Parallel Algorithm for Natural Neighbour Interpolation

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This document describes a parallel algorithm for natural neighbor interpolation that was presented at the 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists. The algorithm allows for 3D modeling from 2D seismic data through interpolation. It uses an incremental Voronoi diagram construction approach and natural neighbor interpolation to calculate density values at new points. This lends itself to parallelization, as each new point's calculation depends only on the initial data points and can be performed independently in parallel threads to improve efficiency. Testing showed speedups using OpenMP and CUDA parallelization on a supercomputer.

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2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists
Yekaterinburg, Russia, October 6, 2016
Parallel Algorithm for Natural Neighbor Interpolation (slide 1 / 19)
2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists
Yekaterinburg, Russia, October 6, 2016
Parallel Algorithm for Natural Neighbor Interpolation (slide 1 / 19)
Parallel Algorithm for
Natural Neighbor Interpolation
Alexander Tsidaev
Bulashevich Institute of Geophysics
Ural Federal University (IRIT-RTF)
Yekaterinburg
2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists
Yekaterinburg, Russia, October 6, 2016

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Parallel Algorithm for Natural Neighbour Interpolation

1. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 1 / 19) 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 1 / 19) Parallel Algorithm for Natural Neighbor Interpolation Alexander Tsidaev Bulashevich Institute of Geophysics Ural Federal University (IRIT-RTF) Yekaterinburg 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016

2. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 2 / 19) Initial Data Seismic data is principally two-dimensional. We need to construct 3D models. Typical seismic profile (recalculated to density values): ?

3. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 3 / 19) Interpolation of the 2D data Interpolation is the key step for initial model construction. This model contains 801 layers.

4. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 4 / 19) Interpolation methods

5. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 5 / 19) Natural Neighbor Interpolation Method based on image by Markluﬀel, distributed under CC BY-SA 3.0 𝑓(𝑥𝑖, 𝑦𝑖) is the measured (initial) value in point 𝑥𝑖, 𝑦𝑖 𝑤𝑖 = 𝑄 𝑘 𝑅 𝑘 is ratio of ”stolen” area

6. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 6 / 19) Voronoi Diagrams Voronoi diagram for 13 points The partitioning of a plane with n points into convex polygons such that each polygon contains exactly one generating point and every point in a given polygon is closer to its generating point than to any other.

7. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 7 / 19) Iterative (incremental) algorithm for Voronoi Based on incremental method by Green, Sibson (1978)

8. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 8 / 19) Iterative (incremental) algorithm for Voronoi Searching intersection points for all polygons. 𝑤𝑖 could be calculated on these steps for all the affected points.

9. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 9 / 19) Iterative (incremental) algorithm for Voronoi Searching intersection points for all polygons. 𝑤𝑖 could be calculated on these steps for all the affected points.

10. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 10 / 19) Iterative (incremental) algorithm for Voronoi Searching intersection points for all polygons. 𝑤𝑖 could be calculated on these steps for all the affected points.

11. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 11 / 19) Iterative (incremental) algorithm for Voronoi Searching intersection points for all polygons. 𝑤𝑖 could be calculated on these steps for all the affected points.

12. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 12 / 19) Iterative (incremental) algorithm for Voronoi Searching intersection points for all polygons. 𝑤𝑖 could be calculated on these steps for all the affected points.

13. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 13 / 19) Iterative (incremental) algorithm for Voronoi Searching intersection points for all polygons. 𝑤𝑖 could be calculated on these steps for all the affected points.

14. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 14 / 19) Iterative (incremental) algorithm for Voronoi When we meet some edge for the second time, the algorithm reaches the end.

15. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 15 / 19) Iterative (incremental) algorithm for Voronoi

16. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 16 / 19) Iterative (incremental) algorithm for Voronoi Resulting new polygion is constructed

17. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 17 / 19) Parallelization It can be noted that: 1. Algorithm is non-obstructive: none of the input data values are changed during calculation. This is the feature of any interpolation algorithm, that the interpolated function remains the initial values in the points of initial set. 2. Value in any inserted point depends only on values of the initial set. So the calculations for any number of interpolated points can be performed in any order. This makes possible to implement the simple parallelization case of multiple independent threads. One thread for each interpolation point.

18. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 18 / 19) Efficiency of Parallel Version Test calculations were performed on the URAN supercomputer (Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg). Initial set contained 644 points, interpolation grid was a square of 1024x1024 points. Two parallelization platforms were used – OpenMP and CUDA.

19. 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists Yekaterinburg, Russia, October 6, 2016 Parallel Algorithm for Natural Neighbor Interpolation (slide 19 / 19) Thank you for your attention

Parallel Algorithm for Natural Neighbour Interpolation

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Parallel Algorithm for Natural Neighbour Interpolation