Collaborative Research: Solution of Inverse Problems with Adaptive Models

偏微分方程（PDE）控制的大系统的参数估计、数据同化和最优控制等反问题在大气科学和海洋学、最优流量控制和国土安全等许多领域都具有重要意义。并调整计算模式以控制数值误差和保持解的定性特征。相比之下，迄今为止，大多数逆问题都是使用非自适应方法解决的，因为获得自适应模拟的梯度具有相当大的挑战。该项目的目标是推进在自适应模型的背景下求解逆问题所需的基本算法。本文的研究成果主要体现在以下几个方面：（1）理解自适应网格方法的离散伴随问题 细化、自适应时间步进和自适应计算模式 (e.g.,（2）开发技术以最大限度地减少 离散伴随格式和（连续）伴随偏微分方程;（3）发展先验伴随误差估计;（4）评估不同伴随方法对性能的影响， 几种数值优化方案;（5）将新算法应用于真实的资料同化问题， 自适应正演模型求解反问题的一般算法和方法有望对包括大气科学、海洋学、环境科学、流的最优控制、结构力学、国土安全等在内的许多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.