swMATH ID: 12977
Software Authors: Quarteroni, Alfio; Manzoni, Andrea; Negri, Federico
Description: Reduced basis methods for partial differential equations. An introduction. This book provides a basic introduction to reduced basis (RB) methods for problems involving the repeated solution of partial differential equations (PDEs) arising from engineering and applied sciences, such as PDEs depending on several parameters and PDE-constrained optimization. The book presents a general mathematical formulation of RB methods, analyzes their fundamental theoretical properties, discusses the related algorithmic and implementation aspects, and highlights their built-in algebraic and geometric structures. More specifically, the authors discuss alternative strategies for constructing accurate RB spaces using greedy algorithms and proper orthogonal decomposition techniques, investigate their approximation properties and analyze offline-online decomposition strategies aimed at the reduction of computational complexity. Furthermore, they carry out both a priori and a posteriori error analysis. Reduced basis methods for partial differential equations. An introduction. The whole mathematical presentation is made more stimulating by the use of representative examples of applicative interest in the context of both linear and nonlinear PDEs. Moreover, the inclusion of many pseudocodes allows the reader to easily implement the algorithms illustrated throughout the text. The book will be ideal for upper undergraduate students and, more generally, people interested in scientific computing. All these pseudocodes are in fact implemented in a MATLAB package that is freely available at https://github.com/redbkit.
Homepage: https://github.com/redbkit
Source Code:  https://github.com/redbkit
Dependencies: Matlab
Related Software: rbMIT; Adam; FEniCS; GitHub; Matlab; Loewner; FreeFem++; pyMOR; TensorFlow; DeepONet; PyTorch; PMTK; EnKF; Gmsh; OpenFOAM; NGSolve; ImageNet; AAA; CutFEM; SyFi
Cited in: 350 Documents
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Cited by 601 Authors

38 Rozza, Gianluigi
23 Manzoni, Andrea
20 Quarteroni, Alfio M.
19 Hesthaven, Jan S.
19 Stabile, Giovanni
16 Iliescu, Traian
13 Luo, Zhendong
13 Taddei, Tommaso
9 Ballarin, Francesco
9 Benner, Peter
8 Deparis, Simone
8 Gugercin, Serkan
8 Schwab, Christoph
8 Urban, Karsten
7 Chen, Yanlai
7 Dal Santo, Niccolò
7 Karatzas, Efthymios N.
7 Maday, Yvon
7 Patera, Anthony T.
6 Chen, Peng
6 Guo, Mengwu
6 Koc, Birgul
6 Mola, Andrea
6 Mou, Changhong
6 Ohlberger, Mario
6 Pagani, Stefano
6 Rave, Stephan David
6 Veroy, Karen
5 Chacón Rebollo, Tomás
5 Smetana, Kathrin
5 Wang, Zhu
4 Dede’, Luca
4 Fresca, Stefania
4 Ghattas, Omar N.
4 Girfoglio, Michele
4 Hess, Martin Wilfried
4 Narayan, Akil C.
4 Perotto, Simona
4 Singler, John R.
4 Zhang, Lei
3 Afkham, Babak Maboudi
3 Ali, Mazen
3 Alla, Alessandro
3 Antil, Harbir
3 Brunton, Steven L.
3 Ehrlacher, Virginie
3 Feng, Lihong
3 Gao, Zhen
3 Gómez Mármol, Macarena
3 Grepl, Martin A.
3 Gunzburger, Max D.
3 Héas, Patrick
3 Herzet, Cédric
3 Hijazi, Saddam
3 Ivagnes, Anna
3 Koutsourelakis, Phaedon-Stelios
3 Liao, Qifeng
3 Lorenzi, Stefano
3 Mishra, Siddhartha
3 Mlinarić, Petar
3 Mohebujjaman, Muhammad
3 Nobile, Fabio
3 Pagliantini, Cecilia
3 Pichi, Federico
3 Quaini, Annalisa
3 Rebholz, Leo G.
3 Regazzoni, Francesco
3 Ripamonti, Nicolò
3 Rubino, Samuele
3 Schindler, Felix
3 Schneier, Michael
3 Scovazzi, Guglielmo
3 Sun, Xiang
3 Teng, Fei
3 Unger, Benjamin
3 Wang, Qian
3 Wells, David R.
3 Xie, Xuping
3 Yano, Masayuki
3 Zhang, Zhenying
2 Antolin, Pablo
2 Azaïez, Mejdi
2 Beattie, Christopher A.
2 Benaceur, Amina
2 Bertrand, Fleurianne
2 Bluck, Michael J.
2 Boffi, Daniele
2 Borzacchiello, Domenico
2 Buffa, Annalisa
2 Buhr, Andreas
2 Cammi, Antonio
2 Chellappa, Sridhar
2 Chen, Tianheng
2 Cicci, Ludovica
2 Cohen, Albert
2 Conti, Paolo
2 Delgado Ávila, Enrique
2 Demo, Nicola
2 Desmet, Wim
2 Elman, Howard C.
...and 501 more Authors
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Cited in 68 Serials

