ExaSolvers - Extreme scale solvers for coupled systems

亿级计算机应该表现出十亿的并行性。在这种极端规模上的计算需要具有完美规模和最佳复杂性的方法。该项目提案汇集了极端规模解决方案的几个关键方面。首先，求解器本身必须具有最佳的数值复杂性--这一要求随着问题规模的增加而变得越来越苛刻--并且必须高效地扩展到极端规模的并行性。其次，在艾级系统上进行仿真会消耗大量的电力，需要低功耗的算法和实现。在第一个项目阶段，我们证明了多重网格可以有效地扩展到可用的最大计算机的全部大小，并且一旦这些计算机可用，看起来很有希望达到更大的规模。我们进一步证明，对于相关的应用问题，在保持最优复杂性的同时，仍能保持健壮性。为了进一步提高并行性，我们将这种方法与特殊的时间并行化方法、优化问题的求解器和数据不确定性问题的求解器相结合。所有这些领域都引入了额外的并行化机会，如该项目的四篇成果论文所示，这些机会已经成功使用。所开发的算法已与核心多重网格求解器相结合，并在软件框架UG4中实现，并在单个算例中证明是有效的。在第二个项目阶段，我们将把UG4多重网格扩展为一个快速、可扩展和健壮的一般偏微分方程组的求解器。我们将进一步制定在高和低算法水平上的能效策略。适应性将是提高计算和动力效率的主要关键。除了核心解算器并行自适应多重网格外，我们还将通过在UG4的核心中引入完全时空多重网格来增加并行度。我们将进一步深化形状优化和反求建模的工作，并为此提供通用的并行工具。此外，我们将通过将分层Tucker张量采样与核心多重网格正解相结合来扩展不确定性量化的工作。将评估算法和实现在解决问题时的能效。各种应用问题被用作基准问题。我们将通过包括关于纳米结构的新的实验结果，从第一阶段开始改善皮肤渗透问题。我们将进一步使用密度驱动的多孔介质流动、孔隙弹性、纳维-斯托克斯方程和结构力学问题等系统作为测试用例，用于缩放和验证通用求解器策略。利用不确定性量化的方法，我们将计算几个基准问题和垃圾处理场的现场案例。所有算法都将在我们的仿真框架UG4中实现。

Exascale computers are supposed to exhibit billion way parallelism. Computing on such extreme scale needs methods which scale perfectly and have optimal complexity. This project proposal brings together several crucial aspects of extreme scale solving. First, the solver itself must be of optimal numerical complexity - a requirement becoming more and more severe with increasing problem size - and scale efficiently up to extreme scales of parallelism. Second, simulations on exascale systems will consume a lot of electric power, requiring algorithms and implementations with low power consumption. In the first project phase, we proved that multigrid scales efficiently unto the full size of the largest computers available and looks promising for even larger scales, as soon as such computers become available. We further proved that robustness can be maintained during the scaling process for relevant application problems while still maintaining optimal complexity. To further improve parallelism, we combined this approach with special methods for parallelization in time, solvers for optimization problems and for data uncertainty problems. All these areas introduce additional parallelization opportunites which have been used successfully as demonstrated in the four result papers from the project. The algorithms developed have been combined with the core multigrid solver and implemented in the software framework UG4 and have been proven effective in single examples. In the second project phase, we will extend the UG4 multigrid to a fast, scalable and robust solver for general systems of partial differential equations. We will further develop strategies for power efficiency on on high as well as on low algorithmic level. Adaptivity will be a major key to computational and power effciency. Besides the core solver parallel adaptive multigrid, we will increase parallelism by introducing full space-time multigrid into the core of UG4. We will further deepen the work on shape optimization and inverse modeling and make a general parallel tool available for this purpose. Morever, we will extend the work on uncertainty quantification by combining hierarchical Tucker tensor sampling with the core multigrid forward solver. Algorithms and implementations will be evaluated for energy efficiency in problem solving. Various application problems are used as benchmark problems. We will improve the skin permeation problem from phase one by including novel experimental results on the nano structure. We will further use systems such as density driven flow through porous media, poroelasticity, Navier-Stokes equations and structural mechanics problems as test cases for scaling and validation of the general solver strategy. With the uncertainty quantification approach, we will compute several benchmark problems and field cases for waste disposal sites. All algorithms will be implemented in our simulation framework UG4.