Solution of Inverse Problems with Adaptive Models

偏微分方程组的参数估计、数据同化和最优控制等反问题在大气科学、海洋科学、最优流动控制、国土安全等领域都具有重要的应用价值。目前的大型偏微分方程组的求解方法是对时间步长和网格进行自适应调整，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，迄今为止大多数反问题都是使用非自适应方法解决的，因为获得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的背景下解决逆问题所需的基本算法。这项工作的学术价值通过以下研究内容得到证实：(1)了解使用自适应网格加密、自适应时间推进和自适应计算模式(例如，迎风或通量限制)的方法的离散伴随；(2)开发技术来最小化离散伴随格式与(连续)伴随偏微分方程组之间的不一致性；(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)将新的算法应用于海洋实际资料同化问题。自适应正演模式反问题的通用算法和方法有望在大气科学、海洋学、环境科学、水流最优控制、结构力学、国土安全等领域产生广泛的影响。

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.