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http://hdl.handle.net/10985/23061
From the POD-Galerkin method to sparse manifold models
BRUNTON, Steven; NOACK, Bernd; LOISEAU, Jean-Christophe
Reduced-order models are essential for the accurate and efficient prediction, estimation, and control of complex systems. This is especially true in fluid dynamics, where the fully resolved state space may easily contain millions or billions of degrees of freedom. Because these systems typically evolve on a low-dimensional attractor, model reduction is defined by two essential steps: (1) identify a good state space for the attractor and (2) identifying the dynamics on this attractor. The leading method for model reduction in fluids is Galerkin projection of the Navier–Stokes equations onto a linear subspace of modes obtained via proper orthogonal decomposition (POD). However, there are serious challenges in this approach, including truncation errors, stability issues, difficulty handling transients, and mode deformation with changing boundaries and operating conditions. Many of these challenges result from the choice of a linear POD subspace in which to represent the dynamics. In this chapter, we describe an alternative approach, feature-based manifold modeling (FeMM), in which the low-dimensional attractor and nonlinear dynamics are characterized from typical experimental data: time-resolved sensor data and optional nontime-resolved particle image velocimetry (PIV) snapshots. FeMM consists of three steps: First, the sensor signals are lifted to a dynamic feature space. Second, we identify a sparse human-interpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to the velocity field is performed to reconstruct the full state of the system. We demonstrate this approach, and compare with POD-Galerkin modeling, on the incompressible two-dimensional flow around a circular cylinder. Best practices and perspectives for future research are also included, along with open-source code for this example.
Tue, 01 Jun 2021 00:00:00 GMThttp://hdl.handle.net/10985/230612021-06-01T00:00:00ZBRUNTON, StevenNOACK, BerndLOISEAU, Jean-ChristopheReduced-order models are essential for the accurate and efficient prediction, estimation, and control of complex systems. This is especially true in fluid dynamics, where the fully resolved state space may easily contain millions or billions of degrees of freedom. Because these systems typically evolve on a low-dimensional attractor, model reduction is defined by two essential steps: (1) identify a good state space for the attractor and (2) identifying the dynamics on this attractor. The leading method for model reduction in fluids is Galerkin projection of the Navier–Stokes equations onto a linear subspace of modes obtained via proper orthogonal decomposition (POD). However, there are serious challenges in this approach, including truncation errors, stability issues, difficulty handling transients, and mode deformation with changing boundaries and operating conditions. Many of these challenges result from the choice of a linear POD subspace in which to represent the dynamics. In this chapter, we describe an alternative approach, feature-based manifold modeling (FeMM), in which the low-dimensional attractor and nonlinear dynamics are characterized from typical experimental data: time-resolved sensor data and optional nontime-resolved particle image velocimetry (PIV) snapshots. FeMM consists of three steps: First, the sensor signals are lifted to a dynamic feature space. Second, we identify a sparse human-interpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to the velocity field is performed to reconstruct the full state of the system. We demonstrate this approach, and compare with POD-Galerkin modeling, on the incompressible two-dimensional flow around a circular cylinder. Best practices and perspectives for future research are also included, along with open-source code for this example.Nonlinear stochastic modelling with Langevin regression
http://hdl.handle.net/10985/23069
Nonlinear stochastic modelling with Langevin regression
CALLAHAM, J. L.; RIGAS, G.; BRUNTON, S. L.; LOISEAU, Jean-Christophe
Many physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using forward and adjoint Fokker–Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.
