redbKIT

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.


References in zbMATH (referenced in 208 articles , 1 standard article )

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  1. Benaceur, Amina: Reducing sensors for transient heat transfer problems by means of variational data assimilation (2021)
  2. Benner, Peter; Goyal, Pawan: Interpolation-based model order reduction for polynomial systems (2021)
  3. Berrone, Stefano; Vicini, Fabio: A reduced basis method for a PDE-constrained optimization formulation in discrete fracture network flow simulations (2021)
  4. Bhattacharya, Kaushik; Hosseini, Bamdad; Kovachki, Nikola B.; Stuart, Andrew M.: Model reduction and neural networks for parametric PDEs (2021)
  5. Carr, Arielle; de Sturler, Eric; Gugercin, Serkan: Preconditioning parametrized linear systems (2021)
  6. Chaudhry, Jehanzeb H.; Olson, Luke N.; Sentz, Peter: A least-squares finite element reduced basis method (2021)
  7. Chen, Peng; Ghattas, Omar: Stein variational reduced basis Bayesian inversion (2021)
  8. Dal Santo, Niccolò; Manzoni, Andrea; Pagani, Stefano; Quarteroni, Alfio: Reduced-order modeling for applications to the cardiovascular system (2021)
  9. Discacciati, Niccolò; Hesthaven, Jan S.: Modeling synchronization in globally coupled oscillatory systems using model order reduction (2021)
  10. Dölz, Jürgen; Egger, Herbert; Schlottbom, Matthias: A model reduction approach for inverse problems with operator valued data (2021)
  11. Fresca, Stefania; Dede’, Luca; Manzoni, Andrea: A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs (2021)
  12. Geist, Moritz; Petersen, Philipp; Raslan, Mones; Schneider, Reinhold; Kutyniok, Gitta: Numerical solution of the parametric diffusion equation by deep neural networks (2021)
  13. Gu, Haotian; Xin, Jack; Zhang, Zhiwen: Error estimates for a POD method for solving viscous G-equations in incompressible cellular flows (2021)
  14. Henríquez, Fernando; Schwab, Christoph: Shape holomorphy of the Calderón projector for the Laplacian in (\mathbbR^2) (2021)
  15. Hinze, Michael; Korolev, Denis: A space-time certified reduced basis method for quasilinear parabolic partial differential equations (2021)
  16. Karatzas, Efthymios N.; Rozza, Gianluigi: A reduced order model for a stable embedded boundary parametrized Cahn-Hilliard phase-field system based on cut finite elements (2021)
  17. Keil, Tim; Mechelli, Luca; Ohlberger, Mario; Schindler, Felix; Volkwein, Stefan: A non-conforming dual approach for adaptive trust-region reduced basis approximation of PDE-constrained parameter optimization (2021)
  18. Koc, Birgul; Rubino, Samuele; Schneier, Michael; Singler, John; Iliescu, Traian: On optimal pointwise in time error bounds and difference quotients for the proper orthogonal decomposition (2021)
  19. Lu, Chuan; Zhu, Xueyu: Bifidelity data-assisted neural networks in nonintrusive reduced-order modeling (2021)
  20. Lye, Kjetil O.; Mishra, Siddhartha; Ray, Deep; Chandrashekar, Praveen: Iterative surrogate model optimization (ISMO): an active learning algorithm for PDE constrained optimization with deep neural networks (2021)

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