TY - GEN A1 - Schneck, Jakob A1 - Weiser, Martin A1 - Wende, Florian T1 - Impact of mixed precision and storage layout on additive Schwarz smoothers N2 - The growing discrepancy between CPU computing power and memory bandwidth drives more and more numerical algorithms into a bandwidth- bound regime. One example is the overlapping Schwarz smoother, a highly effective building block for iterative multigrid solution of elliptic equations with higher order finite elements. Two options of reducing the required memory bandwidth are sparsity exploiting storage layouts and representing matrix entries with reduced precision in floating point or fixed point format. We investigate the impact of several options on storage demand and contraction rate, both analytically in the context of subspace correction methods and numerically at an example of solid mechanics. Both perspectives agree on the favourite scheme: fixed point representation of Cholesky factors in nested dissection storage. T3 - ZIB-Report - 18-62 KW - higher order finite elements KW - mixed precision KW - overlapping Schwarz smoother Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-71305 SN - 1438-0064 ER - TY - JOUR A1 - Götschel, Sebastian A1 - Weiser, Martin T1 - Compression Challenges in Large Scale Partial Differential Equation Solvers JF - Algorithms N2 - Solvers for partial differential equations (PDEs) are one of the cornerstones of computational science. For large problems, they involve huge amounts of data that need to be stored and transmitted on all levels of the memory hierarchy. Often, bandwidth is the limiting factor due to the relatively small arithmetic intensity, and increasingly due to the growing disparity between computing power and bandwidth. Consequently, data compression techniques have been investigated and tailored towards the specific requirements of PDE solvers over the recent decades. This paper surveys data compression challenges and discusses examples of corresponding solution approaches for PDE problems, covering all levels of the memory hierarchy from mass storage up to the main memory. We illustrate concepts for particular methods, with examples, and give references to alternatives. Y1 - 2019 U6 - https://doi.org/10.3390/a12090197 VL - 12 IS - 9 SP - 197 ER - TY - JOUR A1 - Götschel, Sebastian A1 - Schiela, Anton A1 - Weiser, Martin T1 - Kaskade 7 - a Flexible Finite Element Toolbox JF - Computers and Mathematics with Applications N2 - Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the \textsc{Dune} interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables. We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way. Y1 - 2021 U6 - https://doi.org/10.1016/j.camwa.2020.02.011 VL - 81 SP - 444 EP - 458 ER - TY - JOUR A1 - Alhaddad, Samer A1 - Förstner, Jens A1 - Groth, Stefan A1 - Grünewald, Daniel A1 - Grynko, Yevgen A1 - Hannig, Frank A1 - Kenter, Tobias A1 - Pfreundt, Franz-Josef A1 - Plessl, Christian A1 - Schotte, Merlind A1 - Steinke, Thomas A1 - Teich, Jürgen A1 - Weiser, Martin A1 - Wende, Florian T1 - HighPerMeshes - A Domain-Specific Language for Numerical Algorithms on Unstructured Grids JF - Euro-Par 2020: Parallel Processing Workshops. N2 - Solving partial differential equations on unstructured grids is a cornerstone of engineering and scientific computing. Nowadays, heterogeneous parallel platforms with CPUs, GPUs, and FPGAs enable energy-efficient and computationally demanding simulations. We developed the HighPerMeshes C++-embedded Domain-Specific Language (DSL) for bridging the abstraction gap between the mathematical and algorithmic formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different parallel programming and runtime models on the other hand. Thus, the HighPerMeshes DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HighPer-Meshes DSL, and demonstrate its usage with three examples, a Poisson and monodomain problem, respectively, solved by the continuous finite element method, and the discontinuous Galerkin method for Maxwell’s equation. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters is presented. Finally, the achievable performance and scalability are demonstrated for a typical example problem on a multi-core CPU cluster. Y1 - 2021 U6 - https://doi.org/10.1007/978-3-030-71593-9_15 SP - 185 EP - 196 PB - Springer ER - TY - GEN A1 - Götschel, Sebastian A1 - Weiser, Martin T1 - Lossy Compression for Large Scale PDE Problems N2 - Solvers for partial differential equations (PDE) are one of the cornerstones of computational science. For large problems, they involve huge amounts of data that needs to be stored and transmitted on all levels of the memory hierarchy. Often, bandwidth is the limiting factor due to relatively small arithmetic intensity, and increasingly so due to the growing disparity between computing power and bandwidth. Consequently, data compression techniques have been investigated and tailored towards the specific requirements of PDE solvers during the last decades. This paper surveys data compression challenges and corresponding solution approaches for PDE problems, covering all levels of the memory hierarchy from mass storage up to main memory. Exemplarily, we illustrate concepts at particular methods, and give references to alternatives. T3 - ZIB-Report - 19-32 KW - partial differential equation KW - data compression KW - floating point compression KW - lossy compression Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-73817 SN - 1438-0064 ER - TY - GEN A1 - Götschel, Sebastian A1 - Schiela, Anton A1 - Weiser, Martin T1 - Kaskade 7 -- a Flexible Finite Element Toolbox N2 - Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the Dune interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables. We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way. T3 - ZIB-Report - 19-48 KW - finite elements KW - generic programming KW - partial differential equations Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-74616 SN - 1438-0064 ER - TY - JOUR A1 - Schneck, Jakob A1 - Weiser, Martin A1 - Wende, Florian T1 - Impact of mixed precision and storage layout on additive Schwarz smoothers JF - Numerical Linear Algebra with Applications N2 - The growing discrepancy between CPU computing power and memory bandwidth drives more and more numerical algorithms into a bandwidth-bound regime. One example is the overlapping Schwarz smoother, a highly effective building block for iterative multigrid solution of elliptic equations with higher order finite elements. Two options of reducing the required memory bandwidth are sparsity exploiting storage layouts and representing matrix entries with reduced precision in floating point or fixed point format. We investigate the impact of several options on storage demand and contraction rate, both analytically in the context of subspace correction methods and numerically at an example of solid mechanics. Both perspectives agree on the favourite scheme: fixed point representation of Cholesky factors in nested dissection storage. Y1 - 2021 U6 - https://doi.org/10.1002/nla.2366 VL - 28 IS - 4 ER - TY - BOOK A1 - Deuflhard, Peter A1 - Weiser, Martin T1 - Numerische Mathematik 3. Adaptive Lösung partieller Differentialgleichungen Y1 - 2020 SN - 978-3-11-069168-9 U6 - https://doi.org/10.1515/9783110689655 PB - de Gruyter ET - 2 ER -