ExaSolvers - Extreme scale solvers for coupled systems

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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.