(a) A simple illustration of how reverse advection is used to create an advection map (b) that contains uniformly distributed information for fluid displacement at the microchannel outlet. The advection map can then be converted into a column-stochastic transition matrix (c), for which every row and column represents a fluid element in the two-dimensional (2D) cross section shown in (a) for the inlet and outlet, respectively. Thus, the displacement for a fluid element that would otherwise be calculated by streamtracing through a 3D domain (d) can be computed using only matrix multiplication (e). The inlet fluid states are discretized in the same way as the transition matrix (e-i), and then reshaped to form a row vector (e-ii). This is then multiplied by a transition matrix (e-iii), which produces the outlet fluid states as a row vector (e-iv). This can then be reshaped into the 2D representation of the domain (e-v).