arXiv:2605.04688v2 Announce Type: replace Abstract: We develop a reduced-order framework for optimizing mixing in two-dimensional incompressible flows. Instead of optimizing the full transport PDE, the method maximizes the length of advected material interfaces, leading to a Hamiltonian control problem with a finite spatial basis based on parametrized stream functions. We derive the continuous adjoint equations and reduced gradients, and discretize the forward and adjoint dynamics with the implicit midpoint rule. The resulting discrete adjoint is algebraically consistent with the derivative of the fully discrete objective when the implicit equations are solved exactly. The approach applies to bounded two-dimensional domains with sufficiently regular finite-dimensional stream-function parametrizations. Numerical experiments on cellular-flow and Doswell frontogenesis benchmarks show interface stretching and mix-norm decay over the simulated horizons. In four comparisons, the interface runs have lower recorded optimization times, whereas the Sobolev controls attain smaller terminal mix-norms. The different constraints, stopping rules and timing scopes preclude an equal-accuracy speedup claim.