Schindler, Felix ; Albrecht, Felix: Model reduction for parametric multi-scale problems. 2016
Inhalt
- Introduction
- 1 Elliptic parametric multiscale problems
- 1.1 Elliptic problems and grid-based approximations
- 1.2 Multiscale problems and numerical multiscale methods
- 1.3 Parametric problems and model order reduction
- 1.4 Parametric multiscale problems and combined approaches
- 2 The localized reduced basis multiscale method (LRBMS)
- 2.1 Detailed discretization
- 2.2 Reduced discretization
- 2.3 Error control
- 2.3.1 Residual based error control of the model reduction error
- 2.3.2 Localized error control of the discretization and the full error
- 2.4 Adaptivity
- 3 Software concepts and implementations
- 3.1 Discretization framework
- 3.1.1 Mathematical foundation and theoretical requirements
- 3.1.1.1 Approximating the solution of a PDE
- 3.1.1.2 Error estimation
- 3.1.1.3 Projections and prolongations.
- 3.1.2 Abstract design principles and technical requirements
- 3.1.3 Existing implementations
- 3.1.4 A new discretization framework
- 3.2 Model reduction framework
- 4 Numerical experiments
- 4.1 The localized reduced basis (multiscale) method
- 4.2 A new discretization framework: dune-gdt
- 4.2.1 A first online enrichment experiment
- 4.2.2 A first validation of the new localized estimator
- 4.2.3 A first localization study of the new estimator
- 4.2.4 Detailed study of the parametric localized error estimator
- 4.3 A new model reduction framework: pyMOR
- 4.4 The online adaptive LRBMS
- Bibliography