Studying natural processes for real life applications often involves the modeling and computational simulations of complex processes occurring on many time or spatial scales. The governing equations that represent the dynamics are therefore typically high-dimensional and nonlinear. On the one hand, that makes using tools from dynamical systems theory difficult. On the other hand, each numerical simulation will inevitably fail to resolve so-called unresolved scales or subgrid-scales.
Model reduction aims at simplifying the governing equations by finding low-dimensional models that approximate the full high-dimensional dynamics.
This seminar will discuss model reduction strategies often used in fluid dynamics, climate sciences, molecular dynamics, and control theory.
Methods and conceptual frameworks to be discussed include Proper Orthogonal Decomposition (POD), balanced truncation in control, the Mori-Zwanzig formalism for stochastic parameterization of fast degrees of freedom, and data-driven model reduction techniques such as Markov State Models (MSM) and Hidden Markov Models (HMM).
Antoulas AC 2005. Approximation of Large-Scale Dynamical Systems, SIAM Publications.
Bittracher A, Koltai P, Klus S et al 2018. Transition Manifolds of Complex Metastable Systems: Theory and Data-Driven Computation of Effective Dynamics. Journal of Nonlinear Science28:471. https://doi.org/10.1007/s00332-017-9415-0
Chorin AJ, Hald OH, Kupferman R 2000. Optimal prediction and the Mori–Zwanzig representation of irreversible processes. Proceedings of the National Academy of Sciences, 97(7):2968–2973.