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 58 articles )

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  1. Benaceur, Amina; Ehrlacher, Virginie; Ern, Alexandre; Meunier, Sébastien: A progressive reduced basis/empirical interpolation method for nonlinear parabolic problems (2018)
  2. Benner, Peter; Khoromskaia, Venera; Khoromskij, Boris N.: Range-separated tensor format for many-particle modeling (2018)
  3. Buhr, Andreas; Smetana, Kathrin: Randomized local model order reduction (2018)
  4. Cohen, Albert; Schwab, Christoph; Zech, Jakob: Shape holomorphy of the stationary Navier-Stokes equations (2018)
  5. Drmač, Zlatko; Saibaba, Arvind Krishna: The discrete empirical interpolation method: canonical structure and formulation in weighted inner product spaces (2018)
  6. 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)
  7. Héas, Patrick; Herzet, Cédric: Reduced modeling of unknown trajectories (2018)
  8. Heinkenschloss, Matthias; Jando, Dörte: Reduced order modeling for time-dependent optimization problems with initial value controls (2018)
  9. Hesthaven, J. S.; Ubbiali, S.: Non-intrusive reduced order modeling of nonlinear problems using neural networks (2018)
  10. Kärcher, Mark; Tokoutsi, Zoi; Grepl, Martin A.; Veroy, Karen: Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls (2018)
  11. 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)
  12. Lukassen, Axel Ariaan; Kiehl, Martin: Parameter estimation with model order reduction for elliptic differential equations (2018)
  13. Musharbash, Eleonora; Nobile, Fabio: Dual dynamically orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions (2018)
  14. Santo, Niccolò Dal; Deparis, Simone; Manzoni, Andrea; Quarteroni, Alfio: Multi space reduced basis preconditioners for large-scale parametrized pdes (2018)
  15. Stabile, Giovanni; Rozza, Gianluigi: Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations (2018)
  16. Stefanescu, Razvan; Moosavi, Azam; Sandu, Adrian: Parametric domain decomposition for accurate reduced order models: applications of MP-LROM methodology (2018)
  17. Taddei, Tommaso; Patera, Anthony T.: A localization strategy for data assimilation; application to state estimation and parameter estimation (2018)
  18. Taddei, T.; Penn, J. D.; Yano, M.; Patera, A. T.: Simulation-based classification; a model-order-reduction approach for structural health monitoring (2018)
  19. Xie, X.; Mohebujjaman, M.; Rebholz, L. G.; Iliescu, T.: Data-driven filtered reduced order modeling of fluid flows (2018)
  20. Xie, Xuping; Wells, David; Wang, Zhu; Iliescu, Traian: Numerical analysis of the Leray reduced order model (2018)

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