  We study the inverse boundary value problem for the Helmholtz equation using
the Dirichlet-to-Neumann map at selected frequency as the data. We develop an
explicit reconstruction of the wavespeed using a multi-level nonlinear
projected steepest descent iterative scheme in Banach spaces. We consider
wavespeeds containing discontinuities. A conditional Lipschitz stability
estimate for the inverse problem holds for wavespeeds of the form of a linear
combination of piecewise constant functions with an underlying domain
partitioning, and gives a framework in which the scheme converges. The
stability constant grows exponentially as the number of subdomains in the
domain partitioning increases. To mitigate this growth of the stability
constant, we introduce hierarchical compressive approximations of the solution
to the inverse problem with piecewise constant functions. We establish an
optimal bound of the stability constant, which leads to a condition on the
compression rate pertaining to these approximations.
