We discuss locality and non-locality in frequency and space in the optimization of systems governed by differential equations. These analytical features have their origin in the structure of the nonlinear program for such systems, in particular, the nature of the reduced Hessian. These analytical features, in turn, have consequences for the development of multi-level schemes for the optimization of systems governed by differential equations. The analytical ideas we discuss have been developed in the context of systems governed by differential equations, but may also be applicable to other systems whose physical state is described by the solutions of large, coupled systems of equations.
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