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Reduced-order Model Approach for Data-driven Uncertainty Quantification of Wave

Filtering is concerned with the sequential updating of Markovian systems, given noisy, partial observations of the system state. The problem has many significant and practical applications in science and engineering, for example navigational and guidance systems, radar tracking, sonar ranging, satellite and airplane orbit determination, the spread of hazardous plumes or pollutants, prediction of weather and climate in atmosphere-ocean dynamics. Due to the increasing prevalence of data in all areas of science and engineering, and due to the inherent complexity of physical models developed for the description of many phenomena arising in science and engineering, the need for accurate and speedy filters is paramount. However in its full form filtering requires the description of a time-evolving probability distribution on the system state, conditioned on data, which for many systems can be hard to represent in a computationally tractable way. This is a particular challenge for the complex physical models arising in areas such as atmospheric sciences, oceanography and oil reservoir simulation. However a recent body of work by Andrew Majda and coworkers has demonstrated the possibility of using drastic simplifications of the models for complex turbulent phenomena in order to construct effective filters which are computationally tractable in real-time. The underlying philosophy of this work is to replace the true underlying Markovian model (often deterministic, but chaotic) with a simplified stochastic model which captures the key physical phenomena at the statistical level yet is amenable to closed form expressions for the purpose of filtering. It is possible to interpret this work as providing an important step towards the physics-informed machine learning, going beyond traditional machine learning methodologies which often attempt to build models from the data alone. In this talk we shed further light on this body of work, through analysis, through the derivation of new methods in the same spirit, and through careful numerical experiments.