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Percolation

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Simulation of percolation models by Guillermo Amaral, ESUG09, Brest, France

Published in: Technology, Business

Percolation

  1. 1. Percolation<br />Simulating percolation models<br />Guillermo Amaral<br />CaesarSystems - Argentina<br />
  2. 2. Guillermo Amaral<br />2<br />
  3. 3. Guillermo Amaral<br />3<br />
  4. 4. Guillermo Amaral<br />4<br />
  5. 5. A virtual lab<br />Guillermo Amaral<br />5<br />
  6. 6. Percolation deals with…<br />
  7. 7. Propagation of diseases<br />Guillermo Amaral<br />7<br />
  8. 8. Propagation of fire<br />Guillermo Amaral<br />8<br />
  9. 9. Oil & gas in reservoirs<br />Guillermo Amaral<br />9<br />
  10. 10. Gelation & Polymerization<br />Guillermo Amaral<br />10<br />
  11. 11. The problem<br />
  12. 12. Original problem (Broadbent - Hammersley, 1957)<br />Guillermo Amaral<br />12<br />What is the probability that the water reaches the center of the rock?<br />
  13. 13. The simulation<br />
  14. 14. The mathematical model<br />
  15. 15. vϵℤ2<br />u<br />v<br />u at distance 1 fromv<br />v<br />Open pathfrom<br />u tov<br />u<br />Open clusterfromv<br />e<br />Percolatingcluster<br />v<br />P(e“open”) = p<br />P(e“close”) = 1 - p<br />v<br />The simplest model<br />Guillermo Amaral<br />15<br />
  16. 16. Dimensions<br />Elementbeing open/close<br />Structure<br />Direction<br />3-D<br />Square<br />Bow-tie<br />p1<br />p<br />Bond<br />p2<br />p<br />Hexagonal<br />Kagomé<br />Site<br />Model types<br />Guillermo Amaral<br />16<br />Isotropic<br />Anisotropic<br />2-D<br />n-D…<br />Other…<br />Both…<br />
  17. 17. <ul><li>θ(p) = Pp(a givenvertexbelongsto a percolatingcluster)
  18. 18. θ(p) = 0 si p = 0
  19. 19. θ(p) = 1 si p = 1
  20. 20. θ(p) ismonotonically non-decrescent
  21. 21. ThereispcЄ[0, 1] suchthat:
  22. 22. θ(p) = 0 if p < pc
  23. 23. θ(p) > 0 if p > pc
  24. 24. Whenis p = pc?</li></ul>Phase transition: Critical probability<br />Guillermo Amaral<br />17<br />θ(p)<br />1<br />pc?<br />p<br />pc<br />0<br />1<br />
  25. 25. Known critical probabilities<br />Guillermo Amaral<br />18<br />
  26. 26. Why simulation?<br />Problems very hard to prove analytically<br />Square bond model critical probability = 0.5<br />Clues for a formal proof<br />Application to practical cases<br />Guillermo Amaral<br />19<br />
  27. 27. Areas of interest<br />Large-graph representation<br />Pseudo-random numbers<br />Graph exploration<br />Analysis of connected components<br />Guillermo Amaral<br />20<br />
  28. 28. Simulation variables<br />Guillermo Amaral<br />21<br />Simulation<br />
  29. 29. Simulation process<br />Guillermo Amaral<br />22<br />2. Generate a “random” configuration<br />1. Build the model<br />3. Search for percolating clusters<br />4. Collect results of output variables<br />
  30. 30. The simulator<br />
  31. 31. My experience… <br />
  32. 32. Guillermo Amaral<br />25<br />Programming with a solution in mind leads to answers, but modeling the problem also raises new questions<br />
  33. 33. Questions<br />
  34. 34. A case of study<br />
  35. 35. pH<br />x0<br /><ul><li>(x0↔v)
  36. 36. (x0↔v’ )</li></ul>pv<br />v<br /><ul><li>v = (x, y)
  37. 37. v’ = (y, x)</li></ul>IfpH &lt; pv,<br />P(x0↔v) &lt;P(x0↔v’)?<br />v’ <br />Scope analysis<br />Guillermo Amaral<br />28<br />
  38. 38. Scope analysis visualization<br />Guillermo Amaral<br />29<br />Mirrorcoloring<br />Scalecoloring<br />&gt;<br />=<br />
  39. 39. Object design<br />
  40. 40. Objects (1)<br />Guillermo Amaral<br />31<br />PercolationModel<br />OpenPolicy<br />BondPercolation<br />SitePercolation<br />SiteOpenPolicy<br />BondOpenPolicy<br />LatticeGraph<br />IsotropicPolicy<br />AnisotropicPolicy<br />Lattice<br />SquareVerticalHorizontal<br />…<br />GraphPattern<br />SquareLattice<br />CubicLattice<br />SubgraphPattern<br />NodeBasedPattern<br />AdjacencySolver<br />Square1KVertical1Horizontal<br />Square1Vertical1KHorizontal<br />…<br />PatternAdjacencySolver<br />MatrixAdjacencySolver<br />Caesar<br />
  41. 41. Objects (2)<br />Guillermo Amaral<br />32<br />GraphAlgorithm<br />AdjacencyMatrix<br />PSBitMatix<br />PSSparseFloatMatrix<br />GraphSearchAlgorithm<br />QuickUnionFind<br />PSFloatMatrix<br />PSSparseMatrix<br />BreathFirstSearch<br />DepthFirstSearch<br />WeightedQuickUnionFind<br />WQUFPC<br />ModelSampler<br />CriticalRangeFinder<br />NodeScopeAnalizer<br />…<br />ModelEvaluator<br />CompositeSampler<br />ModelHistory<br />VariableWalker<br />UnionFindAnalizer<br />…<br />Caesar<br />
  42. 42. Objects (3)<br />Guillermo Amaral<br />33<br />ChartObject<br />ChartAxis<br />XYSerieMarker<br />Chart<br />ChartSerie<br />RangeMark<br />DrawerTool<br />PieChar<br />XYChart<br />NodeLocator<br />XYChartPointLocator<br />ClusterPainter<br />EdgeLocator<br />PSDrawer<br />CriticalRangeDrawer<br />ChartDrawer<br />SquareLatticeGraphDrawer<br />BondPercolationGraphDrawer<br />PieChartDrawer<br />XYChartDrawer<br />SitePercolationGraphDrawer<br />Caesar<br />

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