  We consider large-scale linear inverse problems in Bayesian settings. Our
general approach follows a recent line of work that applies the approximate
message passing (AMP) framework in multi-processor (MP) computational systems
by storing and processing a subset of rows of the measurement matrix along with
corresponding measurements at each MP node. In each MP-AMP iteration, nodes of
the MP system and its fusion center exchange lossily compressed messages
pertaining to their estimates of the input. There is a trade-off between the
physical costs of the reconstruction process including computation time,
communication loads, and the reconstruction quality, and it is impossible to
simultaneously minimize all the costs. We pose this minimization as a
multi-objective optimization problem (MOP), and study the properties of the
best trade-offs (Pareto optimality) in this MOP. We prove that the achievable
region of this MOP is convex, and conjecture how the combined cost of
computation and communication scales with the desired mean squared error. These
properties are verified numerically.
