Research Summaries
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Koopman Operator Theory and Approximation
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Koopman operator is a linear representation of nonlinear dynamics by an infinite-dimensional composition operator. Due to linearity, Koopman operator approach to analysis of nonlinear dynamics allows one to use familiar notions from spectral theory: spectrum, eigenfunctions, eigenvalues, etc. (Un)fortunately, Koopman operators are typically infinite-dimensional, even when the underlying systems are finite-dimensional, as they act on function spaces over the state space of the original dynamics. |
My interests lie in understanding how choices of function spaces affect the approximations to the Koopman operator, how accurately certain features can be approximated, and how different approximation approaches “zero-in” on different aspects of dynamics, e.g., regular behavior vs. irregular behavior.
The flavor of my research here is a mixture of computational and theoretical investigation, following R. Hamming's quote: “The purpose of computing is insight, not numbers.”
Collaboration with Igor Mezić, Ryan Mohr, and Mihai Putinar.
Braid dynamics
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Braids are mathematical structures, originating in algebraic topology, that capture topology of dynamical trajectories “dancing” around each other, without storing their precise locations. This makes them both computationally efficient and tolerant to errors in measurement. An important application of braids is in analyzing behavior of oceans from very limited set of measurements. |
Collaboration with Jean-Luc Thiffeault (UW Madison).
Classifying finite-time deformation in material advection in 3D
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Instantaneous deformation by a vector field can be determined by looking at eigenvalues the velocity field Jacobian. Long-term (asymptotic) deformation is indicated by the Lyapunov exponents. In this project we provide numerical criteria for the intermediate-time deformation based on the Jacobian of mesochronic (time-averaged) velocity fields. |
Collaboration with Stefan Siegmund, Doan Thai Son and Igor Mezić.
Inverse problems for singular distributions
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When dynamics settles on a low-dimensional attractor, the probability of finding a trajectory is zero almost-everywhere. That is, the invariant measure is singular, and difficult to measure directly, or approximate. Moments of these invariant measures can be computed by averaging basis functions along trajectories. |
Reconstructing the measure from its moments is known as “inverse moment problem”. While there exist techniques for approximately solving these problems when measures are very singular (Pade reconstruction), or not singular at all (entropy maximization), the intermediate cases that appear in dynamics are difficult to treat.
In this project, we demonstrated that the “difficult” moment problem can be converted (regularized) to the easier moment problem for continuous measures in a computationally tractable way, making it accessible to entropy optimization. More challenging is to convert the approximation to the “regularized” continuous measure into the original singular problem. Nevertheless, this can be done as well in low dimensions.
Collaboration with Mihai Putinar:
Construction of invariants from simulated data
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Computer simulations of dynamical systems often produce curves (trajectories) that show us how one initial state of the system evolves over time. However, humans think of dynamics in terms of larger objects, such as ocean eddies, currents, and streams. In this project I described how individual trajectories can be aggregated into the larger objects. The algorithm combines ergodic theory and machine learning to describe individual trajectories not in our physical space, but rather in a space where the distance between them signifies how similar they are (ergodic quotient). |
This description of dynamics has a natural connection with the so-called Koopman operator description of dynamics. Indeed, coordinates that demonstrate similarity between trajectories are precisely eigenfunctions of the Koopman operator. As a result, the analysis of the ergodic quotient is a natural counterpart to the Koopman Mode Decomposition, which studies how the dynamics are reflected on measurements taken from dynamical systems.
This project is the outgrowth of my PhD work with Igor Mezić at UC Santa Barbara.