redbKIT 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; GitHub; Adam; FEniCS; Loewner; FreeFem++; pyMOR; Matlab; PMTK; NGSolve; TensorFlow; hapod; SINDy; PRMLT; ITHACA-FV; SyFi; LifeV; RBniCS; libMesh; Gmsh Cited in: 272 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Reduced basis methods for partial differential equations. An introduction. Zbl 1337.65113Quarteroni, Alfio; Manzoni, Andrea; Negri, Federico 2016 all top 5 Cited by 460 Authors 27 Rozza, Gianluigi 19 Quarteroni, Alfio M. 17 Hesthaven, Jan S. 17 Manzoni, Andrea 15 Stabile, Giovanni 12 Luo, Zhendong 10 Iliescu, Traian 10 Taddei, Tommaso 8 Urban, Karsten 7 Ballarin, Francesco 7 Benner, Peter 7 Dal Santo, Niccolò 7 Deparis, Simone 7 Patera, Anthony T. 7 Schwab, Christoph 6 Chen, Yanlai 6 Karatzas, Efthymios N. 6 Maday, Yvon 6 Pagani, Stefano 5 Chen, Peng 5 Gugercin, Serkan 5 Guo, Mengwu 5 Ohlberger, Mario 4 Dede’, Luca 4 Mola, Andrea 4 Mou, Changhong 4 Narayan, Akil C. 4 Rave, Stephan David 4 Smetana, Kathrin 4 Veroy, Karen 4 Wang, Zhu 4 Zhang, Lei 3 Afkham, Babak Maboudi 3 Ali, Mazen 3 Alla, Alessandro 3 Chacón Rebollo, Tomás 3 Gómez Mármol, Macarena 3 Gunzburger, Max D. 3 Héas, Patrick 3 Herzet, Cédric 3 Hijazi, Saddam 3 Koc, Birgul 3 Koutsourelakis, Phaedon-Stelios 3 Mishra, Siddhartha 3 Mohebujjaman, Muhammad 3 Nobile, Fabio 3 Perotto, Simona 3 Quaini, Annalisa 3 Regazzoni, Francesco 3 Ripamonti, Nicolò 3 Schindler, Felix 3 Schneier, Michael 3 Scovazzi, Guglielmo 3 Singler, John R. 3 Teng, Fei 3 Wang, Qian 3 Wells, David R. 3 Xie, Xuping 3 Yano, Masayuki 3 Zhang, Zhenying 2 Antil, Harbir 2 Beattie, Christopher A. 2 Benaceur, Amina 2 Bluck, Michael J. 2 Brunton, Steven L. 2 Buhr, Andreas 2 Chen, Tianheng 2 Cohen, Albert 2 Delgado Ávila, Enrique 2 Ehrlacher, Virginie 2 Elman, Howard C. 2 Farthing, Matthew W. 2 Feng, Xinlong 2 Gao, Zhen 2 Georgaka, Sokratia 2 Ghattas, Omar N. 2 Girfoglio, Michele 2 Gosea, Ion Victor 2 Greif, Constantin 2 Grepl, Martin A. 2 Grigo, Constantin 2 Heinkenschloss, Matthias 2 Hess, Martin Wilfried 2 Jiang, Lijian 2 Jiang, Wenrui 2 Kamilis, Dimitris 2 Kärcher, Mark 2 Khurshudyan, Asatur Zh. 2 Kressner, Daniel 2 Kutyniok, Gitta 2 Kutz, J. Nathan 2 Le Clainche, Soledad 2 Li, Qiuqi 2 Li, Richen 2 Liao, Qifeng 2 Longo, Marcello 2 Lorenzi, Stefano 2 Ma, Lingling 2 Nouveau, Léo 2 Nouy, Anthony ...and 360 more Authors all top 5 Cited in 56 Serials 38 Computer Methods in Applied Mechanics and Engineering 37 SIAM Journal on Scientific Computing 32 Journal of Computational Physics 17 Journal of Scientific Computing 16 Advances in Computational Mathematics 14 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 11 Computers & Mathematics with Applications 11 SIAM/ASA Journal on Uncertainty Quantification 9 SIAM Journal on Numerical Analysis 5 Journal of Mathematical Analysis and Applications 4 Computers and Fluids 4 Journal of Computational and Applied Mathematics 4 Communications in Computational Physics 3 Applied Numerical Mathematics 3 Computational Geosciences 3 Archives of Computational Methods in Engineering 3 Communications in Applied and Industrial Mathematics 2 Computational Mechanics 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 International Journal of Computational Fluid Dynamics 2 Comptes Rendus. Mathématique. Académie des Sciences, Paris 2 Advances in Difference Equations 2 SMAI Journal of Computational Mathematics 1 Acta Mechanica 1 Classical and Quantum Gravity 1 Wave Motion 1 Mathematics of Computation 1 Integral Equations and Operator Theory 1 Numerische Mathematik 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 Constructive Approximation 1 COMPEL 1 Journal of Nonlinear Science 1 Continuum Mechanics and Thermodynamics 1 Journal of Inverse and Ill-Posed Problems 1 Vietnam Journal of Mathematics 1 Chaos 1 Discrete Dynamics in Nature and Society 1 Journal of Numerical Mathematics 1 SIAM Journal on Applied Dynamical Systems 1 Multiscale Modeling & Simulation 1 Acta Numerica 1 Inverse Problems in Science and Engineering 1 Unitext 1 Journal of Nonlinear Science and Applications 1 SIAM Journal on Financial Mathematics 1 Mathematical Control and Related Fields 1 Stochastic and Partial Differential Equations. Analysis and Computations 1 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings and Surveys all top 5 Cited in 33 Fields 227 Numerical analysis (65-XX) 105 Partial differential equations (35-XX) 81 Fluid mechanics (76-XX) 29 Computer science (68-XX) 21 Statistics (62-XX) 19 Mechanics of deformable solids (74-XX) 19 Systems theory; control (93-XX) 14 Approximations and expansions (41-XX) 13 Biology and other natural sciences (92-XX) 11 Calculus of variations and optimal control; optimization (49-XX) 10 Linear and multilinear algebra; matrix theory (15-XX) 10 Probability theory and stochastic processes (60-XX) 9 Dynamical systems and ergodic theory (37-XX) 8 Optics, electromagnetic theory (78-XX) 8 Operations research, mathematical programming (90-XX) 6 Classical thermodynamics, heat transfer (80-XX) 4 Functional analysis (46-XX) 3 Real functions (26-XX) 3 Statistical mechanics, structure of matter (82-XX) 3 Geophysics (86-XX) 2 Potential theory (31-XX) 2 Ordinary differential equations (34-XX) 2 Integral equations (45-XX) 2 Mechanics of particles and systems (70-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Mathematics education (97-XX) 1 Functions of a complex variable (30-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Operator theory (47-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Quantum theory (81-XX) 1 Relativity and gravitational theory (83-XX) 1 Information and communication theory, circuits (94-XX) Citations by Year