Certified Reduced Basis Methods for Parametrized Partial Differential Equations

By Jan S. Hesthaven, Gianluigi Rozza, Benjamin Stamm. 2015

1,144 citations recorded by the source corpus.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a2adc8e3d635cf8d6242b2d0e58409d2bcf0309a

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In Computational Mechanics

In Electrical and Electronic Engineering

In Numerical Analysis

Citations in the corpus

A deep learning-based hybrid approach for the solution of multiphysics problems in electrosurgery (2019, methodology): “Another family of MOR techniques uses a reduced basis constructed by leveraging a greedy algorithm and a posteriori error indicators [17].”
A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regime (2019, positive): “reduced-basis (RB) methods Hesthaven et al. (2016); Quarteroni et al.”
Decay of the Kolmogorov $N$-width for wave problems (2019, neutral): “to the recent surveys [4, 5, 10].”
Non-intrusive reduced order modeling of nonlinear problems using neural networks (2018, methodology): “As a result, a greedy strategy is typically more effective and efficient than POD, as it enables the exploration of a wider region of the parameter space while entailing the computation of many fewer high-fidelity solutions [23].”
Giga-voxel computational morphogenesis for structural design (2017, methodology): “O.S. contributed with the original idea, method development, analytical studies and manuscript preparation.”
A localized reduced-order modeling approach for PDEs with bifurcating solutions (2018, background): “, [20, 39]) and proper orthogonal decomposition (see, e.”
Data-driven reduced order modeling for time-dependent problems (2019, methodology): “For time-dependent problems, required to capture the causality with the evolution over time, a combined version has been proposed as the POD-greedy approach [16, 18].”
The Future of Sensitivity Analysis: An essential discipline for systems modeling and policy support (2020, positive): “Example methods arising from ML that have been used in SA include Gaussian processes (Rasmussen, 2004; Yang et al., 2018), generalized polynomial chaos expansions (Sudret, 2008), reduced basis methods (Hesthaven et al., 2016) and ANNs (Beh et al., 2017).”
Projection-based model reduction: Formulations for physics-based machine learning (2019, background): “See for example [2, 3, 4, 5, 6, 7, 8] for summaries of state-of-the-art in projection-based parametric model reduction methods and applications.”
Digital Twin: Values, Challenges and Enablers From a Modeling Perspective (2019, neutral): “[248] J.”
POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder (2017, methodology): “In this method the governing equations, describing the phenomenon, are projected onto a low dimensional space called the reduced basis space [16, 30] that is optimally constructed starting from high fidelity simulations.”
Bayesian Model and Dimension Reduction for Uncertainty Propagation: Applications in Random Media (2017, background): “A prominent role in these efforts, at least in the context of PDE-based models, is held by reduced-basis techniques [56, 63, 29] which are based on the identification of a low-dimensional linear subspace in the solution vector space on which a Galerkin projection of the governing equations is attemp”
Fast uncertainty quantification of activation sequences in patient‐specific cardiac electrophysiology meeting clinical time constraints (2018, background): “A reasonable approximation is to ignore this fact or, in other words, to ignore the infimum in formula (21) and to select γ(t; S) = γ(t; S̄), where S̄ = E is the mean value for the random field.”