  Over-the-air computation (AirComp) seamlessly integrates communication and
computation by exploiting the waveform superposition property of
multiple-access channels. Different from the existing works that focus on
transceiver design of AirComp over static networks, this paper considers an
unmanned aerial vehicle (UAV) aided AirComp system, where the UAV as a flying
base station aggregates data from mobile sensors. The trajectory design of the
UAV provides an additional degree of freedom to improve the performance of
AirComp. Our goal is to minimize the time-averaged mean-squared error (MSE) of
AirComp by jointly optimizing the UAV trajectory, receive normalizing factors,
and sensors' transmit power. To this end, we first propose a novel and
equivalent problem transformation by introducing intermediate variables. This
reformulation leads to a convex subproblem when fixing any other two blocks of
variables, thereby enabling efficient algorithm design based on the principle
of block coordinate descent and alternating direction method of multipliers
(ADMM) techniques. In particular, we derive the optimal closed-form solutions
for normalizing factors and intermediate variables optimization subproblems. We
also recast the convex trajectory design subproblem into an ADMM form and
obtain the closed-form expressions for each variable updating. Simulation
results show that the proposed algorithm achieves a smaller time-averaged MSE
while reducing the simulation time by orders of magnitude compared to
state-of-the-art algorithms.
