Inverse problems like parameter estimation, data assimilation, optimal engineering design, and optimal control of large scale systems governed by partial differential equations, are of considerable importance in many fields including atmospheric science and oceanography, aerospace engineering, and fluid and structural mechanics.
State-of-the-art solvers for large scale partial differential equations adaptively refine the time step and the mesh, and adjust the computational pattern according to the features of the solution. Adaptivity is necessary to control the numerical errors introduced by temporal and spatial discretizations and to preserve the qualitative features of the solution (e.g., avoid the formation of spurious wiggles). In contrast, most inverse problems to date have been solved using non-adaptive methods (e.g., fixed grids and timesteps).
This project develops a fully discrete framework for solving inverse problems in the context of adaptive models. The framework fills the gap between the state-of-the-art adaptive methods used in (forward) simulations and the computational tools currently available for the solution of inverse problems. The specific research objectives are to develop discrete algorithms for inverse problems with models that employ refinement of the spatial discretization, and adaptive time stepping, to guarantee that the discrete inversion process leads to convergent numerical approximations, and to control the accuracy of the inverse solution.
The results of this work are general algorithms and methodologies that will advance the field of inverse problems by developing the capability to adapt time steps, grid sizes, and computational patterns, such as to control the quality and accuracy of the inverse solutions. These results have the potential to impact any mature field which relies on adaptive simulations, such as data assimilation in atmospheric sciences, oceanography, and environmental sciences; optimal control of flows, and optimal engineering design.
The algorithmic and software tools developed during this research will be largely disseminated through specialized journals and conferences. This project provides an excellent opportunity for training graduate students in the areas of inverse problems and adaptive computations.
