TY  - JOUR
A1  - Froyland, Gary
A1  - Koltai, Peter
A1  - Stahn, Martin
T1  - Computation and optimal perturbation of finite-time coherent sets for aperiodic flows without trajectory integration
JF  - SIAM journal on applied dynamical systems
N2  - Understanding the macroscopic behavior of dynamical systems is an important tool to unravel transport mechanisms in complex flows. A decomposition of the state space into coherent sets is a popular way to reveal this essential macroscopic evolution. To compute coherent sets from an aperiodic time-dependent dynamical system we consider the relevant transfer operators and their infinitesimal generators on an augmented space-time manifold. This space-time generator approach avoids trajectory integration and creates a convenient linearization of the aperiodic evolution. This linearization can be further exploited to create a simple and effective spectral optimization methodology for diminishing or enhancing coherence. We obtain explicit solutions for these optimization problems using Lagrange multipliers and illustrate this technique by increasing and decreasing mixing of spatial regions through small velocity field perturbations.
KW  - coherent set
KW  - mixing
KW  - transfer operator
KW  - infinitesimal generator
KW  - unsteady flow
KW  - mixing optimization
Y1  - 2020
U6  - https://doi.org/10.1137/19M1261791
SN  - 1536-0040
VL  - 19
IS  - 3
SP  - 1659
EP  - 1700
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Koltai, Peter
A1  - Lie, Han Cheng
A1  - Plonka, Martin
T1  - Frechet differentiable drift dependence of Perron-Frobenius and Koopman operators for non-deterministic dynamics
JF  - Nonlinearity
N2  - We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron–Frobenius and Koopman operators.
KW  - stochastic differential equations
KW  - transfer operator
KW  - Koopman operator
KW  - Perron-Frobenius operator
KW  - smooth drift dependence
KW  - linear response
KW  - pathwise expectations
Y1  - 2019
U6  - https://doi.org/10.1088/1361-6544/ab1f2a
SN  - 0951-7715
SN  - 1361-6544
VL  - 32
IS  - 11
SP  - 4232
EP  - 4257
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  -