Computational Advances in Operator Theoretic Approach to Dynamical Systems, with Application to Data Assimilation
Abstract
We plan to bring theoretical and numerical advances to the data assimilation (DA) methods, in particular the 4d-Var and the 4d-EnVar, by pursuing them in the operator-theoretic, probabilistic and numerical linear algebra frameworks. The proposal comprises several interwoven threads of research, addressing the main difficulties in real world applications of data assimilation, all rooted in the curse of dimensionality.We plan to extend our recent work on the numerical modal decompositions for the Koopman operator, appropriately formulated, to the Perron-Frobenius operator. These two operators provide tools for studying the evolution of the observables and of the densities, which are key ingredients of the 4d-Var assimilation. Further, we will introduce a new entirely orthogonal transformations based approach for both operators based modal decompositions. As an output of these efforts, we envisage a set of computational tools that will be used to tackle the problems of estimating (and updating with the flow) the covariance matrices, and providing good subspaces for model reduction techniques. We adapt our new uncertainty quantification (UQ) technology (based on spectraldecompositions of the Perron-Frobenius and the Koopman operator and the corresponding eigenmodes based representations) to compute the input statistical information (error covariance matrices) that is essential for the success of DA. Further, the UQ is planned for guiding the model reduction approaches.Our planned outcomes can be summarized as follows:- New concepts and numerical methods for efficient implementation of operator theoretic techniques in data assimilation. This includes new numerical methods for spectral decompositions of finite dimensional approximations of the Koopman and Perron-Frobenius operators, and the corresponding modal decompositions. - Numerical methods for approximating the pseudospectrum of the finite sections of the Koopman operator.- New methods for estimating and updating error covariance matrices, using Peron-Frobenius operator based forward uncertainty quantification, and the predictive skills of the Koopman mode decomposition. These results are targeted at the ensemble based DA methods. - New theoretical framework with operator theoretic, probabilistic and numerical pillars for the 4d-Var and related data assimilation methods.The choiceof this research theme has been strongly motivated by our recent contacts with the Navy Research Lab. Since our proposed contribution will be a mathematically defined and modularly organized framework, we expect it to be used (as a sophisticated general purpose technology) in a wide spectrum of other applications of data assimilation that are of potential relevance not only for the Navy (atmospheric and ocean modeling) but for other DoD related applications (target tracking, assessment of other environmental variables etc.).
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- May 05, 2021
- Source ID
- N000142112384
Entities
People
- Maria Fonoberova
Organizations
- Office of Naval Research
- United States Navy