Zachary J. Grey

Zachary J. Grey, Paul G. Constantine

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.

Zachary J. Grey, Susanna Mosleh, Jacob D. Rezac et al.

To meet the ever-growing demands of data throughput for forthcoming and deployed wireless networks, new wireless technologies like Long-Term Evolution License-Assisted Access (LTE-LAA) operate in shared and unlicensed bands. However, the LAA network must co-exist with incumbent IEEE 802.11 Wi-Fi systems. We consider a coexistence scenario where multiple LAA and Wi-Fi links share an unlicensed band. We aim to improve this coexistence by maximizing the key performance indicators (KPIs) of these networks simultaneously via dimension reduction and multi-criteria optimization. These KPIs are network throughputs as a function of medium access control protocols and physical layer parameters. We perform an exploratory analysis of coexistence behavior by approximating active subspaces to identify low-dimensional structure in the optimization criteria, i.e., few linear combinations of parameters for simultaneously maximizing KPIs. We leverage an aggregate low-dimensional subspace parametrized by approximated active subspaces of throughputs to facilitate multi-criteria optimization. The low-dimensional subspace approximations inform visualizations revealing convex KPIs over mixed active coordinates leading to an analytic Pareto trace of near-optimal solutions.