the numerical solution of high frequency wave propagation has been a longstanding challenge in computational science and engineering. this talk addresses this problem in the time-harmonic regime. we consider a sequence of examples with important applications, and for each we construct an efficient preconditioner (approximate inverse) that allows one to solve the system with a small number of iterations. from these examples emerges a new framework, where sparsity, geometry of wave phenomenon, and highly accurate discretizations are combined together to address this challenging topic.