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 174 articles , 1 standard article )

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  1. Benner, Peter; Goyal, Pawan: Interpolation-based model order reduction for polynomial systems (2021)
  2. Gu, Haotian; Xin, Jack; Zhang, Zhiwen: Error estimates for a POD method for solving viscous G-equations in incompressible cellular flows (2021)
  3. Lu, Chuan; Zhu, Xueyu: Bifidelity data-assisted neural networks in nonintrusive reduced-order modeling (2021)
  4. Novo, Julia; Rubino, Samuele: Error analysis of proper orthogonal decomposition stabilized methods for incompressible flows (2021)
  5. Sun, Xiang; Choi, Jung-Il: Non-intrusive reduced-order modeling for uncertainty quantification of space-time-dependent parameterized problems (2021)
  6. 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)
  7. 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)
  8. Afkham, Babak Maboudi; Ripamonti, Nicolò; Wang, Qian; Hesthaven, Jan S.: Conservative model order reduction for fluid flow (2020)
  9. Ali, Mazen; Nouy, Anthony: Singular value decomposition in Sobolev spaces: part I (2020)
  10. 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)
  11. Burkovska, Olena; Gunzburger, Max: Affine approximation of parametrized kernels and model order reduction for nonlocal and fractional Laplace models (2020)
  12. Busto, Saray; Stabile, G.; Rozza, G.; Vázquez-Cendón, M. Elena: POD-Galerkin reduced order methods for combined Navier-Stokes transport equations based on a hybrid FV-FE solver (2020)
  13. Dal Santo, Niccolò; Deparis, Simone; Pegolotti, Luca: Data driven approximation of parametrized PDEs by reduced basis and neural networks (2020)
  14. DeCaria, Victor; Iliescu, Traian; Layton, William; McLaughlin, Michael; Schneier, Michael: An artificial compression reduced order model (2020)
  15. Degen, Denise; Veroy, Karen; Wellmann, Florian: Certified reduced basis method in geosciences. Addressing the challenge of high-dimensional problems (2020)
  16. Fareed, Hiba; Singler, John R.: Error analysis of an incremental proper orthogonal decomposition algorithm for PDE simulation data (2020)
  17. Fu, Guosheng; Wang, Zhu: POD-(H)DG method for incompressible flow simulations (2020)
  18. Glau, Kathrin; Kressner, Daniel; Statti, Francesco: Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing (2020)
  19. Héas, P.: Selecting reduced models in the cross-entropy method (2020)
  20. Heinkenschloss, Matthias; Kramer, Boris; Takhtaganov, Timur: Adaptive reduced-order model construction for conditional value-at-risk estimation (2020)

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