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Hierarchical Line Integration for Fast FTLE Computation

This paper presents a hierarchical approach for efficiently computing finite-time Lyapunov exponent (FTLE) fields by integrating and concatenating trajectories at different levels. The method reduces computational costs, provides proven error orders, and is ideal for modern multicore architectures. Results show significant speedup and low error rates, making it a promising solution for dense trajectory seeding in time-dependent flows. Future work may focus on improving error orders and accelerating integrators for higher-order data.

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Hierarchical Line Integration for Fast FTLE Computation

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  1. Hierarchical Line IntegrationTVCG Papers Marcel Hlawatsch, Filip Sadlo, Daniel Weiskopf University of Stuttgart, Germany

  2. Motivation Dense sets of trajectories required for, e.g., delocalized 2 < -5000 [Fuchs et al. 2008] Line integral convolution (LIC) Finite-Time Lyapunov Exponent (FTLE) Other Lagrangian concepts(delocalized vortex criteria)

  3. Motivation • Integration of many trajectories is expensive! • Different trajectories pass same region • Reuse parts of trajectories • Fast LIC[Stalling and Hege 1995, 1998] • Shift convolution kernel on long trajectories • Collect LIC contributions at pixels • Grid Advection for FTLE computation[Sadlo et al. 2010] • Reuse part of path lines for FTLE time series  advectsampling grid • Fast Computation of FTLE Fields[Brunton and Rowley, Chaos 2010] • Concurrent to this work, similar idea • No quantities along trajectories (no LIC etc.) • Higher memory consumption, no proven error order

  4. Concept • Coordinate maps : : start point of traj. : end point of traj. : hierarchylevel • obtained, e.g., by integration • constructedby „concatenation“ • All levels have same resolution (no pyramid) • Overwrite (store only highest level) : : : nodes : • general case: end points not at nodes  interpolation • repeated interpolation  source of error (see later)

  5. Procedure traditional approach (n integration steps) our approach(h levels) integration of initial trajectories  one catenation (s = 2) for next level O(n)  O(h) = O(log n)

  6. Computational Cost Better than “optimum”? • Theory: speedup >2 if steps >16 • Concatenation steps: no integration 2D Time-independent FTLE(in milliseconds)

  7. Computation of Quantities: LIC • Perform operations inside hierarchical scheme • Combine quantities • , min, max, etc. • LIC: convolutionofGaussianwithGaussianisGaussian straightforward hierarchical

  8. Scheme with Time-Dependent Data • level 0: from data set (by integration, blue) • green: required for result at time t1 (at level 3) • bold outlines: blocks kept in memory (overwrite) • hatched: next time blocks • integration range  number of blocks in memory • scheme pays off for time series, i.e., dense trajectory seeding in time • no temporal interpolation needed

  9. Results: FTLE in Time-Dependent 2D Flow FTLE ridge error • Lagrangian coherent structures (LCS) • avg. error = 0.014 cells FTLE error • 95th percentile norm. error (max. = 1.16%) • max. error = 47.33%(atisolatedpoints) hierarchical straightforward • speedup  61 • 512 x 512 resol. • 100 time frames

  10. Results: FTLE in Time-Dependent 3D Flow FTLE ridge error hierarchical straightforward FTLE error • speedup  22 • 1283resol. • 64 time frames

  11. Error Analysis Error order of scheme: : numberof hier. levels : global erroratnode : cellsize : maximumsecond derivative over all coordinatemaps : Lipschitzconstantfromcontinuityofcoordinatemaps • Scheme is second order in cell size (see 2-page proofinsidepaper ;-) )

  12. Conclusion • Acceleration scheme for spacetime-dense sets of solutions (end points of traj.) • Accelerated computation of quantities along trajectories • Logarithmic computational complexity • Well suited for modern multicore or many-core architectures • Proven error order • Future work • Higher-order interpolation schemes  better error order? • Costly integrators for higher-order data  higher acceleration

  13. Hierarchical Line Integration Thank you for your attention! Acknowledgements:

  14. Results: Comparison to IBFV IBFV straightforward hierarchical

  15. Results: LIC coordinate map error • 95th percentile error (max. = 0.1%) • max. error = 1.23% • longer advection time than LIC straightforward hierarchical

  16. Illustration • We produce end points, not complete trajectories • colored points: our approach • white curves: trajectories • background: coordinate map error • larger error in regions of high FTLE(predictability …)

  17. Boundaries • Closed Boundaries / Periodic Domains • No problems (no accesses outside coordinate map) • Open Boundaries (outflow) • Design choice: stop trajectories or continue? • Stopping often preferred • Achieved by adding a zero-velocity border • Repeated interpolation against zero border affects maps • Conservative approach: propagate flag, check flag

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