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 189 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. Fresca, Stefania; Dede’, Luca; Manzoni, Andrea: A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs (2021)
  4. Gu, Haotian; Xin, Jack; Zhang, Zhiwen: Error estimates for a POD method for solving viscous G-equations in incompressible cellular flows (2021)
  5. Lu, Chuan; Zhu, Xueyu: Bifidelity data-assisted neural networks in nonintrusive reduced-order modeling (2021)
  6. 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)
  7. Mou, Changhong; Koc, Birgul; San, Omer; Rebholz, Leo G.; Iliescu, Traian: Data-driven variational multiscale reduced order models (2021)
  8. Novo, Julia; Rubino, Samuele: Error analysis of proper orthogonal decomposition stabilized methods for incompressible flows (2021)
  9. Sun, Xiang; Choi, Jung-Il: Non-intrusive reduced-order modeling for uncertainty quantification of space-time-dependent parameterized problems (2021)
  10. Teng, Fei; Luo, Zhendong: A reduced-order extrapolated approach to solution coefficient vectors in the Crank-Nicolson finite element method for the uniform transmission line equation (2021)
  11. Williamson, Kevin; Cho, Heyrim; Sousedík, Bedřich: Application of adaptive ANOVA and reduced basis methods to the stochastic Stokes-Brinkman problem (2021)
  12. Zancanaro, Matteo; Ballarin, Francesco; Perotto, Simona; Rozza, Gianluigi: Hierarchical model reduction techniques for flow modeling in a parametrized setting (2021)
  13. 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)
  14. Afkham, Babak Maboudi; Ripamonti, Nicolò; Wang, Qian; Hesthaven, Jan S.: Conservative model order reduction for fluid flow (2020)
  15. Ali, Mazen; Nouy, Anthony: Singular value decomposition in Sobolev spaces. I (2020)
  16. 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)
  17. Beattie, Christopher; Gugercin, Serkan; Tomljanović, Zoran: Sampling-free model reduction of systems with low-rank parameterization (2020)
  18. Benner, Peter; Du, Xin; Yang, Guanghong; Ye, Dan: Balanced truncation of linear time-invariant systems over finite-frequency ranges (2020)
  19. Bigoni, Caterina; Zhang, Zhenying; Hesthaven, Jan S.: Systematic sensor placement for structural anomaly detection in the absence of damaged states (2020)
  20. Black, Felix; Schulze, Philipp; Unger, Benjamin: Projection-based model reduction with dynamically transformed modes (2020)

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