Tue, 01 Jun 2021 00:00:00 GMThttp://hdl.handle.net/10985/230692021-06-01T00:00:00ZCALLAHAM, J. L.RIGAS, G.BRUNTON, S. L.LOISEAU, Jean-ChristopheMany physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using forward and adjoint Fokker–Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.Deep Recurrent Encoder: an end-to-end network to model magnetoencephalography at scale
http://hdl.handle.net/10985/23068
Deep Recurrent Encoder: an end-to-end network to model magnetoencephalography at scale
CHEHAB, Omar; DEFOSSEZ, Alexandre; GRAMFORT, Alexandre; KING, Jean-Remi; LOISEAU, Jean-Christophe
Understanding how the brain responds to sensory inputs from non-invasive brain recordings like magnetoencephalography (MEG) can be particularly challenging: (i) the high-dimensional dynamics of mass neuronal activity are notoriously difficult to model, (ii) signals can greatly vary across subjects and trials and (iii) the relationship between these brain responses and the stimulus features is non-trivial. These challenges have led the community to develop a variety of preprocessing and analytical (almost exclusively linear) methods, each designed to tackle one of these issues. Instead, we propose to address these challenges through a specific end-to-end deep learning architecture, trained to predict the MEG responses of multiple subjects at once. We successfully test this approach on a large cohort of MEG recordings acquired during a one-hour reading task. Our Deep Recurrent Encoder (DRE) reliably predicts MEG responses to words with a three-fold improvement over classic linear methods. We further describe a simple variable importance analysis to investigate the MEG representations learnt by our model and recover the expected evoked responses to word length and word frequency. Last, we show that, contrary to linear encoders, our model captures modulations of the brain response in relation to baseline fluctuations in the alpha frequency band. The quantitative improvement of the present deep learning approach paves the way to a better characterization of the complex dynamics of brain activity from large MEG datasets.
Sat, 01 Oct 2022 00:00:00 GMThttp://hdl.handle.net/10985/230682022-10-01T00:00:00ZCHEHAB, OmarDEFOSSEZ, AlexandreGRAMFORT, AlexandreKING, Jean-RemiLOISEAU, Jean-ChristopheUnderstanding how the brain responds to sensory inputs from non-invasive brain recordings like magnetoencephalography (MEG) can be particularly challenging: (i) the high-dimensional dynamics of mass neuronal activity are notoriously difficult to model, (ii) signals can greatly vary across subjects and trials and (iii) the relationship between these brain responses and the stimulus features is non-trivial. These challenges have led the community to develop a variety of preprocessing and analytical (almost exclusively linear) methods, each designed to tackle one of these issues. Instead, we propose to address these challenges through a specific end-to-end deep learning architecture, trained to predict the MEG responses of multiple subjects at once. We successfully test this approach on a large cohort of MEG recordings acquired during a one-hour reading task. Our Deep Recurrent Encoder (DRE) reliably predicts MEG responses to words with a three-fold improvement over classic linear methods. We further describe a simple variable importance analysis to investigate the MEG representations learnt by our model and recover the expected evoked responses to word length and word frequency. Last, we show that, contrary to linear encoders, our model captures modulations of the brain response in relation to baseline fluctuations in the alpha frequency band. The quantitative improvement of the present deep learning approach paves the way to a better characterization of the complex dynamics of brain activity from large MEG datasets.On the role of nonlinear correlations in reduced-order modelling
http://hdl.handle.net/10985/23072
On the role of nonlinear correlations in reduced-order modelling
CALLAHAM, Jared L.; BRUNTON, Steven L.; LOISEAU, Jean-Christophe
This work investigates nonlinear dimensionality reduction as a means of improving the accuracy and stability of reduced-order models of advection-dominated flows. Nonlinear correlations between temporal proper orthogonal decomposition (POD) coefficients can be exploited to identify latent low-dimensional structure, approximating the attractor with a minimal set of driving modes and a manifold equation for the remaining modes. By viewing these nonlinear correlations as an invariant manifold reduction, this least-order representation can be used to stabilize POD–Galerkin models or as a state space for data-driven model identification. In the latter case, we use sparse polynomial regression to learn a compact, interpretable dynamical system model from the time series of the active modal coefficients. We demonstrate this perspective on a quasiperiodic shear-driven cavity flow and show that the dynamics evolves on a torus generated by two independent Stuart–Landau oscillators. The specific approach to nonlinear correlations analysis used in this work is applicable to periodic and quasiperiodic flows, and cannot be applied to chaotic or turbulent flows. However, the results illustrate the limitations of linear modal representations of advection-dominated flows and motivate the use of nonlinear dimensionality reduction more broadly for exploiting underlying structure in reduced-order models.
