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 151 articles , 1 standard article )

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  1. Abbasi, M. H.; Iapichino, L.; Besselink, B.; Schilders, W. H. A.; van de Wouw, N.: Error estimation in reduced basis method for systems with time-varying and nonlinear boundary conditions (2020)
  2. Ali, Mazen; Nouy, Anthony: Singular value decomposition in Sobolev spaces: part I (2020)
  3. Ballarin, Francesco; Chacón Rebollo, Tomás; Delgado Ávila, Enrique; Gómez Mármol, Macarena; Rozza, Gianluigi: Certified reduced basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height (2020)
  4. Burkovska, Olena; Gunzburger, Max: Affine approximation of parametrized kernels and model order reduction for nonlocal and fractional Laplace models (2020)
  5. Dal Santo, Niccolò; Deparis, Simone; Pegolotti, Luca: Data driven approximation of parametrized PDEs by reduced basis and neural networks (2020)
  6. DeCaria, Victor; Iliescu, Traian; Layton, William; McLaughlin, Michael; Schneier, Michael: An artificial compression reduced order model (2020)
  7. Degen, Denise; Veroy, Karen; Wellmann, Florian: Certified reduced basis method in geosciences. Addressing the challenge of high-dimensional problems (2020)
  8. Fareed, Hiba; Singler, John R.: Error analysis of an incremental proper orthogonal decomposition algorithm for PDE simulation data (2020)
  9. Héas, P.: Selecting reduced models in the cross-entropy method (2020)
  10. Heinkenschloss, Matthias; Kramer, Boris; Takhtaganov, Timur: Adaptive reduced-order model construction for conditional value-at-risk estimation (2020)
  11. Herzet, C.; Diallo, M.: Performance guarantees for a variational “multi-space” decoder (2020)
  12. Hijazi, Saddam; Stabile, Giovanni; Mola, Andrea; Rozza, Gianluigi: Data-driven POD-Galerkin reduced order model for turbulent flows (2020)
  13. Karatzas, Efthymios N.; Stabile, Giovanni; Atallah, Nabil; Scovazzi, Guglielmo; Rozza, Gianluigi: A reduced order approach for the embedded shifted boundary FEM and a heat exchange system on parametrized geometries (2020)
  14. Kast, Mariella; Guo, Mengwu; Hesthaven, Jan S.: A non-intrusive multifidelity method for the reduced order modeling of nonlinear problems (2020)
  15. Khurshudyan, As. Zh.: A mesoscopic model for particle-reinforced composites (2020)
  16. Le Clainche, Soledad; Vega, José M.: A review on reduced order modeling using DMD-based methods (2020)
  17. Li, Qiuqi; Zhang, Pingwen: A variable-separation method for nonlinear partial differential equations with random inputs (2020)
  18. Lung, Robert; Wu, Yue; Kamilis, Dimitris; Polydorides, Nick: A sketched finite element method for elliptic models (2020)
  19. Luo, Zhendong; Jiang, Wenrui: A reduced-order extrapolated Crank-Nicolson finite spectral element method for the 2D non-stationary Navier-Stokes equations about vorticity-stream functions (2020)
  20. Luo, Zhendong; Jiang, Wenrui: A reduced-order extrapolated technique about the unknown coefficient vectors of solutions in the finite element method for hyperbolic type equation (2020)

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