Lagrangian Reduced Order Modeling Using Finite Time Lyapunov Exponents

There are two main strategies for improving the projection-based reduced order model (ROM) accuracy—(i) improving the ROM, that is, adding new terms to the standard ROM; and (ii) improving the ROM basis, that is, constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose two new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct two new Lagrangian ROMs, which we denote α-ROM and λ-ROM. We show that both Lagrangian ROMs are more accurate than the standard Eulerian ROMs, that is, ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). In particular, the α-ROM can be orders of magnitude more accurate than the standard Eulerian ROMs. We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs’ accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.

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Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

Chaos An Interdisciplinary Journal of Nonlinear Science ◽

10.1063/1.4984627 ◽

2017 ◽

Vol 27 (6) ◽

pp. 063103 ◽

Cited By ~ 15

Author(s):

Hessam Babaee ◽

Mohamad Farazmand ◽

George Haller ◽

Themistoklis P. Sapsis

Keyword(s):

Lyapunov Exponents ◽

Finite Time ◽

Reduced Order ◽

Finite Time Lyapunov Exponents

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Domains of Attraction for an Autoparametric Mass-Pendulum System: A Lyapunov Exponent Map Approach

Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2003/vib-48594 ◽

2003 ◽

Author(s):

Khalid El-Rifai ◽

George Haller ◽

Anil K. Bajaj

Keyword(s):

Lyapunov Exponent ◽

Finite Time ◽

Initial Conditions ◽

Transient Behavior ◽

System Response ◽

Pendulum System ◽

Domains Of Attraction ◽

Nonlinear Dynamical ◽

Finite Time Lyapunov Exponents ◽

Finite Time Lyapunov Exponent

Many recent studies have been performed on resonantly excited mass-pendulum systems with autoparametric (internal) resonance capturing interesting local steady state phenomena. The objective of this work is to explore the transient behavior in such systems. The domains of attraction of the time-periodic system provide some help in understanding the transient dynamics, and these are sought using a recently developed algorithm that solves for the finite-time Lyapunov exponent over a grid of initial conditions. Though the use of finite-time Lyapunov exponents in nonlinear dynamical analyses is not novel, its application to multi-degree-offreedom forced nonlinear systems has not been reported in the literature. In addition to identifying regions of different final states, the technique used captures different levels of attraction within a domain. This sheds some light on the role played by other modes present in a multi-degree-of-freedom system in shaping the overall system response.

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New Method for Computing Finite-Time Lyapunov Exponents

Physical Review Letters ◽

10.1103/physrevlett.91.254101 ◽

2003 ◽

Vol 91 (25) ◽

Cited By ~ 15

Author(s):

T. Okushima

Keyword(s):

Lyapunov Exponents ◽

Finite Time ◽

New Method ◽

Finite Time Lyapunov Exponents

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All about the ⊥ with its applications in the linear statistical models

Open Mathematics ◽

10.1515/math-2015-0005 ◽

2015 ◽

Vol 13 (1) ◽

Cited By ~ 4

Author(s):

Augustyn Markiewicz ◽

Simo Puntanen

Keyword(s):

Statistical Models ◽

Inner Product ◽

Real Matrix ◽

Inner Products ◽

Column Space ◽

The Matrix ◽

Linear Statistical Models ◽

Extensive List

Abstract For an n x m real matrix A the matrix A⊥ is defined as a matrix spanning the orthocomplement of the column space of A, when the orthogonality is defined with respect to the standard inner product ⟨x, y⟩ = x'y. In this paper we collect together various properties of the ⊥ operation and its applications in linear statistical models. Results covering the more general inner products are also considered. We also provide a rather extensive list of references

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MORSE SPECTRUM FOR NONAUTONOMOUS DIFFERENTIAL EQUATIONS

Stochastics and Dynamics ◽

10.1142/s0219493708002342 ◽

2008 ◽

Vol 08 (03) ◽

pp. 351-363 ◽

Cited By ~ 4

Author(s):

FRITZ COLONIUS ◽

PETER E. KLOEDEN ◽

MARTIN RASMUSSEN

Keyword(s):

Differential Equations ◽

Finite Time ◽

Accumulation Point ◽

Morse Decomposition ◽

Finite Time Lyapunov Exponents ◽

The Past ◽

Accumulation Points ◽

Morse Spectrum ◽

Nonautonomous Differential Equations ◽

Linear Cocycle

The concept of a Morse decomposition consisting of nonautonomous sets is reviewed for linear cocycle mappings w.r.t. the past, future and all-time convergences. In each case, the set of accumulation points of the finite-time Lyapunov exponents corresponding to points in a nonautonomous set is shown to be an interval. For a finest Morse decomposition, the Morse spectrum is defined as the union of all of the above accumulation point intervals over the different nonautonomous sets in such a finest Morse decomposition. In addition, Morse spectrum is shown to be independent of which finest Morse decomposition is used, when more than one exists.

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Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

Journal of Fluid Mechanics ◽

10.1017/s0022112009006363 ◽

2009 ◽

Vol 629 ◽

pp. 41-72 ◽

Cited By ~ 66

Author(s):

ALEXANDER HAY ◽

JEFFREY T. BORGGAARD ◽

DOMINIQUE PELLETIER

Keyword(s):

Sensitivity Analysis ◽

Proper Orthogonal Decomposition ◽

Stokes Equations ◽

Orthogonal Decomposition ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Reduced Order ◽

Selection Step ◽

Low Dimensional ◽

Proper Orthogonal

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

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Non-intrusive nonlinear model reduction via machine learning approximations to low-dimensional operators

Advanced Modeling and Simulation in Engineering Sciences ◽

10.1186/s40323-021-00213-5 ◽

2021 ◽

Vol 8 (1) ◽

Author(s):

Zhe Bai ◽

Liqian Peng

Keyword(s):

Machine Learning ◽

Nonlinear Dynamical Systems ◽

Reduced Order Models ◽

Simulation Code ◽

Nonlinear Dynamical ◽

Reduced Order ◽

Nonlinear Model Reduction ◽

Regression Techniques ◽

Low Dimensional ◽

Modern Machine

AbstractAlthough projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to $$10^3{\times }$$ 10 3 × in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs. The domain of applications include both parabolic and hyperbolic PDEs, regardless of the dimension of full-order models (FOMs).

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