[Paper Review] Improved Fast Iterative Algorithm for Eikonal Equation for GPU Computing

Yuhao Huang|arXiv (Cornell University)|Jun 30, 2021

Numerical Methods and Algorithms12 references4 citations

TL;DR

This paper proposes an improved fast iterative method (iFIM) for solving the Eikonal equation on GPUs by eliminating convergence checks during value updates and introducing a separate remedy step to correct errors. The method achieves faster performance than FIM, FMM, and FSSM in GPU-parallelized implementations due to efficient use of SIMD architecture and reduced dependency on sorting structures.

ABSTRACT

In this paper we propose an improved fast iterative method to solve the Eikonal equation, which can be implemented in parallel. We improve the fast iterative method for Eikonal equation in two novel ways, in the value update and in the error correction. The new value update is very similar to the fast iterative method in that we selectively update the points, chosen by a convergence measure, in the active list. However, in order to reduce running time, the improved algorithm does not run a convergence check of the neighboring points of the narrow band as the fast iterative method usually does. The additional error correction step is to correct the errors that the previous value update step may cause. The error correction step consists of finding and recalculating the point values in a separate remedy list which is quite easy to implement on a GPU. In contrast to the fast marching method and the fast sweeping method for the Eikonal equation, our improved method does not need to compute the solution with any special ordering in neither the remedy list nor the active list. Therefore, our algorithm can be implemented in parallel. In our experiments, we implemente our new algorithm in parallel on a GPU and compare the elapsed time with other current algorithms. The improved fast iterative method runs faster than the other algorithms in most cases through our numercal studies.

Motivation & Objective

• To develop a parallelizable algorithm for solving the Eikonal equation that outperforms existing methods on GPU architectures.
• To eliminate the computationally expensive convergence checks in the fast iterative method (FIM) that hinder parallelization.
• To introduce a separate remedy list for correcting errors introduced during value updates, enabling efficient GPU execution.
• To achieve higher performance than fast marching method (FMM) and fast sweeping method (FSSM) in GPU environments by avoiding sorting and sequential dependencies.
• To enable scalable, data-parallel computation of the Eikonal equation using modern GPU streaming multiprocessors.

Proposed method

• The improved fast iterative method (iFIM) uses an active list to select points for value updates based on a convergence measure, similar to FIM, but omits convergence checks on neighboring points.
• A new remedy list is introduced after the update step to identify and recompute points with residual errors, enabling error correction in parallel.
• The algorithm avoids maintaining a sorted data structure (e.g., heapsort), which is incompatible with SIMD execution, thus improving GPU parallelism.
• The local solver uses first-order Godunov upwind finite difference discretization of the Eikonal equation, solving quadratic equations per grid point.
• The method leverages GPU kernel launches with configurable thread blocks, and performance is tuned by adjusting threads per block for optimal memory coalescing and occupancy.
• The remedy step is implemented as a separate kernel that efficiently locates and corrects erroneous points using a simple error threshold check.

Experimental results

Research questions

• RQ1 Can the convergence check step in the fast iterative method be removed without degrading solution accuracy?
• RQ2 How can error propagation from early updates be corrected efficiently in a parallel GPU setting?
• RQ3 Can a GPU-optimized Eikonal solver be designed without relying on sorting or sequential ordering?
• RQ4 What is the performance gain of removing convergence checks and adding a remedy step in a GPU-parallel context?
• RQ5 How does the iFIM algorithm compare in speed and accuracy to FMM, FSSM, and FIM on various Eikonal problem configurations?

Key findings

• The iFIM algorithm consistently runs faster than the original FIM on GPU, with performance gains attributed to reduced kernel overhead and better memory access patterns.
• The fast marching method (FMM) is significantly slower than iFIM due to the use of a heap sort data structure, which is not amenable to GPU parallelization.
• For variable speed functions, FMM’s performance degrades further due to increased heap operations, while iFIM maintains high efficiency.
• The remedy step successfully corrects errors introduced during the update phase, as confirmed by visual and quantitative comparisons with exact solutions in Examples 2 and 5.
• Optimal performance in iFIM is achieved by tuning the number of threads per block, with Figure 12 showing a clear performance peak at specific thread counts for N=2000² grids.
• The iFIM achieves O(N) complexity in practice on GPU, outperforming FMM’s O(N log N) and matching the theoretical efficiency of FSSM, while being more easily parallelizable.