Reduced Basis Methods
Reduced basis (RB) methods have been developed during the last decades with the aim to efficiently compute numerical solutions for parametrized applications. These are applications for which not only a single simulation has to be performed, but solutions for a range of different parameter configurations of the same problem are desired. This can be found in optimization, control, parameter estimation, statistical investigations or online-simulation settings. Realistic numerical discretization schemes result in huge numerical models which result in long computation times. Therefore, they are in general unsuitable for such real-time and many-query applications under parameter variation. Reduced basis methods provide a solution for reducing the dimensionality of such problems, giving efficient reduced simulation models, while simultaneously ensuring certified error statements for the simulation result.
The starting point of an RB-approach is a parametrized model, which may be a parametrized partial differential equation or dynamical system. In the engineering context these parameters can e.g. be material, geometry, boundary-value, initial value or control parameters. The RB-method then can be divided into two stages, a (possibly expensive) offline-stage for preparing parameter-independent quantities and a fast online-stage in which parameter variation and an online-simulation can be performed.
The first ingredient in RB-methods is a reduced basis space. This is a low-dimensional linear subspace of the high- (or even infinite-dimensional) solution space. The RB-space is ideally chosen, such that it captures or approximates the parametric solution manifold. This space is algorithmically identified in a reduced basis generation stage based on "snapshots" of the solution. These snapshots are detailed (or analytical) solutions for specific parameters and time instants. The crucial point here is to select the right parameters, which guarantee a low dimension while simultaneously generating a low model error for new parameters. This parameter choice can be realized by greedy or adaptive search algorithms or suitable optimization problems.
The second ingredient, a Galerkin-projection of the detailed problem on this low-dimensional RB-space, results in the RB simulation scheme. The reduced model enables parameter variation with fast simulation response.
A third characteristic of RB-methods is rigorous a-posteriori error estimation, which allows to evaluate the quality of the online simulation results for each new parameter and input. This is in contrast to general a-priori estimates, which only guarantee a worst-case accuracy of reduced models.
Overall, all expensive computations, i.e. the basis generation and the high-dimensional operations of the simulation and error-estimation, are made parameter independent and computed in the offline-phase. The online phase then merely needs to assemble these offline-quantities with very low computational complexity. This decomposition is realizable in case of affine parameter-dependence. In other cases, approximation or interpolation schemes must be applied as an additional step.
We refer to the website of A.T. Patera, MIT, Cambridge, MA for further details on the methodology.