Transfer operator method for control in fluid flows

2009 ◽  
Author(s):  
Umesh Vaidya ◽  
Baskar Ganapathysubramanian ◽  
Arvind Raghunathan
Keyword(s):  
Transfer Operator ◽  
Operator Method ◽  
Fluid Flows

scholarly journals RELATIVELY COHERENT SETS AS A HIERARCHICAL PARTITION METHOD

2013 ◽  
Vol 23 (07) ◽  
pp. 1330026 ◽  
Author(s):  
TIAN MA ◽  
ERIK M. BOLLT
Keyword(s):  
Computational Complexity ◽  
Objective Function ◽  
Oil Spill ◽  
Empirical Data ◽  
Finite Time ◽  
Transfer Operator ◽  
Operator Method ◽  
Hierarchical Partition ◽  
Partition Method ◽  
Double Gyre

Finite time coherent sets [Froyland et al., 2010] have recently been defined by a measure-based objective function describing the degree that sets hold together, along with a Frobenius–Perron transfer operator method to produce optimally coherent sets. Here, we present an extension to generalize the concept to hierarchically define relatively coherent sets based on adjusting the finite time coherent sets to use relative measures restricted to sets which are developed iteratively and hierarchically in a tree of partitions. Several examples help clarify the meaning and expectation of the techniques, as they are the nonautonomous double gyre, the standard map, an idealized stratospheric flow, and empirical data from the Mexico Gulf during the 2010 oil spill. Also for the sake of analysis of computational complexity, we include an Appendix concerning the computational complexity of developing the Ulam–Galerkin matrix estimates of the Frobenius–Perron operator centrally used here.

scholarly journals Spectral Early-Warning Signals for Sudden Changes in Time-Dependent Flow Patterns

Fluids ◽  
2021 ◽  
Vol 6 (2) ◽  
pp. 49
Author(s):  
Moussa Ndour ◽  
Kathrin Padberg-Gehle ◽  
Martin Rasmussen
Keyword(s):  
Transfer Operator ◽  
Polar Vortex ◽  
Fluid Flows ◽  
Flow Patterns ◽  
Time Dependent ◽  
Mixing Processes ◽  
Dependent Flow ◽  
Antarctic Polar Vortex ◽  
Transport And Mixing ◽  
The Antarctic

Lagrangian coherent sets are known to crucially determine transport and mixing processes in non-autonomous flows. Prominent examples include vortices and jets in geophysical fluid flows. Coherent sets can be identified computationally by a probabilistic transfer-operator-based approach within a set-oriented numerical framework. Here, we study sudden changes in flow patterns that correspond to bifurcations of coherent sets. Significant changes in the spectral properties of a numerical transfer operator are heuristically related to critical events in the phase space of a time-dependent system. The transfer operator approach is applied to different example systems of increasing complexity. In particular, we study the 2002 splitting event of the Antarctic polar vortex.

PRECISE CALCULATION OF HAUSDORFF DIMENSION OF APOLLONIAN GASKET

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850050
Author(s):  
ZAI-QIAO BAI ◽  
STEVEN R. FINCH
Keyword(s):  
Hausdorff Dimension ◽  
Transfer Operator ◽  
Operator Method ◽  
Matrix Approximation ◽  
Infinite Dimensional ◽  
Precise Calculation ◽  
Finite Matrix ◽  
Finite Set ◽  
Infinite Set ◽  
Infinite Sum

A transfer operator method is proposed to calculate [Formula: see text], the Hausdorff dimension of the Apollonian gasket. Compared with previous operator-based methods, we make two improvements in this paper. We adopt an infinite set of contractive Möbius transformations (rather than a finite set of parabolic ones) to generate the Apollonian gasket. We also apply an efficient finite matrix approximation of an infinite sum of infinite-dimensional operators. By using this method, a high precision estimate of [Formula: see text] is obtained: [Formula: see text]

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