RB methods: Basic principles, basic properties

Abstract

Reduced basis methods are introduced for elliptic linear parametrized PDEs. Any reduced basis (RB) approximation is, in a nutshell, a (Petrov-)Galerkin projection onto an N-dimensional space VN (the RB space) that approximates the high-fidelity (say, finite element) solution of the given PDE, for any choice of the parameter within a prescribed parameter set. We illustrate the main steps needed to set up such methods efficiently. We discuss in detail projection methods, which represent the main feature of these techniques, and highlight the difference between Galerkin and least-squares RB methods. We show how to obtain a suitable offline/online decomposition meant to lower the computational complexity and then derive a posteriori error estimates for bounding the error of the RB solution with respect to the underlying high-fidelity solution. We consider the rather general case of inf-sup stable operators, of which coercive operators can be regarded as a particular — yet very relevant — instance. Proper orthogonal decomposition (POD) and greedy algorithms, two major techniques employed to build reduced spaces, are described thoroughly in Chaps. 6 and 7.