Hierarchical Line Integration

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.