Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models

By Francisco Chinesta, Amine Ammar, Elías Cueto. 2010

410 citations recorded by the source corpus.

This paper revisits a powerful discretization technique, the Proper Generalized Decomposition—PGD, illustrating its ability for solving highly multidimensional models. This technique operates by constructing a separated representation of the solution, such that the solution complexity scales linearly with the dimension of the space in which the model is defined, instead the exponentially-growing complexity characteristic of mesh based discretization strategies. The PGD makes possible the efficient solution of models defined in multidimensional spaces, as the ones encountered in quantum chemistry, kinetic theory description of complex fluids, genetics (chemical master equation), financial mathematics, … but also those, classically defined in the standard space and time, to which we can add new extra-coordinates (parametric models, …) opening numerous possibilities (optimization, inverse identification, real time simulations, …).

Subfields: Numerical Analysis, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/103d1fc7af6fb91f913c55e5e090f1248690c260

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Citations in the corpus

A complete dynamic approach to the Generalized Beam Theory cross-section analysis including extension and shear modes (2014, methodology): “In this sense it presents strong formal similarities with the technique of the Proper Generalized Decomposition, which is proven to be very suitable for solving highly multi-dimensional models [30], although, at the present state of research, the interest of the GBT formulation has not yet specifica”
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.”
PGD-Based Computational Vademecum for Efficient Design, Optimization and Control (2013, background): “There are other constructors of the separated representation more efﬁcient when applying to non-symmetric differential operators, as the one based on the residual minimization, among many others [24].”
Parametric solutions involving geometry: A step towards efficient shape optimization (2014, background): “The interested reader can refer to [10, 11] and the references therein for a complete review of PGD techniques.”