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

References in zbMATH (referenced in 227 articles , 1 standard article )

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  1. Feng, Xiaobing; Luo, Yan; Vo, Liet; Wang, Zhu: An efficient iterative method for solving parameter-dependent and random convection-diffusion problems (2022)
  2. Guo, Mengwu; Manzoni, Andrea; Amendt, Maurice; Conti, Paolo; Hesthaven, Jan S.: Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantities (2022)
  3. Héas, Patrick; Herzet, Cédric: Low-rank dynamic mode decomposition: an exact and tractable solution (2022)
  4. Benaceur, Amina: Reducing sensors for transient heat transfer problems by means of variational data assimilation (2021)
  5. Benner, Peter; Goyal, Pawan: Interpolation-based model order reduction for polynomial systems (2021)
  6. Berrone, Stefano; Vicini, Fabio: A reduced basis method for a PDE-constrained optimization formulation in discrete fracture network flow simulations (2021)
  7. Bhattacharya, Kaushik; Hosseini, Bamdad; Kovachki, Nikola B.; Stuart, Andrew M.: Model reduction and neural networks for parametric PDEs (2021)
  8. Carere, Giuseppe; Strazzullo, Maria; Ballarin, Francesco; Rozza, Gianluigi; Stevenson, Rob: A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences (2021)
  9. Carr, Arielle; de Sturler, Eric; Gugercin, Serkan: Preconditioning parametrized linear systems (2021)
  10. Chaudhry, Jehanzeb H.; Olson, Luke N.; Sentz, Peter: A least-squares finite element reduced basis method (2021)
  11. Chellappa, Sridhar; Feng, Lihong; Benner, Peter: A training set subsampling strategy for the reduced basis method (2021)
  12. Chen, Peng; Ghattas, Omar: Stein variational reduced basis Bayesian inversion (2021)
  13. Dal Santo, Niccolò; Manzoni, Andrea; Pagani, Stefano; Quarteroni, Alfio: Reduced-order modeling for applications to the cardiovascular system (2021)
  14. Danczul, Tobias; Schöberl, Joachim: A reduced basis method for fractional diffusion operators. II (2021)
  15. Discacciati, Niccolò; Hesthaven, Jan S.: Modeling synchronization in globally coupled oscillatory systems using model order reduction (2021)
  16. Dölz, Jürgen; Egger, Herbert; Schlottbom, Matthias: A model reduction approach for inverse problems with operator valued data (2021)
  17. Fresca, Stefania; Dede’, Luca; Manzoni, Andrea: A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs (2021)
  18. Gadalla, Mahmoud; Cianferra, Marta; Tezzele, Marco; Stabile, Giovanni; Mola, Andrea; Rozza, Gianluigi: On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis (2021)
  19. Gao, Zhen; Liu, Qi; Hesthaven, Jan S.; Wang, Bao-Shan; Don, Wai Sun; Wen, Xiao: Non-intrusive reduced order modeling of convection dominated flows using artificial neural networks with application to Rayleigh-Taylor instability (2021)
  20. Geist, Moritz; Petersen, Philipp; Raslan, Mones; Schneider, Reinhold; Kutyniok, Gitta: Numerical solution of the parametric diffusion equation by deep neural networks (2021)

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