Acoustic and elastic metamaterials have shown potential for many different applications in wave manipulation and structural isolation. Minimization of a scattered pressure field has been identified as an important use for such materials, and a wide variety of designs have been proposed for this purpose. Several of these designs are difficult to manufacture (e.g., pentamode designs), and many others involve numerous tunable parameters and a large design parameter space (e.g., distributions of resonators). This presents a challenge for global search-based optimization and raises the need for gradient-based methods. In this work, adjoint-based, PDE-constrained optimization is used to improve the feasibility of minimizing a scattered pressure field via an optimal set of channels in an annulus surrounding a rigid scatterer. Two different optimization strategies are investigated: material identification, where the physical properties of each channel are treated as unknown, and force identification, where the magnitude and phase of the reflected field is treated as unknown at various patches on the annulus. Results are presented for both a single plane wave direction and multiple plane wave directions, as well as for both a single frequency and multiple frequencies. [S.N.L. is managed and operated by NTESS under DOE NNSA Contract No. DE-NA0003525.]