The Reference Point Method, a ``hyperreduction'' technique: Application to PGD-based nonlinear model reduction

A new approximation technique, called Reference Point Method (RPM), is proposed in order to reduce the computational complexity of algebraic operations for constructing reduced-order models in the case of time dependent and/or parametrized nonlinear partial differential equations. Even though model reduction techniques enableone to decrease the dimension of the initial problem in the sense that far fewer degrees of freedom are needed to represent the solution, the complexity of evaluating the nonlinear terms and assembling the low dimensional operator associated with the reduced-order model still scales with the size of the original high-dimensional model. This point can be critical, especially when the reduced-order basis changes throughout the solution strategy as it is the case for model reduction techniques based on Proper Generalized Decomposition (PGD). Based on the concept of spatial, parameter/time reference points and influence patches, the RPM defines a compressed version of the data from which an approximate low-rank separated representation by patch of the operators can be constructed by explicit formulas at low-cost without resorting to SVD-based techniques. An application of the RPM to PGD-based model reduction for a nonlinear parametrized elliptic PDE previously studied by other authors with reduced-basis method and EIM is proposed. It is shown that computational complexity to construct the reduced-order model can be divided in practice by one order of magnitude compared with the classical PGD approach.