Hierarchical Line Integration

Abstract

This paper presents an acceleration scheme for the numerical computation of sets of trajectories in vector fields or iterated solutions in maps, possibly with simultaneous evaluation of quantities along the curves such as integrals or extrema. It addresses cases with a dense evaluation on the domain, where straightforward approaches are subject to redundant calculations. These are avoided by first calculating short solutions for the whole domain. From these, longer solutions are then constructed in a hierarchical manner until the designated length is achieved. While the computational complexity of the straightforward approach depends linearly on the length of the solutions, the computational cost with the proposed scheme grows only logarithmically with increasing length. Due to independence of subtasks and memory locality, our algorithm is suitable for parallel execution on many-core architectures like GPUs. The trade-offs of the method--lower accuracy and increased memory consumption--are analyzed, including error order as well as numerical error for discrete computation grids. The usefulness and flexibility of the scheme are demonstrated with two example applications: line integral convolution and the computation of the finite-time Lyapunov exponent. Finally, results and performance measurements of our GPU implementation are presented for both synthetic and simulated vector fields from computational fluid dynamics.

Interpolation-Based Pathline Tracing in Particle-Based Flow Visualization

“…Garth et al [6] construct integral surfaces more efficiently and accurately by decoupling the surface approximation from the graphical representation and by approximating and refining time lines. Hlawatsch et al [7] use a hierarchical scheme to decrease the number of integration steps by constructing longer integral lines from previously computed partial solutions, reducing the complexity from linear to logarithmic. Krishnan et al [8] construct time and streak surfaces from time-varying vector fields by building and adapting a surface mesh and representing tracer trajectories as a sequence of fourth order polynomials to allow for easier refinement.…”

Lagrangian representations of flow fields with parameter curves

“…Hlawatsch et al [15] and Agranovsky et al [1] use the flow map directly. Hlawatsch et al utilize a hierarchical scheme to decrease the number of integration steps by constructing longer integral lines from previously computed partial solutions.…”

Graph-based seed scheduling for out-of-core FTLE and pathline computation

“…Kasten et al [15] introduce localized FTLE that reuses the FTLE values from the previous steps. To gain further speed up, though with more error introduced, Brunton and Rowley [2] and Hlawatsch et al [13] both present similar ideas to approximate the flow map by hierarchically interpolating short integrals over time.…”

“…Moreover, particles are seeded at every time step to obtain a sequence of FTLE fields. Many methods have been proposed to speed up the flow map generation [22,10,2,13]. Without trading accuracy for speed, some methods compute densely seeded pathlines in parallel using hardware acceleration such as GPGPU [8,11] or clusters [17].…”