  Design and optimization benefit from understanding the dependence of a
quantity of interest (e.g., a design objective or constraint function) on the
design variables. A low-dimensional active subspace, when present, identifies
important directions in the space of design variables; perturbing a design
along the active subspace associated with a particular quantity of interest
changes that quantity more, on average, than perturbing the design orthogonally
to the active subspace. This low-dimensional structure provides insights that
characterize the dependence of quantities of interest on design variables.
Airfoil design in a transonic flow field with a parameterized geometry is a
popular test problem for design methodologies. We examine two particular
airfoil shape parameterizations, PARSEC and CST, and study the active subspaces
present in two common design quantities of interest, transonic lift and drag
coefficients, under each shape parameterization. We mathematically relate the
two parameterizations with a common polynomial series. The active subspaces
enable low-dimensional approximations of lift and drag that relate to physical
airfoil properties. In particular, we obtain and interpret a two-dimensional
approximation of both transonic lift and drag, and we show how these
approximation inform a multi-objective design problem.
