Multigrid Optimization Methods for High Performance Computing

Christian Wagner
University Trier
University Trier, 2012


   title={Multigrid Optimization Methods for High Performance Computing},

   author={Wagner, Christian},



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The aim of this work was the investigation of implementability and efficiency of an algorithm for solving optimal control problems on a new hardware architecture. For an academic test problem the collective smoothing multigrid method (CSMG) was realized on a commodity graphics card (GPU) and the performance in term of elapsed time compared to those on a recent CPU. For dealing with large problem size, new algorithms were designed and two different approaches considered: on the one hand, a recursive approach and on the other hand, a simultaneous approach. Both are based on a nonoverlapping domain decomposition of the entire domain into two subdomains, where a discrete approximation of the Steklov-Poincare operator is derived by a Schur complement method. This so-called capacitance matrix is computed efficiently and inverted analytically. Numerical results show the performance of the CSMG for the one domain case and for both of the developed domain decomposition methods, comparing GPU and CPU. For large-scale optimization, an optimal control problem with ~248 Mio. unknowns was solved by dividing the entire domain into 8 subdomains and processed on 8 GPUs/CPUs in parallel as a proof on concept.
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