EXAHD - An Exa-Scalable Two-Level Sparse Grid Approach for Higher-Dimensional Problems in Plasma Physics and Beyond

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Higher-dimensional problems are among the most compute-hungry problems, with an inherent need for future exascale resources. Caused by the exponential dependency of the number of degrees of freedom of classical discretization schemes on the problem's dimensionality, the solution of higher-dimensional problems (beyond the classical four dimensions from continuum mechanics) is a challenging and difficult task. The sparse grid combination technique, introduced to HPC environments by our project, allows to overcome this "curse of dimensionality" to a large extent. It can be employed in a wide range of applications (simulation, optimization, inverse problems,...) in domains as diverse as medicine, finance or astrophysics. The combination technique furthermore provides intriguing approaches to deal with challenges posed by future exascale systems. The combination technique introduces a second, numerically decoupled level of parallelism that ensures high scalability beyond domain decomposition. It is based on a superposition of solutions obtained on significantly coarser and anisotropic full grids, an approach that can also be exploited to deal with faults in an algorithm-based way without the need for expensive checkpoint-restart. In our project, we have demonstrated the feasibility of our approach by applying it to a highly visible and relevant application: turbulence simulations of hot fusion plasmas. We have shown algorithm-based fault tolerance, its scalability properties, and advances in the underlying numerics. Building on the foundations of the first funding period, we will advance the state of the art in three of SPPEXA's research topics towards exascale computing. First, we introduce new algorithmic approaches to the exascale challenges. We will extend fault tolerance for higher-dimensional problems to all levels of parallelization and to the detection and treatment even of silent failures due to data corruption. We propose a third layer in our approach to scale even beyond the boundaries of one single HPC system and even for the solution of time-dependent PDEs. This will be accompanied by hierarchical communication schemes to reduce communication even further. Furthermore, we will advance the frontiers of the numerics of higher-dimensional problems. Second, our software framework will provide a general tool for the solution of higher-dimensional problems, with efficient adaptive and dynamic load balancing. And third, these developments will drive our exemplary application code to high-impact scenarios that are far beyond what is currently feasible with conventional parallelizations even on the fastest systems.