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，具有对未来百亿亿级资源的内在需求。由于经典离散化方法的自由度数与问题维数呈指数关系，求解高维问题（超越经典连续介质力学的四维问题）是一项具有挑战性和难度的任务。我们的项目将稀疏网格组合技术引入到HPC环境中，可以在很大程度上克服这种“维度诅咒”。它可以在医学、金融或天体物理学等不同领域的广泛应用（模拟、优化、逆问题等）中使用。这种组合技术进一步为应对未来百亿亿级系统带来的挑战提供了有趣的方法。组合技术引入了第二种数字解耦的并行性，确保了超越域分解的高可伸缩性。它是基于在非常粗糙和各向异性的全网格上获得的解的叠加，这种方法也可以用于以基于算法的方式处理故障，而不需要昂贵的检查点重新启动。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，它的可伸缩性特性，以及底层数值的进展。在第一个资助期的基础上，我们将在SPPEXA的三个面向百亿亿次计算的研究课题中推进最先进的技术。首先，我们介绍了解决百亿亿次挑战的新算法方法。我们将把高维问题的容错扩展到所有级别的并行化，甚至检测和处理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，甚至可以超越单个HPC系统的边界，甚至可以解决时间相关的pde。这将伴随着分层通信方案，以进一步减少通信。此外，我们将推进高维问题的数值前沿。其次，我们的软件框架将为解决高维问题提供一个通用的工具，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典型应用程序代码驱动到高影响的场景中，这远远超出了目前在最快的系统上使用传统并行化的可行性。

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.