoint approaches on the performance of several numerical optimization schemes; and(5) apply the new algorithms to real data assimilation problems in oceanography.The general algorithms and methodologies for solving inverse problems withadaptive forward models are expected to have a broad impact on many fieldsincluding atmospheric sciences, oceanography, environmental sciences,optimal control of flows, structural mechanics, homeland security, etc.