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Stabilizing unstable flows by coarse mesh observables and actuators - a pavement to data assimilation

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One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics are determined by finitely many parameters—finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I will show how to explore this finite-dimensional feature, of the long-time behavior of infinite-dimensional dissipative systems, to design finite-dimensional feedback control for stabilizing their solutions. Moreover, based on this approach I will also present a data assimilation (downscaling) algorithm for weather and climate predictions employing discrete coarse mesh measurements. Notably, numerical implementation of this algorithm yields errors that are bounded uniformly in time; consequently it can be reliably used for long-time integration and statistics. Finally, computational demonstrations implementing this algorithm will exhibit that its performance remarkably exceeds what is suggested by the theory.

This talk is part of the Applied and Computational Analysis series.

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