Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems

By Anthony Nouy. 2010

208 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…Hartree-Fock or DFT models [23,24,31,44,45,47,87,141–146,150,156–159]; • the solution of stochastic and parametric PDEs [68,69,76,77,161,165,171,189,202]; • approximation of Green’s functions of high-dimensional PDEs [30,113]; • the solution of the Boltzmann equation [129, 149], and the chemical…”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “We refer to the review papers [46, 49], the literature [6, 45, 50, 77, 153, 154] and the references therein for details on PGD and its applications.”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig on Minimal Subspaces in Tensor Representations on Minimal Subspaces in Tensor Representations (unknown): “…solution of problems defined in high-dimensional tensor spaces, such as PDEs arising in stochastic calculus [1] [4] [9] (e.g., Fokker-Planck equation), stochastic parametric PDEs arising in uncertainty quantification with spectral approaches [18] [8] [19], and quantum chemistry (cf., e.g., [25]).”
A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations (2010, methodology): “…separation of physical variables and parameters (or random variables) in the context of parametrized (or stochastic ) PDEs [27] [28] [32] [29] 3 , with a possible additional separation of parameters [11] [30] 4 , separation of coordinate variables in multi-dimensional PDEs [6] [1] [2] [8] 5 .”
Computational Reduction for Parametrized PDEs: Strategies and Applications (2012, methodology): “Two additional techniques – indeed quite close to POD – for generating reduced spaces are the Centroidal Voronoi Tessellation [7–9] and the Proper Generalized Decomposition [10,11,16,36].”
A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach (2011, background): “Still in the context of stochastic PDEs, a further separation of parameters have also been introduced, by exploiting the tensor product structure of stochastic function spaces [9,22].”
Low-rank methods for high-dimensional approximation and model order reduction (2015, unknown): “nt model, so that parameter-independent equations have the structure of standard problems which can be solved with available solution codes for parameter-independent models. The reader is referred to [63, 64, 66, 20] for practical implementations and illustrations of the behavior of these algorithms”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, result): “Similar representations, although with different solution strategies, have been later investigated in [60, 41, 28].”
PGD-Based Computational Vademecum for Efficient Design, Optimization and Control (2013, methodology): “Later, separated representations were employed in the context of stochastic modelling [64] as well as for solving multidimensional models suffering the so-called curse of dimensionality, some of them never solved before [1].”
A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties (2013, methodology): “Here the classical canonical decompositions and the hierarchical canonical decompositions as in [26, 27] are introduced, the latter ones giving very low ranks representations.”
Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, unknown): “Another family of methods, called Proper Generalized Decomposition (PGD) methods [9,16, 27,33,34], have been introduced for the direct construction of representations of type (1).”
Tensor-based methods for numerical homogenization from high-resolution images (2013, methodology): “Note that the proposed adaptive strategy could also be used in other context for goal-oriented approximation of PDEs in tensor formats.”