Data Assimilation and Parameter Estimation for Parametric Partial Differential Equations
Technical Report,01 Jun 2016,30 May 2019
Texas A and M College Station United States
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Parametric partial differential equations pdes are used to model complex physical and biological systems and arise in optimal control and design. Their solution varies with the parameters in a complex way, especially when there are a large number of parameters. This project studied novel ways to understand the effect of changing parameters by using a technique called model reduction which isolates the important parameters. Various methods for model reduction were studied and evaluated for performance. This included representation by high dimensional polynomials and interpolation of certain judiciously chosen parameter snapshots. A new class of algorithms for model reduction based on nonlinear approximation were introduced and evaluated for performance. The project also studied the best way to incorporate data observations of the solution to improve efficiency. Certain algorithms for data assimilation were proven to be optimal. Several new methods were introduced to speed up computation. These included extracting random snapshots and employing various approaches to optimization. The project also studied the problem of how well the parameters can be determined when as observation of the state is given. Sufficient conditions on the coefficients of the pde were proved to ensure that the parameters are uniquely determined by the state. This was then employed to build algorithms for parameter estimation with certified error bounds.
- Theoretical Mathematics