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 138 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. Burkovska, Olena; Gunzburger, Max: Affine approximation of parametrized kernels and model order reduction for nonlocal and fractional Laplace models (2020)
  3. DeCaria, Victor; Iliescu, Traian; Layton, William; McLaughlin, Michael; Schneier, Michael: An artificial compression reduced order model (2020)
  4. Degen, Denise; Veroy, Karen; Wellmann, Florian: Certified reduced basis method in geosciences. Addressing the challenge of high-dimensional problems (2020)
  5. Fareed, Hiba; Singler, John R.: Error analysis of an incremental proper orthogonal decomposition algorithm for PDE simulation data (2020)
  6. Héas, P.: Selecting reduced models in the cross-entropy method (2020)
  7. Herzet, C.; Diallo, M.: Performance guarantees for a variational “multi-space” decoder (2020)
  8. 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)
  9. Kast, Mariella; Guo, Mengwu; Hesthaven, Jan S.: A non-intrusive multifidelity method for the reduced order modeling of nonlinear problems (2020)
  10. Le Clainche, Soledad; Vega, José M.: A review on reduced order modeling using DMD-based methods (2020)
  11. Li, Qiuqi; Zhang, Pingwen: A variable-separation method for nonlinear partial differential equations with random inputs (2020)
  12. Lung, Robert; Wu, Yue; Kamilis, Dimitris; Polydorides, Nick: A sketched finite element method for elliptic models (2020)
  13. 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)
  14. Luo, Zhendong; Wang, Hui: A highly efficient reduced-order extrapolated finite difference algorithm for time-space tempered fractional diffusion-wave equation (2020)
  15. Pivovarov, Dmytro; Steinmann, Paul; Willner, Kai: Acceleration of the spectral stochastic FEM using POD and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry (2020)
  16. Saibaba, Arvind K.: Randomized discrete empirical interpolation method for nonlinear model reduction (2020)
  17. Stabile, Giovanni; Rosic, Bojana: Bayesian identification of a projection-based reduced order model for computational fluid dynamics (2020)
  18. Taddei, Tommaso: A registration method for model order reduction: data compression and geometry reduction (2020)
  19. Teng, Fei; Luo, Zhendong: A reduced-order extrapolation technique for solution coefficient vectors in the mixed finite element method for the 2D nonlinear Rosenau equation (2020)
  20. Yang, Feng-Lian; Yan, Liang: A non-intrusive reduced basis EKI for time fractional diffusion inverse problems (2020)

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