  We present a distributed active subspace method for training surrogate models
of complex physical processes with high-dimensional inputs and function valued
outputs. Specifically, we represent the model output with a truncated
Karhunen-Lo\`eve (KL) expansion, screen the structure of the input space with
respect to each KL mode via the active subspace method, and finally form an
overall surrogate model of the output by combining surrogates of individual
output KL modes. To ensure scalable computation of the gradients of the output
KL modes, needed in active subspace discovery, we rely on adjoint-based
gradient computation. The proposed method combines benefits of active subspace
methods for input dimension reduction and KL expansions used for spectral
representation of the output field. We provide a mathematical framework for the
proposed method and conduct an error analysis of the mixed KL active subspace
approach. Specifically, we provide an error estimate that quantifies errors due
to active subspace projection and truncated KL expansion of the output. We
demonstrate the numerical performance of the surrogate modeling approach with
an application example from biotransport.
