[Paper Review] Parallel Algorithms for Constructing Data Structures for Fast Multipole Methods

Qi Hu, Nail A. Gumerov+1 more|arXiv (Cornell University)|Jan 8, 2013

Electromagnetic Scattering and Analysis21 references3 citations

TL;DR

This paper presents novel parallel algorithms for constructing data structures in Fast Multipole Methods (FMM), enabling efficient O(N) construction on CPUs, GPUs, and distributed heterogeneous systems. The approach achieves 20–100× speedup on GPUs and demonstrates strong scalability across multiple nodes, significantly reducing the data structure overhead in large-scale FMM simulations with dynamic particle distributions.

ABSTRACT

We present efficient algorithms to build data structures and the lists needed for fast multipole methods. The algorithms are capable of being efficiently implemented on both serial, data parallel GPU and on distributed architectures. With these algorithms it is possible to map the FMM efficiently on to the GPU or distributed heterogeneous CPU-GPU systems. Further, in dynamic problems, as the distribution of the particles change, the reduced cost of building the data structures improves performance. Using these algorithms, we demonstrate example high fidelity simulations with large problem sizes by using FMM on both single and multiple heterogeneous computing facilities equipped with multi-core CPU and many-core GPUs.

Motivation & Objective

• To reduce the computational cost of building FMM data structures, which is a major bottleneck in dynamic simulations with changing particle distributions.
• To enable efficient, scalable construction of FMM data structures on heterogeneous CPU-GPU architectures and distributed systems.
• To achieve O(N) time complexity for data structure construction, making it negligible compared to the overall FMM computation.
• To balance workloads across multiple nodes and minimize communication overhead in distributed FMM implementations.
• To support high-fidelity, large-scale simulations by optimizing data structure construction for both single and multi-node heterogeneous computing environments.

Proposed method

• Proposes a parallel algorithm for constructing octree-based spatial data structures using well-separated pair decomposition (WSPD) to manage near- and far-field interactions.
• Introduces a GPU-optimized implementation of the FMM data structure construction, achieving 20–100× speedup over CPU-only versions.
• Develops a distributed algorithm that assigns spatial boxes to nodes with load balancing and minimizes communication via a data manager for box import/export.
• Uses a many-to-many communication model to reduce bottlenecks in multi-node systems, with each node computing its own communication targets.
• Integrates box type computation with octree and neighbor list construction to reduce overall data structure cost.
• Employs a hierarchical, multi-level FMM framework with P2M (particle-to-multipole), M2M (multipole-to-multipole), M2L (multipole-to-local), L2P (local-to-particle), and final summation steps, all accelerated via parallel data structure construction.

Experimental results

Research questions

• RQ1 Can FMM data structure construction be parallelized efficiently on GPUs and distributed CPU-GPU systems to reduce its O(N) cost to a negligible fraction of total FMM time?
• RQ2 How does the performance of the proposed parallel data structure construction scale across multiple GPUs and nodes, especially under dynamic particle distributions?
• RQ3 What is the impact of communication overhead and workload imbalance on the scalability of distributed FMM data structure construction?
• RQ4 Can the proposed algorithms achieve strong scalability with minimal communication cost and balanced CPU-GPU workloads?
• RQ5 How does the integration of box type computation with other data structures affect overall performance and scalability?

Key findings

• The proposed parallel algorithms achieve O(N) time complexity for FMM data structure construction, making it scalable and efficient for large-scale simulations.
• GPU-accelerated data structure construction yields 20–100× speedup over CPU-only implementations, significantly reducing the data structure step's contribution to total FMM time.
• Strong scalability is demonstrated across multiple nodes with 1–32 GPUs, showing near-perfect scaling for 16M and 8M problem sizes when GPU work is the bottleneck.
• Communication overhead for the data manager is minimal—1–15% of total time—indicating it is negligible in large-scale problems where kernel evaluations dominate.
• The algorithm achieves good load balancing in distributed systems, with two GPUs per node showing optimal performance due to balanced CPU and GPU workloads.
• The communication bottleneck is mitigated by a many-to-many communication model, with each node computing its own communication targets, and future work may further reduce traffic through improved communication patterns.