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 39 articles )

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  1. Heinkenschloss, Matthias; Jando, Dörte: Reduced order modeling for time-dependent optimization problems with initial value controls (2018)
  2. 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)
  3. Musharbash, Eleonora; Nobile, Fabio: Dual dynamically orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions (2018)
  4. Xie, Xuping; Wells, David; Wang, Zhu; Iliescu, Traian: Numerical analysis of the Leray reduced order model (2018)
  5. Afkham, Babak Maboudi; Hesthaven, Jan S.: Structure preserving model reduction of parametric Hamiltonian systems (2017)
  6. Ali, Mazen; Steih, Kristina; Urban, Karsten: Reduced basis methods with adaptive snapshot computations (2017)
  7. Alla, A.; Falcone, M.; Volkwein, S.: Error analysis for POD approximations of infinite horizon problems via the dynamic programming approach (2017)
  8. 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)
  9. Bachmayr, Markus; Cohen, Albert: Kolmogorov widths and low-rank approximations of parametric elliptic PDEs (2017)
  10. Buhr, Andreas; Engwer, Christian; Ohlberger, Mario; Rave, Stephan: ArbiLoMod, a simulation technique designed for arbitrary local modifications (2017)
  11. Chacón Rebollo, Tomás; Delgado Ávila, Enrique; Mármol, Macarena Gómez; Ballarin, Francesco; Rozza, Gianluigi: On a certified Smagorinsky reduced basis turbulence model (2017)
  12. Dal Santo, Niccolò; Deparis, Simone; Manzoni, Andrea: A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion equations (2017)
  13. Horger, Thomas; Wohlmuth, Barbara; Dickopf, Thomas: Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems (2017)
  14. Kazeev, Vladimir; Oseledets, Ivan; Rakhuba, Maxim; Schwab, Christoph: QTT-finite-element approximation for multiscale problems. I: Model problems in one dimension (2017)
  15. Lee, Kookjin; Elman, Howard C.: A preconditioned low-rank projection method with a rank-reduction scheme for stochastic partial differential equations (2017)
  16. Powell, C.E.; Silvester, D.; Simoncini, V.: An efficient reduced basis solver for stochastic Galerkin matrix equations (2017)
  17. Quarteroni, A.; Manzoni, A.; Vergara, C.: The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications (2017)
  18. Soldner, Dominic; Brands, Benjamin; Zabihyan, Reza; Steinmann, Paul; Mergheim, Julia: A numerical study of different projection-based model reduction techniques applied to computational homogenisation (2017)
  19. Son, Nguyen Thanh; Stykel, Tatjana: Solving parameter-dependent Lyapunov equations using the reduced basis method with application to parametric model order reduction (2017)
  20. Stabile, Giovanni; Hijazi, Saddam; Mola, Andrea; Lorenzi, Stefano; Rozza, Gianluigi: POD-Galerkin reduced order methods for CFD using finite volume discretisation: vortex shedding around a circular cylinder (2017)

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