  Bayesian optimal sensor placement, in its full generality, seeks to maximize
the mutual information between uncertain model parameters and the predicted
data to be collected from the sensors for the purpose of performing Bayesian
inference. Equivalently, the expected information entropy of the posterior of
the model parameters is minimized over all possible sensor configurations for a
given sensor budget. In the context of structural dynamical systems, this
minimization is computationally expensive because of the large number of
possible sensor configurations. Here, a very efficient convex relaxation scheme
is presented to determine informative and possibly-optimal solutions to the
problem, thereby bypassing the necessity for an exhaustive, and often
infeasible, combinatorial search. The key idea is to relax the binary sensor
location vector so that its components corresponding to all possible sensor
locations lie in the unit interval. Then, the optimization over this vector is
a convex problem that can be efficiently solved. This method always yields a
unique solution for the relaxed problem, which is often binary and therefore
the optimal solution to the original problem. When not binary, the relaxed
solution is often suggestive of what the optimal solution for the original
problem is. An illustrative example using a fifty-story shear building model
subject to sinusoidal ground motion is presented, including a case where there
are over 47 trillion possible sensor configurations. The solutions and
computational effort are compared to greedy and heuristic methods.
