Proper Orthogonal Decomposition
Proper orthogonal decomposition (POD) provides a method for deriving low order models of dynamical systems. This approach is currently often used successfully as a model reduction technique for nonlinear PDEs, where the basis functions correspond to solutions of the dynamical system at pre-specified time-instances or parameter values, for a control that is selected by the user. These are called snapshots. Due to possible linear dependency or almost linear dependency, the snapshots themselves are not useful as a basis. Therefore, a singular value decomposition is carried out and the leading generalized eigenfunctions are chosen as a basis, referred to as the POD basis. Then, a POD-Galerkin approximation is applied in the spatial variable utilizing the POD basis functions as ansatz and test functions in the variational formulation of the PDE. POD was successfully used in a variety of fields including signal analysis and pattern recognition, fluid dynamics and coherent structures and inverse problems.
There are a-priori error estimates available for POD Galerkin schemes applied to different types of PDES including the two-dimensional Navier-Stokes equations, semi-linear parabolic and elliptic equations and recently also for coupled elliptic-parabolic PDEs. Furthermore, using nonlinear optimal control theory one can optimize the choice of the snapshots in such a manner that the error between the POD solution and the associated solution to the continuous PDE is minimized. In the context of optimal control the POD approach may suffer from the fact that the basis elements are computed from a reference configuration containing features different from those of the optimally controlled trajectory. Therefore, there are a lot of research activities avoiding the problem of unmodelled dynamics. Let us mention here the approaches Optimality-System POD (see e.g. K. Kunisch and S. Volkwein), Trust-Region POD (see e.g. E.W. Sachs et al.), adaptive POD computations (see Hinze et al. or S.S. Ravindran et al., for instance) and a-posteriori error analysis (see e.g. F. Tröltzsch and S. Volkwein).
For more details we refer to the book Turbulence, Coherent Structures, Dynamical Systems and Symmetry of P. Holmes, J.L. Lumley, G. Berkooz (Monographs in Mechanics, Cambridge University Press, 1996) and the Lecture notes Model Reduction using Proper Orthogonal Decomposition by S. Volkwein (University of Constance, 2011).