Optimal Mixing Enhancement by Local Perturbation

Froyland, Gary; González-Tokman, Cecilia; Watson, Thomas M.

doi:10.1137/15m1023221

Cited by 27 publications

(49 citation statements)

References 35 publications

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“…Stabilization problems have been successfully solved via the Lyapunov measure equation [184,147], where the problem can be formulated as a linear program, extended for stochastic systems [39], and applied to high-dimensional fluid flows [182]. A different perspective is assumed in [60], where a convex optimization problem is formulated to determine local perturbations in the context of optimal mixing. In particular, the stochastic kernel is designed as a perturbation to the deterministic kernel of the Perron-Frobenius operator, which controls the diffusion and thus the mixing properties of the underlying system.…”

Section: Control Designmentioning

confidence: 99%

Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser

Kutz

Brunton

2020

Lecture Notes in Control and Information Sciences

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The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite-dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.

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Section: Control Designmentioning

confidence: 99%

Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser

Kutz

Brunton

2020

Lecture Notes in Control and Information Sciences

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“…The mixing norm remains constant, meaning that the color distribution never approaches the average color. Of course, since cutting and shuffling merely redistributes the color pieces, without changing their individual colors, the distribution cannot approach the average (see also [37,Sect. 4]).…”

Section: Visualizing Mixing: Space-time Plotsmentioning

confidence: 99%

“…Contrary to expectation (and the results of Ashwin et al [35]), increasing the diffusion coefficient leads to a deceleration of the mixing rate when both stretching and folding and cutting and shuffling are present. Given just four detailed studies [35][36][37][38] on this topic exist, the dynamics of cutting and shuffling a line segment in the presence of diffusion remain largely unexplored.…”

Section: Introductionmentioning

confidence: 99%

Cutting and shuffling with diffusion: Evidence for cut-offs in interval exchange maps

Wang

Christov

2018

Phys. Rev. E

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Low-dimensional dynamical systems are fruitful models for mixing in fluid and granular flows. We study a one-dimensional discontinuous dynamical system (termed "cutting and shuffling" of a line segment), and we present a comprehensive computational study of its finite-time mixing properties including the effect of diffusion. To explore a large parameter space, we introduce fit functions for two mixing metrics of choice: the number of cutting interfaces (a standard quantity in dynamical systems theory of interval exchange transformations) and a mixing norm (a more physical measure of mixing). We compute averages of the mixing metrics across different permutations (shuffling protocols), showing that the latter averages are a robust descriptor of mixing for any permutation choice. If the decay of the normalized mixing norm is plotted against the number of map iterations rescaled by the characteristic e-folding time, then universality emerges: mixing norm decay curves across all cutting and shuffling protocols collapse onto a single stretched-exponential profile. Next, we predict this critical number of iterations using the average length of unmixed subsegments of continuous color during cutting and shuffling and a Batchelor-scale-type diffusion argument. This prediction, called a "stopping time" for finite Markov chains, compares well with the e-folding time of the stretched-exponential fit. Finally, we quantify the effect of diffusion on cutting and shuffling through a Péclet number (a dimensionless inverse diffusivity), showing that the system transitions more sharply from an unmixed initial state to a mixed final state as the Péclet number becomes large. Our numerical investigation of cutting and shuffling of a line segment in the presence of diffusion thus presents evidence for the latter phenomenon, known as a "cut-off" for finite Markov chains, in interval exchange maps.

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“…IETs with diffusion have been used as toy models to investigate mixing [26,27] but there has been no investigation of the mixing rates of this larger parameter space. Bounds have been found on the mixing rates for permutations composed with expanding maps on the unit interval, with the conclusion that permutations do not improve the mixing rate and typically make it worse [28].…”

Section: Introductionmentioning

confidence: 99%

Deceleration of one-dimensional mixing by discontinuous mappings

2017

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We present a computational study of a simple one-dimensional map with dynamics composed of stretching, permutations of equal sized cells, and diffusion. We observe that the combination of the aforementioned dynamics results in eigenmodes with long-time exponential decay rates. The decay rate of the eigenmodes is shown to be dependent on the choice of permutation and changes non-monotonically with the diffusion coefficient for many of the permutations. The global mixing rate of the map M in the limit of vanishing diffusivity approximates well the decay rates of the eigenmodes for small diffusivity, however this global mixing rate does not bound the rates for all values of the diffusion coefficient. This counter-intuitively predicts a deceleration in the asymptotic mixing rate with increasing diffusivity rate. The implication of the results on finite time mixing are discussed.

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Optimal Mixing Enhancement by Local Perturbation

Cited by 27 publications

References 35 publications

Data-Driven Approximations of Dynamical Systems Operators for Control

Data-Driven Approximations of Dynamical Systems Operators for Control

Cutting and shuffling with diffusion: Evidence for cut-offs in interval exchange maps

Deceleration of one-dimensional mixing by discontinuous mappings

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