Lead: Erik Thiede, Flatiron Institute
Topic: Variational Koopman & Galerkin Approximation
Please see two families of algorithms below that are potentially relevant.
1) First, there are algorithms that attempt to get at the spectral quantities (eigenfunctions, eigenvalues) of the random process. For the assumptions we have (reversible Markov Chain), the most relevant formalism is probably VAC. Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations has a good introduction: Section II through IIIB are recommended. If you only have time to read one reference, this is a good one.
Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations
This is a very-widely used formalism in chemical kinetics, and consequently a lot has been written about it (mostly in the context of two other, very closely related algorithms: Markov State Modelling and Time-lagged Independent Component Analysis). This review article by Stefan Klus has a more general review, as well as a bigger discussion of the connection with the Koopman analysis on other settings.
Data-Driven Model Reduction and Transfer Operator Approximation
2) The second family instead attempts to solve Feynman-Kac problems. These include quantities such as mean-first-passage times, hitting probabilities, and the solution to equation 6 in last week’s Mira et al. Not saying that solving that equation this way is necessarily a good idea: just that it’s covered in the family. The general theory was first laid down in our paper last year:
Galerkin Approximation of Dynamical Quantities using Trajectory Data
Note that “transition operator” is used rather than “Koopman operator.”
If you want an application paper in this family (it predates us, so the theoretical connections with Koopman are implicit in it), you can check out this paper:
Transition path theory analysis of c-Src kinase activation
Please email Sara Mejias Gonzalez at email@example.com for Zoom information if you would like to participate.