  The efficiency of a fluid mixing device is often limited by fundamental laws
and/or design constraints, such that a perfectly homogeneous mixture cannot be
obtained in finite time. Here, we address the natural corollary question: Given
the best available mixer, what is the optimal initial tracer pattern that leads
to the most homogeneous mixture after a prescribed finite time? For ideal
passive tracers, we show that this optimal initial condition coincides with the
right singular vector (corresponding to the smallest singular value) of a
suitably truncated Perron-Frobenius (PF) operator. The truncation of the PF
operator is made under the assumption that there is a small length-scale
threshold $\ell_\nu$ under which the tracer blobs are considered, for all
practical purposes, completely mixed. We demonstrate our results on two
examples: a prototypical model known as the sine flow and a direct numerical
simulation of two-dimensional turbulence. Evaluating the optimal initial
condition through this framework only requires the position of a dense grid of
fluid particles at the final instance and their preimages at the initial
instance of the prescribed time interval. As such, our framework can be readily
applied to flows where such data is available through numerical simulations or
experimental measurements.