Tue, 01 Mar 2022 00:00:00 GMThttp://hdl.handle.net/10985/230722022-03-01T00:00:00ZCALLAHAM, Jared L.BRUNTON, Steven L.LOISEAU, Jean-ChristopheThis work investigates nonlinear dimensionality reduction as a means of improving the accuracy and stability of reduced-order models of advection-dominated flows. Nonlinear correlations between temporal proper orthogonal decomposition (POD) coefficients can be exploited to identify latent low-dimensional structure, approximating the attractor with a minimal set of driving modes and a manifold equation for the remaining modes. By viewing these nonlinear correlations as an invariant manifold reduction, this least-order representation can be used to stabilize POD–Galerkin models or as a state space for data-driven model identification. In the latter case, we use sparse polynomial regression to learn a compact, interpretable dynamical system model from the time series of the active modal coefficients. We demonstrate this perspective on a quasiperiodic shear-driven cavity flow and show that the dynamics evolves on a torus generated by two independent Stuart–Landau oscillators. The specific approach to nonlinear correlations analysis used in this work is applicable to periodic and quasiperiodic flows, and cannot be applied to chaotic or turbulent flows. However, the results illustrate the limitations of linear modal representations of advection-dominated flows and motivate the use of nonlinear dimensionality reduction more broadly for exploiting underlying structure in reduced-order models.Data-driven modeling of the chaotic thermal convection in an annular thermosyphon
http://hdl.handle.net/10985/23063
Data-driven modeling of the chaotic thermal convection in an annular thermosyphon
LOISEAU, Jean-Christophe
Identifying accurate and yet interpretable low-order models from data has gained renewed interest over the past decade. In the present work, we illustrate how the combined use of dimensionality reduction and sparse system identification techniques allows us to obtain an accurate model of the chaotic thermal convection in a two-dimensional annular thermosyphon. Taking as guidelines the derivation of the Lorenz system, the chaotic thermal convection dynamics simulated using a high-fidelity computational fluid dynamics solver are first embedded into a low-dimensional space using dynamic mode decomposition. After
having reviewed the physical properties the reduced-order model should exhibit, the latter is identified using SINDy, an increasingly popular and flexible framework for the identification of nonlinear continuous-time dynamical systems from data. The identified model closely resembles the canonical Lorenz system, having the same structure and exhibiting the same physical properties. It moreover accurately predicts a bifurcation of the high-dimensional system (corresponding to the onset of steady convection cells) occurring at a much lower Rayleigh number than the one considered in this study.
Wed, 01 Jul 2020 00:00:00 GMThttp://hdl.handle.net/10985/230632020-07-01T00:00:00ZLOISEAU, Jean-ChristopheIdentifying accurate and yet interpretable low-order models from data has gained renewed interest over the past decade. In the present work, we illustrate how the combined use of dimensionality reduction and sparse system identification techniques allows us to obtain an accurate model of the chaotic thermal convection in a two-dimensional annular thermosyphon. Taking as guidelines the derivation of the Lorenz system, the chaotic thermal convection dynamics simulated using a high-fidelity computational fluid dynamics solver are first embedded into a low-dimensional space using dynamic mode decomposition. After
having reviewed the physical properties the reduced-order model should exhibit, the latter is identified using SINDy, an increasingly popular and flexible framework for the identification of nonlinear continuous-time dynamical systems from data. The identified model closely resembles the canonical Lorenz system, having the same structure and exhibiting the same physical properties. It moreover accurately predicts a bifurcation of the high-dimensional system (corresponding to the onset of steady convection cells) occurring at a much lower Rayleigh number than the one considered in this study.System Identification of Two-Dimensional Transonic Buffet
