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 objective 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 find a Linear/Ridge expansion as an efficient nonlinear model reduction.

MATHMOD 2022 Discussion Contribution Volume

“…Given directions and offsets For fixed directions a j ∈ R d and offsets b j ∈ R, the coefficients α i ∈ R and c j ∈ R are just given as the solution of a linear system of equations (Lemma 2.5 in Greif et al (2022)). Therefore we reformulate the approximation to just search for the optimal (a j , b j ) ∈ R d+1 .…”

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

“…Given directions and offsets For fixed directions a j ∈ R d and offsets b j ∈ R, the coefficients α i ∈ R and c j ∈ R are just given as the solution of a linear system of equations (Lemma 2.5 in Greif et al (2022)). Therefore we reformulate the approximation to just search for the optimal (a j , b j ) ∈ R d+1 .…”