[Paper Review] Semi-analytic integration for a parallel space-time boundary element method modeling the heat equation

Jan Zapletal, Raphael Watschinger+2 more|arXiv (Cornell University)|Feb 19, 2021

Electromagnetic Scattering and Analysis19 references4 citations

TL;DR

This paper presents semi-analytical integration techniques for space-time boundary element methods solving the 3D heat equation, enabling efficient matrix assembly and representation formula evaluation through analytically integrated time components. The approach is implemented in the open-source C++ library besthea, achieving high parallel efficiency on multi-core systems with strong scalability up to 18 threads per socket, though NUMA effects limit scaling across sockets.

ABSTRACT

The presented paper concentrates on the boundary element method (BEM) for the heat equation in three spatial dimensions. In particular, we deal with tensor product space-time meshes allowing for quadrature schemes analytic in time and numerical in space. The spatial integrals can be treated by standard BEM techniques known from three dimensional stationary problems. The contribution of the paper is twofold. First, we provide temporal antiderivatives of the heat kernel necessary for the assembly of BEM matrices and the evaluation of the representation formula. Secondly, the presented approach has been implemented in a publicly available library besthea allowing researchers to reuse the formulae and BEM routines straightaway. The results are validated by numerical experiments in an HPC environment.

Motivation & Objective

• To develop a robust, reusable framework for solving the 3D heat equation using space-time boundary element methods with high efficiency.
• To derive and provide exact temporal antiderivatives of the heat kernel for accurate and fast Galerkin matrix assembly.
• To implement the method in a publicly available C++ library (besthea) to accelerate adoption and reuse in future BEM projects.
• To validate the approach through numerical experiments in an HPC environment, demonstrating scalability and correctness.
• To address the lack of detailed, implementable guidance for space-time BEM in 3D, especially for parallel and vectorized execution.

Proposed method

• The method employs tensor product space-time meshes, enabling analytical integration in time and numerical quadrature in space.
• It derives exact antiderivatives of the heat kernel and its normal derivative for use in assembling Galerkin matrices and evaluating the representation formula.
• The boundary integral operators (single-layer, double-layer, and their duals) are discretized using Galerkin BEM with standard spatial BEM techniques.
• The implementation leverages threading via OpenMP and AVX512 vectorization for performance on shared-memory systems.
• The besthea library provides reusable C++ routines and formulae, with a focus on near-field computation for future integration with fast multipole methods (FMM).
• The approach supports both piecewise constant and linear spatial basis functions, with performance evaluated under varying thread counts and memory access patterns.

Experimental results

Research questions

• RQ1 How can the time-integration of space-time boundary element matrices for the 3D heat equation be performed analytically to reduce computational cost?
• RQ2 What are the exact semi-analytical expressions for the antiderivatives of the heat kernel and its normal derivative required for matrix assembly and representation formula evaluation?
• RQ3 How efficiently can the resulting BEM system be assembled and solved in parallel on modern multi-core architectures?
• RQ4 What are the scalability limits of the implementation when crossing NUMA nodes (sockets)?
• RQ5 How can the method be made readily reusable by the research community through a well-documented, open-source library?

Key findings

• The derived semi-analytical antiderivatives of the heat kernel and its normal derivative are correct and enable exact time integration in space-time BEM.
• The besthea library successfully implements the method, providing ready-to-use formulae and BEM routines for immediate reuse in new projects.
• Matrix assembly achieves over 90% efficiency on a single socket (up to 18 threads), with performance scaling well within a socket for both piecewise constant and linear basis functions.
• Evaluation of potentials shows similar scalability, with efficiency above 90% within a socket and a drop when accessing both sockets due to non-NUMA-aware memory allocation.
• The implementation does not scale optimally across sockets due to the use of std::vector with standard allocator, but this limitation is expected to be resolved in future FMM-based, distributed-memory versions of the library.
• Numerical experiments confirm the correctness and performance of the method in an HPC environment, validating the theoretical derivations.