1 / 11

A cache-aware algorithm for PDEs on hierarchical data structures

A cache-aware algorithm for PDEs on hierarchical data structures. Frank Günther June 2004. Efficiency in memory. Multigrid. Parallelization. Hierarchical data structures. Adaptivity. Amount of data. Motivation. Solver for PDEs. Space-filling curves. G. Cantor: cardinality of manifolds

mufutau-frye
Download Presentation

A cache-aware algorithm for PDEs on hierarchical data structures

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. A cache-aware algorithm for PDEs on hierarchical data structures Frank GüntherJune 2004

  2. Efficiency in memory Multigrid Parallelization Hierarchical data structures Adaptivity Amount of data Motivation Solver forPDEs A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

  3. Space-filling curves • G. Cantor: cardinality of manifolds • Is there a continuous and bijective ? • E. Netto: No! • Search for continous and surjective mappings • Peano, Hilbert, Sierpinski, Moore, Lebesgue A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

  4. Finite-Element-Discretization Squares on Cartesian grids Multigrid Hierarchical generating system (a-priori-) adaptivity Embedding of arbitrary geometries in the unit square Poisson’s equation Stokes’ equation Dirichlet-boundary-conditions Mathematical ingredients and demands A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

  5. Peano’s curve and stacks • Formation of “lines of points”, which are processed monotonously • Lines are conserved for all depths of recursion • Stacks as data structure • “Coloring” depends on basis:nodal basis on finest level: 2 colors, 2 stacks hierarchical generating system: 4 colors, 8 stacks A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

  6. The Algorithm I • Develop “rule sets” for stack access • Deterministic • No unnecessary access • All kinds of points (inner, outer, on boundary, hanging) must be covered • Efficient programming of stacks and stack access A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

  7. The Algorithm II • Input: linearized geometry-based tree • Recursive cell-oriented programming • Optimizations for cell types • “OO by hand” A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

  8. Results for Poisson’s equation Regular grids Adaptive grids A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

  9. Performance • Simulation • Prediction • “What happens, and where does it happen?” • Measuring • Hardware Performance Counter • Confirmation of prediction Cache hit rate in L2-Cache beyond 99,0% A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

  10. What are the costs of a variable? • Within regular or adaptive grid? • Within an embedded geometry? Nearly always the same! A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

  11. Technology with high potential 3D Numerical efficiency Parallelization Efficiency inhardware More equations Efficiency by methodology Full adaptivityadv. treatment ofboundaries Fluid structureinteraction Conclusion and outlook PDE-solver with space-filling curves A cache-aware algorithm for PDEs on hierarchical data structures Frank Günther

More Related