Low-dimensional reduced-order models for statistical response and uncertainty quantification: Barotropic turbulence with topography

2017 ◽  
Vol 343 ◽  
pp. 7-27 ◽  
Author(s):  
Di Qi ◽  
Andrew J. Majda
Keyword(s):  
Uncertainty Quantification ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Low Dimensional

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

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).

scholarly journals GENETIC ALGORITHM-BASED CALIBRATION OF REDUCED ORDER GALERKIN MODELS

2011 ◽  
Vol 16 (1) ◽  
pp. 233-247 ◽  
Author(s):  
Witold Stankiewicz ◽  
Robert Roszaka ◽  
Marek Morzyńskia
Keyword(s):  
Genetic Algorithm ◽  
Bluff Body ◽  
Calibration Procedure ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Data Set ◽  
Reduced Order ◽  
Low Dimensional ◽  
Fluid Behaviour ◽  
Proper Orthogonal

Low-dimensional models, allowing quick prediction of fluid behaviour, are key enablers of closed-loop flow control. Reduction of the model's dimension and inconsistency of high-fidelity data set and the reduced-order formulation lead to the decrease of accuracy. The quality of Reduced-Order Models might be improved by a calibration procedure. It leads to global optimization problem which consist in minimizing objective function like the prediction error of the model. In this paper, Reduced-Order Models of an incompressible flow around a bluff body are constructed, basing on Galerkin Projection of governing equations onto a space spanned by the most dominant eigenmodes of the Proper Orthogonal Decomposition (POD). Calibration of such models is done by adding to Galerkin System some linear and quadratic terms, which coefficients are estimated using Genetic Algorithm.

scholarly journals Low-Dimensional Reduced-Order Models for Statistical Response and Uncertainty Quantification: Two-Layer Baroclinic Turbulence

2016 ◽  
Vol 73 (12) ◽  
pp. 4609-4639 ◽  
Author(s):  
Di Qi ◽  
Andrew J. Majda
Keyword(s):  
Uncertainty Quantification ◽  
Model Parameters ◽  
Mathematical Framework ◽  
Order Method ◽  
Model Errors ◽  
Reduced Order ◽  
Homogeneous Flow ◽  
Mean And Variance ◽  
The Mean ◽  
Low Dimensional

Abstract Accurate uncertainty quantification for the mean and variance about forced responses to general external perturbations in the climate system is an important subject in understanding Earth’s atmosphere and ocean in climate change science. A low-dimensional reduced-order method is developed for uncertainty quantification and capturing the statistical sensitivity in the principal model directions with largest variability and in various regimes in two-layer quasigeostrophic turbulence. Typical dynamical regimes tested here include the homogeneous flow in the high latitudes and the anisotropic meandering jets in the low latitudes and/or midlatitudes. The idea in the reduced-order method is from a self-consistent mathematical framework for general systems with quadratic nonlinearity, where crucial high-order statistics are approximated by a systematic model calibration procedure. Model efficiency is improved through additional damping and noise corrections to replace the expensive energy-conserving nonlinear interactions. Model errors due to the imperfect nonlinear approximation are corrected by tuning the model parameters using linear response theory with an information metric in a training phase before prediction. Here a statistical energy principle is adopted to introduce a global scaling factor in characterizing the higher-order moments in a consistent way to improve model sensitivity. The reduced-order model displays uniformly high prediction skill for the mean and variance response to general forcing for both homogeneous flow and anisotropic zonal jets in the first 102 dominant low-wavenumber modes, where only about 0.15% of the total spectral modes are resolved, compared with the full model resolution of 2562 horizontal modes.

scholarly journals Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Imran Akhtar ◽  
Jeff Borggaard ◽  
Alexander Hay
Keyword(s):  
Finite Difference ◽  
Standard Method ◽  
Elliptic Cylinder ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Shape Sensitivity ◽  
Reduced Order ◽  
Flow Models ◽  
Low Dimensional ◽  
Parameter Values

Reduced-order models have a number of practical engineering applications for unsteady flows that require either low-dimensional approximations for analysis and control or repeated simulation over a range of parameter values. The standard method for building reduced-order models uses the proper orthogonal decomposition (POD) and Galerkin projection. However, this standard method may be inaccurate when used “off-design” (at parameter values not used to generate the POD). This phenomena is exaggerated when parameter values describe the shape of the flow domain since slight changes in shape can have a significant influence on the flow field. In this paper, we investigate the use of POD sensitivity vectors to improve the accuracy and dynamical system properties of the reduced-order models to problems with shape parameters. To carry out this study, we consider flows past an elliptic cylinder with varying thickness ratios. Shape sensitivities (derivatives of flow variables with respect to thickness ratio) computed by finite-difference approximations are used to compute the POD sensitivity vectors. Numerical studies test the accuracy of the new bases to represent flow solutions over a range of parameter values.

scholarly journals Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators

2021 ◽  
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

Abstract Although 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 $10^3\times$ in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs.

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