Missing Point Estimation in Models Described by Proper Orthogonal Decomposition

By Patricia Astrid, Siep Weiland, Karen E. Willcox, Ton Backx. IEEE Transactions on Automatic Control, 2008

453 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/100ae81b383b379af72a1b3ec8b2c0e569075398

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Citations in the corpus

Nonlinear Model Reduction via Discrete Empirical Interpolation (2010, methodology): “This procedure can be viewed as performing the Galerkin projection onto the truncated POD basis via a specially constructed inner product as defined in [5] which evaluates only at selected grid points instead of computing the usual L2 inner product.”
The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows (2012, unknown): “[29, 20] propose a collocation of the equations followed by a Galerkin projection.”
Proper orthogonal decomposition for optimality systems (2008, background): “A difficulty related to POD-based model reduction may result from the evaluation of the nonlinearity on the POD subspace [3].”
Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations (2011, methodology): “It has also been used for steady [41, 42] and unsteady [43, 44] ﬂow ﬁeld reconstruction, inverse design [42], design variable mapping for multi-ﬁdelity optimization [45], and to some extent, for non-linear computations [26, 46].”
High-performance model reduction techniques in computational multiscale homogenization (2014, background): “…— computational pitfalls, to a wide gamut of problems: nonlinear monotonic elliptic and non-linear parabolic BPVs [33], nonlinear miscible viscous ﬁngering in porous media [17,18], uncertainty quanti-ﬁcation in inverse problems [31], and nonlinear heat conduction problems [5,6], to cite but a few.”
Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media (2011, methodology): “In [21], MPE was used with a greedy algorithm [32] and a sequential QR decomposition approach to improve the choice of selected rows in POD vectors and a clustering technique was applied to optimize snapshots for POD.”
Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency (2014, methodology): “One such approach performs the reconstruction in the least-squares sense using the same ROB used to represent the state vector [22].”
A New Selection Operator for the Discrete Empirical Interpolation Method - Improved A Priori Error Bound and Extensions (2015, background): “Various approaches exist to tackle this issue; see, e.g., [21, 5, 3, 39, 14].”
Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control (2013, methodology): “An error analysis for nonlinear dynamical systems in finite dimensions were carried out in [60] and a missing point estimation in models described by POD was studied in [4].”
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 to linear and nonlinear systems, see, for example, [42]–[46], where theoretical issues, error bounds and applications have been discussed.”
Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective (2014, methodology): “There are some techniques, which provide efﬁcient estimates of f ( V z ( t )) as the (discrete) empirical interpolation methods (DEIM or EIM) [21,57], the missing point estimation method [8], or a best points interpolation method [147].”
Nonlinear model order reduction based on local reduced‐order bases (2012, methodology): “Examples include the hyper reduction method proposed in [24] and the missing point estimation procedure proposed in [25].”
POD-DEIM model order reduction for strain softening viscoplasticity (2017, methodology): “To overcome this inefficiency, hyper-reduction methods, such as Missing Point Estimation [10], Gappy Proper Orthogonal Decomposition [11] and Discrete Empirical Interpolation Method (DEIM) [5], have been proposed to approximate the nonlinear contributions.”
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues (2021, unknown): “63,216,217 Examples of hyperreduction include the empirical interpolation method (EIM), 86 its discrete version (DEIM), 90,218 the gappy POD, 100,111,219 and the missing point estimation (MPE), 108,220 where the approximation is performed using sampling techniques.”
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems (2015, unknown): “Among this class of methods, the missing point estimation [19] and Gauss Newton with approximated tensors (GNAT) [66] methods both build upon the gappy POD interpolation method [87]; the empirical interpolation method (EIM) [26] and its discrete variant, the discrete empirical interpolation method (”
Structure‐preserving, stability, and accuracy properties of the energy‐conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models (2015, background): “Related or similar approaches have been developed specifically for hyper reduction at the semidiscrete [22, 23] and fully discrete [10, 11, 24, 25] levels.”