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Honor Thesis Presentation Mike Bantegui, Hofstra University Advisor: Dr. Xiang Fu, Hofstra University EFFICIENT SOLUTION OF THE N-BODY PROBLEM

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OUTLINE The N-Body Problem Computational challenges Results Existing work IGS Framework Extensions

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THE N-BODY PROBLEM

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THE N-BODY PROBLEM CONT. Must integrate 3N coupled nonlinear second order ordinary differential equations Not analytically possible except for N = 2 Must resort to numerical methods

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CANDIDATE NUMERICAL SOLUTION

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FORCE EVALUATION STEP

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EXAMPLE FORCE EVALUATION

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TIME COMPLEXITY

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ISSUES WITH CANDIDATE SOLUTION

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“GEOMETRIC TRICK” FOR PARALLELIZATION

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TIME COMPLEXITY OF GEOMETRIC TRICK

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ISSUES WITH GEOMETRIC TRICK

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NAÏVE FORCE EVALUATION SPEEDUP

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HANDLING CLOSE ENCOUNTERS

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A MORE EFFICIENT WAY OF EVALUATING FORCE

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EXAMPLE OF CLUSTERING BODIES

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HIERARCHICAL FORCE EVALUATION Apply the clustering principle recursively Consider sub-clusters within a cluster Refine force evaluation via sub-clusters instead of main cluster

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CLUSTERING ALGORITHM Node Cluster(bodies, min_bound, max_bound) If bodies.size == 0 return null if bodies.size == 1 return node containing body Collect bodies into spatial groups For each group Cluster(group.bodies, group.min_bound, group.max_bound) Compute first order multipole expansion, passing up tree return node containing groups

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METHODS OF SPATIAL SUBDIVISION

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FORCE EVALUATION ALGORITHM TreeWalk(body) For each branch in the tree If branch is leaf If branch.body != body Compute force of branch.body on body Else Compute distance between branch and body If body is well separated from cluster Compute force of branch on body Else branch.TreeWalk(body)

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FORCE EVALUATION ALGORITHM, CONT

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SEPARATION CRITERIA

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TIME COMPLEXITY OF TREE WALK

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OVERVIEW OF HIERARCHICAL ALGORITHM

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PARALLELIZING TREE METHODS

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SCALING TO LARGE N, OCTREE

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SCALING TO LARGE N, KD-TREE

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SCALING TO LARGE N, BRUTE FORCE

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BRUTE FORCE VS TREE METHODS

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ENERGY ERRORS

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ENERGY ERRORS CONT.

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RELATED WORK nbody1 – nbody6, Sverre Aarseth ACS toolkit, Jun Makino and Piet Hut Grav-Sim, Mark Ridler Gravit, Gerald Kaszuba et al.

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THE PROPOSED FRAMEWORK Component based Interactive Gravitational Simulator CoreIGS – Core simulation library CmdIGS – Command line driven interface VisIGS – Interactive visualizer

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SOFTWARE ARCHITECTURE

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PROJECT STATS Open Source – Available at IGS.codeplex.com 3 + years development 5670 lines of code 65. cpp,.h files 107 subversion revisions 8 development iterations Many failures before success!

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EXTENSIONS

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CONCLUSIONS Tree methods very efficient Performance vs. Accuracy tradeoff possible Extremely parallelizable Accurate, real-time simulations possible on commodity hardware

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ACKNOWLEDGEMENTS Dr. Xiang Fu for advisement and helpful discussions Dr. Gerda Kamberova for helpful comments on the paper Lukasz Bator for many discussions on algorithmic efficiency

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THANK YOU FOR COMING! Any questions?

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