  This paper considers sequential adaptive estimation of sparse signals under a
constraint on the total sensing effort. The advantage of adaptivity in this
context is the ability to focus more resources on regions of space where signal
components exist, thereby improving performance. A dynamic programming
formulation is derived for the allocation of sensing effort to minimize the
expected estimation loss. Based on the method of open-loop feedback control,
allocation policies are then developed for a variety of loss functions. The
policies are optimal in the two-stage case, generalizing an optimal two-stage
policy proposed by Bashan et al., and improve monotonically thereafter with the
number of stages. Numerical simulations show gains up to several dB as compared
to recently proposed adaptive methods, and dramatic gains compared to
non-adaptive estimation. An application to radar imaging is also presented.
