We present an efficient and scalable solution to estimate uncertain transport behaviors-stochastic flow maps (SFMs)-for visualizing and analyzing uncertain unsteady flows. Computing flow maps from uncertain flow fields is extremely expensive because it requires many Monte Carlo runs to trace densely seeded particles in the flow. We reduce the computational cost by decoupling the time dependencies in SFMs so that we can process shorter sub time intervals independently and then compose them together for longer time periods. Adaptive refinement is also used to reduce the number of runs for each location. We parallelize over tasks-packets of particles in our design-to achieve high efficiency in MPI/thread hybrid programming. Such a task model also enables CPU/GPU coprocessing. We show the scalability on two supercomputers, Mira (up to 256K Blue Gene/Q cores) and Titan (up to 128K Opteron cores and 8K GPUs), that can trace billions of particles in seconds.