Solving parametric PDEs with an enhanced model order reduction method based on Linear/Ridge expansions

Abstract

Classical projection-based model order reduction methods, like the reduced basis method, are popular tools for getting efficiently solvable reduced order models for parametric PDEs. However, for some problems, the error-decay with respect to the dimension of the linear projection space is predetermined to be slow, e.g., for parameterized wave equations with jump discontinuities. In order to cope with this issue, we consider approximations formed by a linear combination of given functions enhanced by ridge functions -a Linear/Ridge expansion. For an explicitly or implicitly solution of a parameter-dependent problem, we reformulate finding a best Linear/Ridge expansion in terms of an optimization problem that we solve with a particle grid algorithm. The linear functions as well as the ridge profiles are built offline with a greedy-type algorithm. By training the directions offline, we can achieve an efficient online evaluation to solve the projected parametric PDE.