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 91 articles , 1 standard article )

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  1. Fareed, Hiba; Singler, John R.: Error analysis of an incremental proper orthogonal decomposition algorithm for PDE simulation data (2020)
  2. 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)
  3. Le Clainche, Soledad; Vega, José M.: A review on reduced order modeling using DMD-based methods (2020)
  4. 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)
  5. Ainsworth, Mark; Tugluk, Ozan; Whitney, Ben; Klasky, Scott: Multilevel techniques for compression and reduction of scientific data -- the multivariate case (2019)
  6. Antil, Harbir; Chen, Yanlai; Narayan, Akil: Reduced basis methods for fractional Laplace equations via extension (2019)
  7. Brunken, Julia; Smetana, Kathrin; Urban, Karsten: (Parametrized) first order transport equations: realization of optimally stable Petrov-Galerkin methods (2019)
  8. Cagniart, Nicolas; Maday, Yvon; Stamm, Benjamin: Model order reduction for problems with large convection effects (2019)
  9. Chen, Tianheng; Rozovskii, Boris; Shu, Chi-Wang: Numerical solutions of stochastic PDEs driven by arbitrary type of noise (2019)
  10. Ghaffari, Rezvan; Ghoreishi, Farideh: Reduced collocation method for time-dependent parametrized partial differential equations (2019)
  11. Greif, Constantin; Urban, Karsten: Decay of the Kolmogorov (N)-width for wave problems (2019)
  12. Grigo, Constantin; Koutsourelakis, Phaedon-Stelios: Bayesian model and dimension reduction for uncertainty propagation: applications in random media (2019)
  13. Gunzburger, M.; Iliescu, T.; Mohebujjaman, M.; Schneier, M.: An evolve-filter-relax stabilized reduced order stochastic collocation method for the time-dependent Navier-Stokes equations (2019)
  14. Iza-Teran, Rodrigo; Garcke, Jochen: A geometrical method for low-dimensional representations of simulations (2019)
  15. Keshavarzzadeh, Vahid; Kirby, Robert M.; Narayan, Akil: Convergence acceleration for time-dependent parametric multifidelity models (2019)
  16. Khurshudyan, Asatur Z.: Derivation of a mesoscopic model for nonlinear particle-reinforced composites from a fully microscopic model (2019)
  17. Liu, Yong; Chen, Tianheng; Chen, Yanlai; Shu, Chi-Wang: Certified offline-free reduced basis (COFRB) methods for stochastic differential equations driven by arbitrary types of noise (2019)
  18. Luo, Zhendong; Teng, Fei; Xia, Hong: A reduced-order extrapolated Crank-Nicolson finite spectral element method based on POD for the 2D non-stationary Boussinesq equations (2019)
  19. Manzoni, Andrea; Pagani, Stefano: A certified RB method for PDE-constrained parametric optimization problems (2019)
  20. Mu, Lin; Zhang, Guannan: A domain decomposition model reduction method for linear convection-diffusion equations with random coefficients (2019)

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