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

高维问题是最需要计算的问题之一，对未来的亿级资源有着内在的需求。由于经典离散格式的自由度数对问题维度的指数依赖性，使得求解高维问题(超越连续介质力学中经典的四维问题)是一项具有挑战性的任务。稀疏网格组合技术，由我们的项目引入到高性能计算环境中，允许在很大程度上克服这种“维度诅咒”。它可用于广泛的应用(模拟、优化、反问题等)。在医学、金融或天体物理学等不同领域。此外，这种组合技术还提供了有趣的方法来应对未来亿级系统带来的挑战。该组合技术引入了第二个数值解耦的并行级别，确保了域分解之外的高可伸缩性。它基于在明显粗糙和各向异性的全网格上获得的解的叠加，这种方法也可以被利用来以基于算法的方式处理故障，而不需要昂贵的检查点重启。在我们的项目中，我们通过将我们的方法应用于一个高度可见性和相关的应用：热聚变等离子体的湍流模拟，证明了我们方法的可行性。我们已经展示了基于算法的容错能力、其可伸缩性以及在底层数值方面的进展。在第一个资助期的基础上，我们将在SPPEXA的三个研究主题中向艾级计算推进最先进的技术。首先，我们引入了新的算法方法来应对艾级挑战。我们将把对高维问题的容错扩展到所有级别的并行化，并扩展到检测和处理甚至是由于数据损坏而导致的静默故障。在我们的方法中，我们提出了第三层，以超越单个HPC系统的边界，甚至用于解决依赖时间的偏微分方程组。这将伴随着分层通信方案，以进一步减少通信。此外，我们还将推进高维问题数值计算的前沿。其次，我们的软件框架将为解决更高维度的问题提供一个通用工具，具有高效的自适应和动态负载平衡。第三，这些开发将把我们的示范性应用程序代码驱动到高影响场景，这远远超出了即使在最快的系统上使用传统并行化目前也是可行的。

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.