Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows

Lessig, Christian

doi:10.1007/s00021-019-0408-7

Cited by 6 publications

(4 citation statements)

References 28 publications

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“…Our example can be used for the decomposition of vector fields into a sum of divergence-free and curl-free vector fields. This result will shed some insight to the wavelet theory for such vector fields ( [7], [9]). A part of the contents of this paper is contained in the second author's doctoral thesis [8], where one can find other concrete examples of admissible vectors.…”

Section: Introductionmentioning

confidence: 73%

Continuous Wavelet Transforms for Vector-Valued Functions

Ishi

Oshiro

2021

Lecture Notes in Computer Science

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Section: Introductionmentioning

confidence: 73%

Continuous Wavelet Transforms for Vector-Valued Functions

Ishi

Oshiro

2021

Lecture Notes in Computer Science

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“…We review results from Arbabi & Sapsis (2019) on climate data based on a six-hourly reanalysis of velocity and temperature in the Earth's atmosphere recorded at sigma level 0.95 from 1981 to 2017 (Berrisford et al 2011). These global fields were expanded in a spherical wavelet basis described by Lessig (2019) and the objective was to extrapolate the PDF tails for the u-velocity, U NP , and temperature, T NP , at the north pole. A sequence of stochastic models was considered for each of the chaotic time series obtained from measurements at the north pole.…”

Section: Dynamics-constrained Data-driven Methods For Extreme Event S...mentioning

confidence: 99%

Statistics of Extreme Events in Fluid Flows and Waves

Sapsis

2021

Annu. Rev. Fluid Mech.

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Extreme events in fluid flows, waves, or structures interacting with them are critical for a wide range of areas, including reliability and design in engineering, as well as modeling risk of natural disasters. Such events are characterized by the coexistence of high intrinsic dimensionality, complex nonlinear dynamics, and stochasticity. These properties severely restrict the application of standard mathematical approaches, which have been successful in other areas. This review focuses on methods specifically formulated to deal with these properties and it is structured around two cases: ( a) problems where an accurate but expensive model exists and ( b) problems where a small amount of data and possibly an imperfect reduced-order model that encodes some physics about the extremes can be employed. Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 53 is January 7, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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“…Our data is based on 6-hourly reanalysis of velocity and temperature in the earth atmosphere recorded at the 100-mbar iso-pressure surface from 1981 to 2017 [5]. These global fields are then expanded in a spherical wavelet basis [37]. Figure 4 shows the time series of the expansion coefficients for a level-1 wavelet envelope centered on top of the North Pole.…”

Section: Reanalysis Climate Data and Tail Extrapolationmentioning

confidence: 99%

Generative stochastic modeling of strongly nonlinear flows with non-Gaussian statistics

Arbabi¹,

Sapsis²

2019

Preprint

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Strongly nonlinear systems, which commonly arise in turbulent flows and climate dynamics, are characterized by persistent and intermittent energy transfer between various spatial and temporal scales. These systems are difficult to model and analyze due to either high-dimensionality or uncertainty and there has been much recent interest in obtaining reduced models, for example in the form of stochastic closures, that can replicate their non-Gaussian statistics in many dimensions. On the other hand, data-driven methods, powered by machine learning and operator theoretic concepts, have shown great utility in modeling nonlinear dynamical systems with various degrees of complexity. Here we propose a data-driven framework to model stationary chaotic dynamical systems through non-linear transformations and a set of decoupled stochastic differential equations (SDEs). Specifically, we first use optimal transport to find a transformation from the distribution of time-series data to a multiplicative reference probability measure such as the standard normal distribution. Then we find the set of decoupled SDEs that admit the reference measure as the invariant measure, and also closely match the spectrum of the transformed data. As such, this framework represents the chaotic time series as the evolution of a stochastic system observed through the lens of a nonlinear map. We demonstrate the application of this framework in Lorenz-96 system, a 10-dimensional model of high-Reynolds cavity flow, and reanalysis climate data. These examples show that SDE models generated by this framework can reproduce the non-Gaussian statistics of systems with moderate dimensions (e.g. 10 and more), and approximate super-Gaussian tails that are not readily computable from the training data. Some advantageous features of our framework are convexity of the main optimization problem, flexibility in choice of reference measure and the SDE model, and interpretability in terms of interaction between variables and the underlying dynamical process.

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Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows

Cited by 6 publications

References 28 publications

Continuous Wavelet Transforms for Vector-Valued Functions

Continuous Wavelet Transforms for Vector-Valued Functions

Statistics of Extreme Events in Fluid Flows and Waves

Generative stochastic modeling of strongly nonlinear flows with non-Gaussian statistics

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