Reduced Basis Methods for Partial Differential Equations: An Introduction

By Alfio Maria Quarteroni, Andrea Manzoni, Federico Negri. 2015

1,008 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/7e4dccef7acbfb881e9f01de975467e4aaf9a473

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In Computational Mechanics

In Electrical and Electronic Engineering

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Citations in the corpus

Geometric Deep Learning for Volumetric Computational Fluid Dynamics (2019, methodology): “For each geometry we apply an input velocity U ∈ [10,100] with angle α in [-π8 , π 8 ] with respect to the horizontal, resulting in two boundary conditions Ux = U · cos(α) and Uy = −Usin(α).”
Model Reduction and Neural Networks for Parametric PDEs (2020, methodology): “This work led to the development of the reduced basis method (RBM) which is widely adopted in engineering; see [3, 26, 43] and the references therein.”
POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition (2021, methodology): “Replacing a full order model (FOM) by a reduced order model (ROM), featuring a much lower dimension, yet capable to express the physical features of the problem at hand, is the main goal of reduced order modeling techniques, among which the reduced basis (RB) method represents one of the most popula”
A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regime (2019, background): “(2015); Afkham and Hesthaven (2017) or regression-based strategies Yu and Hesthaven (2018); Guo and Hesthaven (2018); Hesthaven and Ubbiali (2018).”
A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs (2020, methodology): “POD is one of the most widely employed techniques to generate the linear trial manifold [48].”
Data-driven reduced order modeling for time-dependent problems (2019, methodology): “The Schmidt-Eckart-Young theorem [12, 28, 32] states that the basis consisting of the first L left singular vectors of S minimizes the projection error of the snapshots among all L-dimensional orthogonal bases in R Nh .”
Decay of the Kolmogorov $N$-width for wave problems (2019, neutral): “For more details on the reduced basis method, we refer the reader e.g. to the recent surveys [4, 5, 10].”
Multipole Graph Neural Operator for Parametric Partial Differential Equations (2020, methodology): “• RBM is the reduced basis method [13], a classical reduced order modeling technique that is ubiquitous in applications [32] .”
A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications (2017, methodology): “One of challenges in reduced order modelling is to generate arobust ROM for different parameters/inputs, which can represent the physical dynamics of parameterized partial differential equations (P-PDEs) as the model parameters vary.”