  We provide first the functional analysis background required for reduced
order modeling and present the underlying concepts of reduced basis model
reduction. The projection-based model reduction framework under affinity
assumptions, offline-online decomposition and error estimation is introduced.
Several tools for geometry parametrizations, such as free form deformation,
radial basis function interpolation and inverse distance weighting
interpolation are explained. The empirical interpolation method is introduced
as a general tool to deal with non-affine parameter dependency and non-linear
problems. The discrete and matrix versions of the empirical interpolation are
considered as well. Active subspaces properties are discussed to reduce
high-dimensional parameter spaces as a pre-processing step. Several examples
illustrate the methodologies.
