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 67 articles , 1 standard article )

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  1. Brunken, Julia; Smetana, Kathrin; Urban, Karsten: (Parametrized) first order transport equations: realization of optimally stable Petrov-Galerkin methods (2019)
  2. Luo, Zhendong; Teng, Fei; Xia, Hong: A reduced-order extrapolated Crank-Nicolson finite spectral element method based on POD for the 2D non-stationary Boussinesq equations (2019)
  3. Smetana, Kathrin; Zahm, Olivier; Patera, Anthony T.: Randomized residual-based error estimators for parametrized equations (2019)
  4. Zhou, Yanjie; Luo, Zhendong: An optimized Crank-Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation (2019)
  5. Benaceur, Amina; Ehrlacher, Virginie; Ern, Alexandre; Meunier, Sébastien: A progressive reduced basis/empirical interpolation method for nonlinear parabolic problems (2018)
  6. Benner, Peter; Khoromskaia, Venera; Khoromskij, Boris N.: Range-separated tensor format for many-particle modeling (2018)
  7. Bonizzoni, Francesca; Nobile, Fabio; Perugia, Ilaria: Convergence analysis of Padé approximations for Helmholtz frequency response problems (2018)
  8. Buhr, Andreas; Smetana, Kathrin: Randomized local model order reduction (2018)
  9. Chakir, Rachida; Dapogny, Charles; Japhet, Caroline; Maday, Yvon; Montavon, Jean-Baptiste; Pantz, Olivier; Patera, Anthony: Component mapping automation for parametric component reduced basis techniques (RB-COMPONENT) (2018)
  10. Cohen, Albert; Schwab, Christoph; Zech, Jakob: Shape holomorphy of the stationary Navier-Stokes equations (2018)
  11. Costa, Timothy B.; Kennedy, Kenneth; Peszynska, Malgorzata: Hybrid three-scale model for evolving pore-scale geometries (2018)
  12. Drmač, Zlatko; Saibaba, Arvind Krishna: The discrete empirical interpolation method: canonical structure and formulation in weighted inner product spaces (2018)
  13. Gallinari, Patrick; Maday, Yvon; Sangnier, Maxime; Schwander, Olivier; Taddei, Tommaso: Reduced basis’ acquisition by a learning process for rapid on-line approximation of solution to PDE’s: laminar flow past a backstep (2018)
  14. Héas, Patrick; Herzet, Cédric: Reduced modeling of unknown trajectories (2018)
  15. Heinkenschloss, Matthias; Jando, Dörte: Reduced order modeling for time-dependent optimization problems with initial value controls (2018)
  16. Hesthaven, J. S.; Ubbiali, S.: Non-intrusive reduced order modeling of nonlinear problems using neural networks (2018)
  17. Kamilis, Dimitris; Polydorides, Nick: Uncertainty quantification for low-frequency, time-harmonic Maxwell equations with stochastic conductivity models (2018)
  18. Kärcher, Mark; Tokoutsi, Zoi; Grepl, Martin A.; Veroy, Karen: Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls (2018)
  19. Kazemi, Seyed-Mohammad-Mahdi; Dehghan, Mehdi; Foroush Bastani, Ali: On a new family of radial basis functions: mathematical analysis and applications to option pricing (2018)
  20. Lukassen, Axel Ariaan; Kiehl, Martin: Parameter estimation with model order reduction for elliptic differential equations (2018)

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