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  - Lie, Han Cheng
A1  - Stahn, Martin
A1  - Sullivan, Tim J.
T1  - Randomised one-step time integration methods for deterministic operator differential equations
JF  - Calcolo
N2  - Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065-1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.
KW  - Time integration
KW  - Operator differential equations
KW  - Randomisation
KW  - Uncertainty quantification
Y1  - 2022
U6  - https://doi.org/10.1007/s10092-022-00457-6
SN  - 0008-0624
SN  - 1126-5434
VL  - 59
IS  - 1
PB  - Springer
CY  - Milano
ER  -