Galerkin proper orthogonal decomposition methods for parabolic problems

By Karl Kunisch, Stefan Volkwein. Numerische Mathematik, 2001

707 citations recorded by the source corpus.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/83d70c2a480b1797a7a910fa7f4a9d6be5a33e45

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Citations in the corpus

Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “One common method is the proper orthogonal decomposition (POD) [191, 21, 175, 120, 119], which uses so-called “snapshots”—state vectors of the high-ﬁdelity model at selected inputs—to construct a basis for the low-dimensional subspace.”
Advances in Studies and Applications of Centroidal Voronoi Tessellations (2010, methodology): “In this sense, we are faced with a data compression problem, namely, how do we extract a very low-dimensional basis set that captures the information contained in a snapshot set that may have cardinality in the hundreds or even thousands? One popular approach for this purpose is the proper orthogona”
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.”
Finite element formulation based on proper orthogonal decomposition for parabolic equations (2009, methodology): “Since the matrix K is positive semi-definite and has rank l, the solution of (3.3) and (3.4) can be found, in addition, the following result holds (for example, see [11, 16 , 17]).”
A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics (2015, methodology): “We may consider classical POD [7]; POD is, relative to weak greedy, more readily implemented, more optimal, and also considerably less efficient in the offline stage.”
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.”
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).”
Nonlinear Model Reduction via Discrete Empirical Interpolation (2010, methodology): “The separate POD basis used in DEIM to approximate the nonlinearity is very closely related Kunish-Volkwein's inclusion of finite difference snapshots [49]1.”
Stabilized reduced basis method for parametrized advection-diffusion PDEs (2014, methodology): “We show now an effective procedure to compute the orthonormal basis {χm | 1 ≤ m ≤M} [18, 32].”
Convergence Rates of the POD–Greedy Method (2013, background): “Important fields of applicatio n of the POD are parabolic partial differential equations [14] or problems in fluid dynami cs [8].”
Physically based modeling in catchment hydrology at 50: Survey and outlook (2015, methodology): “Recent research in this field has been spurred by advances in proper orthogonal decomposition and reduced basis finite element methods [Kunisch and Volkwein, 2001; Vermeulen et al., 2004; Grepl and Patera, 2005; Bui-Thanh et al., 2008; Quarteroni et al., 2011; Manzoni, 2014].”
Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control (2005, methodology): “Throughout the remainder of this section we let X denote either the space V or H. 270 Michael Hinze and Stefan Volkwein Remark 10.2.1 (compare [ KV01 , Remark 1]).”
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues (2021, neutral): “102,159 Moreover, instead of collecting snapshot data at arbitrary time intervals and/or parameter values, more effective sampling techniques should be pursued.”