A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems
We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attracto...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Sapsis, Themistoklis P. [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2013transfer abstract |
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| Schlagwörter: |
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| Umfang: |
12 |
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| Übergeordnetes Werk: |
Enthalten in: Role of the Fyn −93A>G polymorphism (rs706895) in acute rejection after liver transplantation - Thude, Hansjörg ELSEVIER, 2015transfer abstract, Amsterdam [u.a.] |
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| Übergeordnetes Werk: |
volume:252 ; year:2013 ; day:1 ; month:06 ; pages:34-45 ; extent:12 |
| Links: |
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| DOI / URN: |
10.1016/j.physd.2013.02.009 |
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| 245 | 1 | 0 | |a A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems |
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| 520 | |a We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. | ||
| 520 | |a We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. | ||
| 650 | 7 | |a Quasilinear Gaussian closure |2 Elsevier | |
| 650 | 7 | |a Modeling of nonlinear energy fluxes |2 Elsevier | |
| 650 | 7 | |a Uncertainty quantification |2 Elsevier | |
| 650 | 7 | |a Turbulent systems |2 Elsevier | |
| 700 | 1 | |a Majda, Andrew J. |4 oth | |
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10.1016/j.physd.2013.02.009 doi GBVA2013005000030.pica (DE-627)ELV038649314 (ELSEVIER)S0167-2789(13)00058-4 DE-627 ger DE-627 rakwb eng 530 530 DE-600 610 VZ 570 VZ BIODIV DE-30 fid Sapsis, Themistoklis P. verfasserin aut A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems 2013transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. Quasilinear Gaussian closure Elsevier Modeling of nonlinear energy fluxes Elsevier Uncertainty quantification Elsevier Turbulent systems Elsevier Majda, Andrew J. oth Enthalten in Elsevier Thude, Hansjörg ELSEVIER Role of the Fyn −93A>G polymorphism (rs706895) in acute rejection after liver transplantation 2015transfer abstract Amsterdam [u.a.] (DE-627)ELV018422527 volume:252 year:2013 day:1 month:06 pages:34-45 extent:12 https://doi.org/10.1016/j.physd.2013.02.009 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA AR 252 2013 1 0601 34-45 12 045F 530 |
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10.1016/j.physd.2013.02.009 doi GBVA2013005000030.pica (DE-627)ELV038649314 (ELSEVIER)S0167-2789(13)00058-4 DE-627 ger DE-627 rakwb eng 530 530 DE-600 610 VZ 570 VZ BIODIV DE-30 fid Sapsis, Themistoklis P. verfasserin aut A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems 2013transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. Quasilinear Gaussian closure Elsevier Modeling of nonlinear energy fluxes Elsevier Uncertainty quantification Elsevier Turbulent systems Elsevier Majda, Andrew J. oth Enthalten in Elsevier Thude, Hansjörg ELSEVIER Role of the Fyn −93A>G polymorphism (rs706895) in acute rejection after liver transplantation 2015transfer abstract Amsterdam [u.a.] (DE-627)ELV018422527 volume:252 year:2013 day:1 month:06 pages:34-45 extent:12 https://doi.org/10.1016/j.physd.2013.02.009 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA AR 252 2013 1 0601 34-45 12 045F 530 |
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10.1016/j.physd.2013.02.009 doi GBVA2013005000030.pica (DE-627)ELV038649314 (ELSEVIER)S0167-2789(13)00058-4 DE-627 ger DE-627 rakwb eng 530 530 DE-600 610 VZ 570 VZ BIODIV DE-30 fid Sapsis, Themistoklis P. verfasserin aut A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems 2013transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. Quasilinear Gaussian closure Elsevier Modeling of nonlinear energy fluxes Elsevier Uncertainty quantification Elsevier Turbulent systems Elsevier Majda, Andrew J. oth Enthalten in Elsevier Thude, Hansjörg ELSEVIER Role of the Fyn −93A>G polymorphism (rs706895) in acute rejection after liver transplantation 2015transfer abstract Amsterdam [u.a.] (DE-627)ELV018422527 volume:252 year:2013 day:1 month:06 pages:34-45 extent:12 https://doi.org/10.1016/j.physd.2013.02.009 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA AR 252 2013 1 0601 34-45 12 045F 530 |
| allfieldsGer |
10.1016/j.physd.2013.02.009 doi GBVA2013005000030.pica (DE-627)ELV038649314 (ELSEVIER)S0167-2789(13)00058-4 DE-627 ger DE-627 rakwb eng 530 530 DE-600 610 VZ 570 VZ BIODIV DE-30 fid Sapsis, Themistoklis P. verfasserin aut A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems 2013transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. Quasilinear Gaussian closure Elsevier Modeling of nonlinear energy fluxes Elsevier Uncertainty quantification Elsevier Turbulent systems Elsevier Majda, Andrew J. oth Enthalten in Elsevier Thude, Hansjörg ELSEVIER Role of the Fyn −93A>G polymorphism (rs706895) in acute rejection after liver transplantation 2015transfer abstract Amsterdam [u.a.] (DE-627)ELV018422527 volume:252 year:2013 day:1 month:06 pages:34-45 extent:12 https://doi.org/10.1016/j.physd.2013.02.009 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA AR 252 2013 1 0601 34-45 12 045F 530 |
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10.1016/j.physd.2013.02.009 doi GBVA2013005000030.pica (DE-627)ELV038649314 (ELSEVIER)S0167-2789(13)00058-4 DE-627 ger DE-627 rakwb eng 530 530 DE-600 610 VZ 570 VZ BIODIV DE-30 fid Sapsis, Themistoklis P. verfasserin aut A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems 2013transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. Quasilinear Gaussian closure Elsevier Modeling of nonlinear energy fluxes Elsevier Uncertainty quantification Elsevier Turbulent systems Elsevier Majda, Andrew J. oth Enthalten in Elsevier Thude, Hansjörg ELSEVIER Role of the Fyn −93A>G polymorphism (rs706895) in acute rejection after liver transplantation 2015transfer abstract Amsterdam [u.a.] (DE-627)ELV018422527 volume:252 year:2013 day:1 month:06 pages:34-45 extent:12 https://doi.org/10.1016/j.physd.2013.02.009 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA AR 252 2013 1 0601 34-45 12 045F 530 |
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Role of the Fyn −93A>G polymorphism (rs706895) in acute rejection after liver transplantation |
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a statistically accurate modified quasilinear gaussian closure for uncertainty quantification in turbulent dynamical systems |
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A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems |
| abstract |
We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. |
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We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. |
| abstract_unstemmed |
We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation. |
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