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.

Summary We address the development of a modular implementation of the MGRIT (MultiGrid‐In‐Time) algorithm to solve linear and nonlinear systems that arise from the discretization of evolutionary models with a parallel‐in‐time approach in the context of the PETSc (the Portable, Extensible Toolkit for Scientific computing) library. Our aim is to give the opportunity of predicting the performance gain achievable when using the MGRIT approach instead of the Time Stepping integrator (TS). To this end, we analyze the performance parameters of the algorithm that provide a‐priori the best number of processing elements and grid levels to use to address the scaling of MGRIT, regarded as a parallel iterative algorithm proceeding along the time dimension.

Scientific computing requires trust in results. In high-performance computing, trust is impeded by silent data corruption (SDC), in other words corruption that remains unnoticed. Numerical integration solvers are especially sensitive to SDCs because an SDC introduced in a certain step affects all the following steps. SDCs can even cause the solver to become unstable. Adaptive solvers can change the step size, by comparing an estimation of the approximation error with an user-defined tolerance. If the estimation exceeds the tolerance, the step is rejected and recomputed. Adaptive solvers have an inherent resilience, because some SDCs might have no consequences on the accuracy of the results, and some SDCs might push the approximation error beyond the tolerance. Our first contribution shows that the rejection mechanism is not reliable enough to reject all SDCs that affect the results' accuracy, because the estimation is also corrupted. We therefore provide another protection mechanism: at the end of each step, a second error estimation is employed to increase the redundancy. Because of the complex dynamics, the choice of the second estimate is difficult: two methods are explored. We evaluated them in HyPar and PETSc, on a cluster of 4,096 cores. We injected SDCs that are large enough to affect the trust or the convergence of the solvers. The new approach can detect 99% of the SDCs, reducing by more than 10 times the number of undetected SDCs. Compared with replication, a classic SDC detector, our protection mechanism reduces the memory overhead by more than 2 times and the computational overhead by more than 20 times in our experiments.