This talk will discuss recent developments in PDE-constrained optimization and KKT preconditioning in the context of trust region SQP methods, where the associated PDE problem has an operator which is nonsymmmetric or indefinite. We will analyze the effectiveness of block diagonal KKT preconditioners with approximate Schur complements developed by Rees, Dollar, and Wathen, which were shown to work for symmetric positive definite problems. We will then extend their results to the indefinite Helmholtz equation. Numerical results for acoustic source inversion problems will be presented.