Space and Time Parallel Multigrid for Optimization and Uncertainty Quantification in PDE Simulations
In this article we present a complete parallelization approach for simulations of PDEs with applications in optimization and uncertainty quantification. The method of choice for linear or nonlinear elliptic or parabolic problems is the geometric multigrid
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stract In this article we present a complete parallelization approach for simulations of PDEs with applications in optimization and uncertainty quantification. The method of choice for linear or nonlinear elliptic or parabolic problems is the geometric multigrid method since it can achieve optimal (linear) complexity in terms of degrees of freedom, and it can be combined with adaptive refinement strategies in order to find the minimal number of degrees of freedom. This optimal solver is parallelized such that weak and strong scaling is possible for extreme scale HPC architectures. For the space parallelization of the multigrid method we use a tree based approach that allows for an adaptive grid refinement and online load balancing. Parallelization in time is achieved by SDC/ISDC or a spacetime formulation. As an example we consider the permeation through human skin which serves as a diffusion model problem where aspects of shape optimization, uncertainty quantification as well as sensitivity to geometry and material parameters are studied. All methods are developed and tested in the UG4 library.
L. Grasedyck () • C. Löbbert IGPM, RWTH Aachen, Aachen, Germany e-mail: [email protected]; [email protected] G. Wittum • A. Nägel G-CSC, University of Frankfurt, Frankfurt, Germany e-mail: [email protected]; [email protected] V. Schulz • M. Siebenborn University of Trier, Trier, Germany e-mail: [email protected]; [email protected] R. Krause • P. Benedusi ICS, University of Lugano, Lugano, Germany e-mail: [email protected]; [email protected] U. Küster • B. Dick HLRS, University of Stuttgart, Stuttgart, Germany e-mail: [email protected]; [email protected] © Springer International Publishing Switzerland 2016 H.-J. Bungartz et al. (eds.), Software for Exascale Computing – SPPEXA 2013-2015, Lecture Notes in Computational Science and Engineering 113, DOI 10.1007/978-3-319-40528-5_23
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1 Introduction From the very beginning of computing, numerical simulation has been the force driving the development. Modern solvers for extremely large scale problems require extreme scalability and low electricity consumption in addition to the properties solvers are always expected to exhibit—like optimal complexity and robustness. Naturally, the larger the system becomes, the more crucial is the asymptotic complexity issue. In this article, in order to get the whole picture, we give a brief review of recent developments towards optimal parallel scaling for the key components of numerical simulation. We consider parallelization in space in Sect. 2, in time in Sect. 4, and with respect to (uncertain) parameters in Sect. 6. These three approaches are designed to be perfectly compatible with each other and can be combined in order to multiply the parallel scalability. At the same time they are kept modular and could in principle also be used in combination with other methods. We address the optimal choice of CPU frequencies for the components of the multigrid method
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