Linear/Ridge expansions: Enhancing linear approximations by ridge functions

Abstract: We consider approximations formed by the sum of a linear combination of given functions enhanced by ridge functions -- a Linear/Ridge expansion. For an explicitly or implicitly given function, we reformulate finding a best Linear/Ridge expansion in terms of an optimization problem. We introduce a particle grid algorithm for its solution. Several numerical results underline the flexibility, robustness and efficiency of the algorithm. One particular source of motivation is model reduction of parameterized transport or wave equations. We show that the particle grid algorithm is able to produce a Linear/Ridge expansion as an efficient nonlinear model reduction.