http://hdl.handle.net/10985/23062
System Identification of Two-Dimensional Transonic Buffet
SANSICA, Andrea; KANAMORI, Masashi; HASHIMOTO, Atsushi; LOISEAU, Jean-Christophe; ROBINET, Jean-Christophe
When modeled within the unsteady Reynolds-Averaged Navier-Stokes framework, the
shock-wave dynamics on a two-dimensional aerofoil at transonic buffet conditions is char-
acterized by time-periodic oscillations. Given the time series of the lift coefficient at different
angles of attack for the OAT15A supercritical profile, the sparse identification of nonlinear dy-
namics (SINDy) technique is used to extract a parametrized, interpretable and minimal-order
description of this dynamics. For all of the operating conditions considered, SINDy infers
that the dynamics in the lift coefficient time series can be modeled by a simple parametrized
Stuart-Landau oscillator, reducing the computation time from hundreds of core hours to sec-
onds. The identified models are then supplemented with equally parametrized measurement
equations and low-rank DMD representation of the instantaneous state vector to reconstruct
the true lift signal and enable real-time estimation of the whole flow field. Simplicity, accuracy
and interpretability make the identified model a very attractive tool towards the construction
of real-time systems to be used during the design, certification and operational phases of the
aircraft life cycle.
Tue, 01 Feb 2022 00:00:00 GMThttp://hdl.handle.net/10985/230622022-02-01T00:00:00ZSANSICA, AndreaKANAMORI, MasashiHASHIMOTO, AtsushiLOISEAU, Jean-ChristopheROBINET, Jean-ChristopheWhen modeled within the unsteady Reynolds-Averaged Navier-Stokes framework, the
shock-wave dynamics on a two-dimensional aerofoil at transonic buffet conditions is char-
acterized by time-periodic oscillations. Given the time series of the lift coefficient at different
angles of attack for the OAT15A supercritical profile, the sparse identification of nonlinear dy-
namics (SINDy) technique is used to extract a parametrized, interpretable and minimal-order
description of this dynamics. For all of the operating conditions considered, SINDy infers
that the dynamics in the lift coefficient time series can be modeled by a simple parametrized
Stuart-Landau oscillator, reducing the computation time from hundreds of core hours to sec-
onds. The identified models are then supplemented with equally parametrized measurement
equations and low-rank DMD representation of the instantaneous state vector to reconstruct
the true lift signal and enable real-time estimation of the whole flow field. Simplicity, accuracy
and interpretability make the identified model a very attractive tool towards the construction
of real-time systems to be used during the design, certification and operational phases of the
aircraft life cycle.PySINDy: A comprehensive Python package for robust sparse system identification
http://hdl.handle.net/10985/23054
PySINDy: A comprehensive Python package for robust sparse system identification
KAPTANOGLU, Alan; DE SILVA, Brian; FASEL, Urban; KAHEMAN, Kadierdan; GOLDSCHMIDT, Andy; CALLAHAM, Jared; DELAHUNT, Charles; NICOLAOU, Zachary; CHAMPION, Kathleen; KUTZ, J.; BRUNTON, Steven; LOISEAU, Jean-Christophe
Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy,
we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differential equations (PDEs), and implicit differential equations. Robust formulations, including the integral form of SINDy and ensembling techniques, are also implemented to improve performance for real-world data. Finally, we provide a range of new optimization algorithms, including several sparse regression techniques and algorithms to enforce and promote inequality constraints and stability. Together, these updates enable entirely new SINDy model discovery capabilities that have not been reported
in the literature, such as constrained PDE identification and ensembling with different sparse regression optimizers.
Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/10985/230542022-01-01T00:00:00ZKAPTANOGLU, AlanDE SILVA, BrianFASEL, UrbanKAHEMAN, KadierdanGOLDSCHMIDT, AndyCALLAHAM, JaredDELAHUNT, CharlesNICOLAOU, ZacharyCHAMPION, KathleenKUTZ, J.BRUNTON, StevenLOISEAU, Jean-ChristopheAutomated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy,
we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differential equations (PDEs), and implicit differential equations. Robust formulations, including the integral form of SINDy and ensembling techniques, are also implemented to improve performance for real-world data. Finally, we provide a range of new optimization algorithms, including several sparse regression techniques and algorithms to enforce and promote inequality constraints and stability. Together, these updates enable entirely new SINDy model discovery capabilities that have not been reported
in the literature, such as constrained PDE identification and ensembling with different sparse regression optimizers.An empirical mean-field model of symmetry-breaking in a turbulent wake
http://hdl.handle.net/10985/23022
An empirical mean-field model of symmetry-breaking in a turbulent wake
CALLAHAM, Jared L.; RIGAS, Georgios; LOISEAU, Jean-Christophe; BRUNTON, Steven L.
Improved turbulence modeling remains a major open problem in mathematical physics. Turbulence is notoriously challenging, in part due to its multiscale nature and the fact that large-scale coherent structures cannot be disentangled from small-scale fluctuations. This closure problem is emblematic of a greater challenge in complex systems, where coarse-graining and statistical mechanics descriptions break down. This work demonstrates an alternative data-driven modeling approach to learn nonlinear models of the coherent structures, approximating turbulent fluctuations as state-dependent stochastic forcing. We demonstrate this approach on a high–Reynolds number turbulent wake experiment, showing that our model reproduces empirical power spectra and probability distributions. The model is interpretable, providing insights into the physical mechanisms underlying the symmetry-breaking behavior in the wake. This work suggests a path toward low-dimensional models of globally unstable turbulent flows from experimental measurements, with broad implications for other multiscale systems.
Wed, 11 May 2022 00:00:00 GMThttp://hdl.handle.net/10985/230222022-05-11T00:00:00ZCALLAHAM, Jared L.RIGAS, GeorgiosLOISEAU, Jean-ChristopheBRUNTON, Steven L.Improved turbulence modeling remains a major open problem in mathematical physics. Turbulence is notoriously challenging, in part due to its multiscale nature and the fact that large-scale coherent structures cannot be disentangled from small-scale fluctuations. This closure problem is emblematic of a greater challenge in complex systems, where coarse-graining and statistical mechanics descriptions break down. This work demonstrates an alternative data-driven modeling approach to learn nonlinear models of the coherent structures, approximating turbulent fluctuations as state-dependent stochastic forcing. We demonstrate this approach on a high–Reynolds number turbulent wake experiment, showing that our model reproduces empirical power spectra and probability distributions. The model is interpretable, providing insights into the physical mechanisms underlying the symmetry-breaking behavior in the wake. This work suggests a path toward low-dimensional models of globally unstable turbulent flows from experimental measurements, with broad implications for other multiscale systems.Global stability, sensitivity and passive control of low-Reynolds-number flows around NACA 4412 swept wings
http://hdl.handle.net/10985/23404
Global stability, sensitivity and passive control of low-Reynolds-number flows around NACA 4412 swept wings
NASTRO, Gabriele; ROBINET, Jean-Christophe; LOISEAU, Jean-Christophe; PASSAGGIA, Pierre-Yves; MAZELLIER, Nicolas
The stability and sensitivity of two- and three-dimensional global modes developing on steady spanwise-homogeneous laminar separated flows around NACA 4412 swept wings are numerically investigated for different Reynolds numbers Re and angles of
attack α. The wake dynamics is driven by the two-dimensional von Kármán mode whose emergence threshold in the α–Re plane is computed with that of the three-dimensional centrifugal mode. At the critical Reynolds number, the Strouhal number, the streamwise wavenumber of the von Kármán mode and the spanwise wavenumber of the leading three-dimensional centrifugal mode scale as a power law of α. The introduction of a sweep angle attenuates the growth of all unstable modes and entails a Doppler effect in the leading modes’ dynamics and a shift towards non-zero frequencies of the three-dimensional centrifugal modes. These are found to be non-dispersive as opposed to the von Kármán modes. The sensitivity of the leading global modes is investigated in the vicinity of the critical conditions through adjoint-based methods. The growth-rate sensitivity map displays a region on the suction side of the wing, wherein a streamwise-oriented force has a net stabilising effect, comparable to what could have been obtained inside the recirculation bubble. In agreement with the predictions of the sensitivity analysis, a spanwise-homogeneous force suppresses the Hopf bifurcation and stabilises the entire branch of von Kármán modes. In the limit of small amplitudes, passive control via spanwise-wavy forcing produces a stabilising effect similar to that of a
spanwise-homogeneous control and is more effective than localised spherical forces.
