  We consider large-scale linear inverse problems in Bayesian settings. We
follow a recent line of work that applies the approximate message passing (AMP)
framework to multi-processor (MP) computational systems, where each processor
node stores and processes a subset of rows of the measurement matrix along with
corresponding measurements. 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. In this setup, we derive the optimal per-iteration
coding rates using dynamic programming. We analyze the excess mean squared
error (EMSE) beyond the minimum mean squared error (MMSE), and prove that, in
the limit of low EMSE, the optimal coding rates increase approximately linearly
per iteration. Additionally, we obtain that the combined cost of computation
and communication scales with the desired estimation quality according to
$O(\log^2(1/\text{EMSE}))$. Finally, we study trade-offs between the physical
costs of the estimation process including computation time, communication
loads, and the estimation quality as a multi-objective optimization problem,
and characterize the properties of the Pareto optimal surfaces.
