Collaborative Research: Solution of Inverse Problems with Adaptive Models

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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.