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 28 articles )

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  1. Kazemi, Seyed-Mohammad-Mahdi; Dehghan, Mehdi; Foroush Bastani, Ali: On a new family of radial basis functions: mathematical analysis and applications to option pricing (2018)
  2. Xie, Xuping; Wells, David; Wang, Zhu; Iliescu, Traian: Numerical analysis of the Leray reduced order model (2018)
  3. Ali, Mazen; Steih, Kristina; Urban, Karsten: Reduced basis methods with adaptive snapshot computations (2017)
  4. Alla, A.; Falcone, M.; Volkwein, S.: Error analysis for POD approximations of infinite horizon problems via the dynamic programming approach (2017)
  5. Azaïez, Mejdi; Ben Belgacem, Faker; Chacón Rebollo, Tomás; Gómez Mármol, Macarena; Sánchez Muñoz, Isabel: Error bounds in high-order Sobolev norms for POD expansions of parameterized transient temperatures (2017)
  6. Bachmayr, Markus; Cohen, Albert: Kolmogorov widths and low-rank approximations of parametric elliptic PDEs (2017)
  7. Buhr, Andreas; Engwer, Christian; Ohlberger, Mario; Rave, Stephan: ArbiLoMod, a simulation technique designed for arbitrary local modifications (2017)
  8. Horger, Thomas; Wohlmuth, Barbara; Dickopf, Thomas: Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems (2017)
  9. Kazeev, Vladimir; Oseledets, Ivan; Rakhuba, Maxim; Schwab, Christoph: QTT-finite-element approximation for multiscale problems. I: Model problems in one dimension (2017)
  10. Powell, C.E.; Silvester, D.; Simoncini, V.: An efficient reduced basis solver for stochastic Galerkin matrix equations (2017)
  11. Son, Nguyen Thanh; Stykel, Tatjana: Solving parameter-dependent Lyapunov equations using the reduced basis method with application to parametric model order reduction (2017)
  12. Ali, Mazen; Urban, Karsten: Reduced basis exact error estimates with wavelets (2016)
  13. Antonietti, Paola F.; Pacciarini, Paolo; Quarteroni, Alfio: A discontinuous Galerkin reduced basis element method for elliptic problems (2016)
  14. Bader, Eduard; Kärcher, Mark; Grepl, Martin A.; Veroy, Karen: Certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints (2016)
  15. Ballani, Jonas; Kressner, Daniel: Reduced basis methods: from low-rank matrices to low-rank tensors (2016)
  16. Chen, Peng; Schwab, Christoph: Sparse-grid, reduced-basis Bayesian inversion: nonaffine-parametric nonlinear equations (2016)
  17. Fumagalli, Ivan; Manzoni, Andrea; Parolini, Nicola; Verani, Marco: Reduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems (2016)
  18. Liao, Qifeng; Lin, Guang: Reduced basis ANOVA methods for partial differential equations with high-dimensional random inputs (2016)
  19. Maday, Yvon; Manzoni, Andrea; Quarteroni, Alfio: An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems (2016)
  20. Manzoni, A.; Pagani, S.; Lassila, T.: Accurate solution of Bayesian inverse uncertainty quantification problems combining reduced basis methods and reduction error models (2016)

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