A discrete-exact adjoint-based optimization is used to adjust a wall shape to reduce the noise of an adjacent spatially developing compressible shear layer. For high-fidelity predictive simulations of this kind, a ‘trial-and-error’ approach to minimize a quantity of interest is prohibitively expensive for high-dimensional control spaces. Adjoint-based methods can point the direction of optimal shape with respect to an arbitrarily large number of control parameters at approximately the same computational cost as a single predictive simulation. A key aspect of this space-time discrete-adjoint method is that the gradient is exact to the numerical precision of the calculation. With the discrete exact gradient, we obtain a noise reduction of ˇ 2.5 dB in our model flow with a Fourier-based wall shape parametrization for a compressible near-wall spatially developing shear layer. We demonstrate that the gradient is exact up to machine precision.