51 Computer Methods in Applied Mechanics and Engineering
42 SIAM Journal on Scientific Computing
41 Journal of Computational Physics
29 Journal of Scientific Computing
18 Advances in Computational Mathematics
15 Computers & Mathematics with Applications
15 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis
11 SIAM/ASA Journal on Uncertainty Quantification
10 SIAM Journal on Numerical Analysis
7 Computers and Fluids
5 Journal of Mathematical Analysis and Applications
4 Journal of Computational and Applied Mathematics
4 Computational Mechanics
4 Computational Geosciences
4 Communications in Computational Physics
3 Applied Mathematics and Computation
3 Applied Numerical Mathematics
3 International Journal of Computational Fluid Dynamics
3 Archives of Computational Methods in Engineering
3 Acta Numerica
3 Communications in Applied and Industrial Mathematics
2 Classical and Quantum Gravity
2 Numerische Mathematik
2 Physica D
2 Applied Mathematics Letters
2 SIAM Journal on Matrix Analysis and Applications
2 Applied Mathematical Modelling
2 SIAM Journal on Mathematical Analysis
2 ETNA. Electronic Transactions on Numerical Analysis
2 Vietnam Journal of Mathematics
2 Comptes Rendus. Mathématique. Académie des Sciences, Paris
2 Advances in Difference Equations
2 Mathematical Control and Related Fields
2 SMAI Journal of Computational Mathematics
1 Acta Mechanica
1 Wave Motion
1 Mathematics of Computation
1 Integral Equations and Operator Theory
1 SIAM Journal on Control and Optimization
1 Zeitschrift für Analysis und ihre Anwendungen
1 Bulletin of the Iranian Mathematical Society
1 Acta Mathematicae Applicatae Sinica. English Series
1 Optimization
1 Constructive Approximation
1 European Journal of Applied Mathematics
1 M\(^3\)AS. Mathematical Models & Methods in Applied Sciences
1 SIAM Journal on Applied Mathematics
1 Journal of Nonlinear Science
1 Continuum Mechanics and Thermodynamics
1 Computational and Applied Mathematics
1 Journal of Inverse and Ill-Posed Problems
1 Computing and Visualization in Science
1 Chaos
1 Discrete Dynamics in Nature and Society
1 European Journal of Mechanics. A. Solids
1 Journal of Numerical Mathematics
1 SIAM Journal on Applied Dynamical Systems
1 Multiscale Modeling & Simulation
1 Boundary Value Problems
1 Inverse Problems in Science and Engineering
1 Unitext
1 Journal of Nonlinear Science and Applications
1 SIAM Journal on Financial Mathematics
1 GEM - International Journal on Geomathematics
1 Stochastic and Partial Differential Equations. Analysis and Computations
1 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings and Surveys
1 Mathematics in Engineering
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Cited in 36 Fields

289 Numerical analysis (65-XX)
135 Partial differential equations (35-XX)
103 Fluid mechanics (76-XX)
47 Computer science (68-XX)
26 Statistics (62-XX)
26 Mechanics of deformable solids (74-XX)
21 Calculus of variations and optimal control; optimization (49-XX)
21 Systems theory; control (93-XX)
19 Approximations and expansions (41-XX)
17 Biology and other natural sciences (92-XX)
15 Probability theory and stochastic processes (60-XX)
14 Dynamical systems and ergodic theory (37-XX)
13 Optics, electromagnetic theory (78-XX)
11 Linear and multilinear algebra; matrix theory (15-XX)
9 Classical thermodynamics, heat transfer (80-XX)
9 Operations research, mathematical programming (90-XX)
6 Functional analysis (46-XX)
6 Geophysics (86-XX)
5 Real functions (26-XX)
3 Ordinary differential equations (34-XX)
3 Operator theory (47-XX)
3 Statistical mechanics, structure of matter (82-XX)
2 Functions of a complex variable (30-XX)
2 Potential theory (31-XX)
2 Several complex variables and analytic spaces (32-XX)
2 Integral equations (45-XX)
2 Mechanics of particles and systems (70-XX)
2 Quantum theory (81-XX)
2 Relativity and gravitational theory (83-XX)
2 Game theory, economics, finance, and other social and behavioral sciences (91-XX)
2 Mathematics education (97-XX)
1 Measure and integration (28-XX)
1 Special functions (33-XX)
1 Harmonic analysis on Euclidean spaces (42-XX)
1 Algebraic topology (55-XX)
1 Global analysis, analysis on manifolds (58-XX)

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