Sun, 01 Jan 2023 00:00:00 GMThttp://hdl.handle.net/10985/234042023-01-01T00:00:00ZNASTRO, GabrieleROBINET, Jean-ChristopheLOISEAU, Jean-ChristophePASSAGGIA, Pierre-YvesMAZELLIER, NicolasThe stability and sensitivity of two- and three-dimensional global modes developing on steady spanwise-homogeneous laminar separated flows around NACA 4412 swept wings are numerically investigated for different Reynolds numbers Re and angles of
attack α. The wake dynamics is driven by the two-dimensional von Kármán mode whose emergence threshold in the α–Re plane is computed with that of the three-dimensional centrifugal mode. At the critical Reynolds number, the Strouhal number, the streamwise wavenumber of the von Kármán mode and the spanwise wavenumber of the leading three-dimensional centrifugal mode scale as a power law of α. The introduction of a sweep angle attenuates the growth of all unstable modes and entails a Doppler effect in the leading modes’ dynamics and a shift towards non-zero frequencies of the three-dimensional centrifugal modes. These are found to be non-dispersive as opposed to the von Kármán modes. The sensitivity of the leading global modes is investigated in the vicinity of the critical conditions through adjoint-based methods. The growth-rate sensitivity map displays a region on the suction side of the wing, wherein a streamwise-oriented force has a net stabilising effect, comparable to what could have been obtained inside the recirculation bubble. In agreement with the predictions of the sensitivity analysis, a spanwise-homogeneous force suppresses the Hopf bifurcation and stabilises the entire branch of von Kármán modes. In the limit of small amplitudes, passive control via spanwise-wavy forcing produces a stabilising effect similar to that of a
spanwise-homogeneous control and is more effective than localised spherical forces.Time-Stepping and Krylov Method for large scale instability problems
http://hdl.handle.net/10985/17840
Time-Stepping and Krylov Method for large scale instability problems
BUCCI, Michele Alessandro; CHERUBINI, Stefania; ROBINET, Jean-Christophe; LOISEAU, Jean-Christophe
With the ever increasing computational power available and the development of high-performances computing, investigating the properties of realistic very large-scale nonlinear dynamical systems has become reachable. It must be noted however that the memory capabilities of computers increase at a slower rate than their computational capabilities. Consequently, the traditional matrix-forming approaches wherein the Jacobian matrix of the system considered is explicitly assembled become rapidly intractable. Over the past two decades, so-called matrix-free approaches have emerged as an efficient alternative. The aim of this chapter is thus to provide an overview of well-grounded matrix-free methods for fixed points computations and linear stability analyses of very large-scale nonlinear dynamical systems.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10985/178402018-01-01T00:00:00ZBUCCI, Michele AlessandroCHERUBINI, StefaniaROBINET, Jean-ChristopheLOISEAU, Jean-ChristopheWith the ever increasing computational power available and the development of high-performances computing, investigating the properties of realistic very large-scale nonlinear dynamical systems has become reachable. It must be noted however that the memory capabilities of computers increase at a slower rate than their computational capabilities. Consequently, the traditional matrix-forming approaches wherein the Jacobian matrix of the system considered is explicitly assembled become rapidly intractable. Over the past two decades, so-called matrix-free approaches have emerged as an efficient alternative. The aim of this chapter is thus to provide an overview of well-grounded matrix-free methods for fixed points computations and linear stability analyses of very large-scale nonlinear dynamical systems.