Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control

By Michael Hinze, Stefan Volkwein. 2005

315 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/dd20ddb995c8bcaa1910b8cd828ebae46bfb2bb8

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In Control and Systems Engineering

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Citations in the corpus

Nonlinear Model Reduction via Discrete Empirical Interpolation (2010, background): “However, effective dimension reduction for the POD-Galerkin approach is usually limited to problems with linear or bilinear terms as in [20, 22, 18, 33].”
An efficient reduced‐order modeling approach for non‐linear parametrized partial differential equations (2008, background): “Nevertheless, its success and thus applications remain very much limited to linear and quadratically non-linear problems such as linearized Euler and parabolic equations [12, 13, 23], Burgers equation [10, 24], and incompressible Navier– Stokes equations [11, 25, 26].”
Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples (2012, methodology): “The reduced basis [20, 21, 23, 24, 27] and the proper orthogonal decomposition (POD) [12,15, 17, 19, 28] are two techniques belonging to this second family.”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “p a posteriori error estimators for the reduced model outputs [107, 86, 92, 182, 206, 215]. Ecient error estimators can also sometimes be provided for other basis generation methods, such as the POD [102, 103]. Data-t low-delity models. Data-t low-delity models are derived directly from inputs and t”
A method for interpolating on manifolds structural dynamics reduced‐order models (2009, methodology): “They are back in vogue for incorporating computational models in design operations [7], designing effective control systems for large-scale flexible structures [8], generating surrogate models for accelerating the speed of optimization procedures [9] and developing realistic uncertainty quantificati”
Model Reduction via Proper Orthogonal Decomposition (2008, methodology): “Instead of solving the eigensystem for the matrixK ∈ Rn×n one only needs to consider a matrix in Rm×m, where m is the number of snapshots considered (for a more detailed discussion we refer to [12, 17] and the references therein).”
Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space (2008, methodology): “Work has been done to address the first issue using adaptive sampling methods to create reduced models for optimal control applications [1, 18, 2, 21].”
Interpolation among reduced‐order matrices to obtain parameterized models for design, optimization and probabilistic analysis (2009, methodology): “Several approaches have been proposed for deriving a basis that spans the parameter space, including POD and Krylov-based sampling methods [10–12], reduced basis methods combined with the use of error estimates and adaptivity [13], and optimization-based approaches [14].”
Generalized multiscale finite element methods (GMsFEM) (2013, unknown): “iques have been developed. The main idea of these methods is to ﬁnd a small dimensional space that can represent the solution space given the input space. Global model reduction techniques (see e.g., [53, 37, 18, 41, 6, 55]) construct a space of global ﬁelds that can approximate the solution space. ”
A New Selection Operator for the Discrete Empirical Interpolation Method - Improved A Priori Error Bound and Extensions (2015, background): “For more details on POD, we refer the reader to [28, 32].”
Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control (2013, methodology): “Proper orthogonal decomposition, model-order reduction, apriori and a-posteriori error analysis, linear-quadratic optimal control, primal-dual active set strategy.”
Hessian‐based model reduction for large‐scale systems with initial‐condition inputs (2008, methodology): “Progress has been made in development and application of these methods to optimization applications with a small number of input parameters, for example optimal control [1,5,23,25] and parametrized design of interconnect circuits [12].”
REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS (2011, background): “The reduced order model can often capture the system behavior accurately; examples from ﬂuid dynamics include [10, 12, 13, 24, 27, 31–33 , 35].”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, methodology): “For optimal control applications, online adaptive sampling has been employed as a systematic way to generate snapshot information (Afanasiev and Hi ze 2001; Fahl and Sachs 2003; Hinze and Volkwein 2005; Kunisch and Volkwein 1999).”
Model Reduction by Moment Matching for Linear and Nonlinear Systems (2010, methodology): “Finally, proper orthogonal decomposition has been proposed as a model reduction method applicable for linear and nonlinear systems, see, for example, [43], [44], [42], [45], [46], where theoretical issues, error bounds and applications have been discussed.”