Extreme Scale FMM-Accelerated Boundary Integral Equation Solver for Wave Scattering

Abstract

Algorithmic and architecture-oriented optimizations are essential for achieving performance worthy of anticipated energy-austere exascale systems. In this paper, we present an extreme scale FMM-accelerated boundary integral equation solver for wave scattering, which uses FMM as a matrix-vector multiplication inside the GMRES iterative method. Our FMM Helmholtz kernels are capable of treating nontrivial singular and near-field integration points. We implement highly optimized kernels for both shared and distributed memory, targeting emerging Intel extreme performance HPC architectures. We extract the potential thread-and data-level parallelism of the key Helmholtz kernels of FMM. Our application code is well optimized to exploit the AVX-512 SIMD units of Intel Skylake and Knights Landing architectures. We provide different performance models for tuning the task-based tree traversal implementation of FMM, and develop optimal architecturespecific and algorithm aware partitioning, load balancing, and communication reducing mechanisms to scale up to 6,144 compute nodes of a Cray XC40 with 196,608 hardware cores. With shared memory optimizations, we achieve roughly 77% of peak single precision floating point performance of a 56-core Skylake processor, and on average 60% of peak single precision floating point performance of a 72-core KNL. These numbers represent nearly 5.4x and 10x speedup on Skylake and KNL, respectively, compared to the the baseline scalar code. With distributed memory optimizations, on the other hand, we report near-optimal efficiency in the weak scalability study with respect to both the O(log P ) communication complexity as well as the theoretical scaling complexity of FMM. In addition, we exhibit up to 85% efficiency in strong scaling. We compute in excess of 2 billion DoF on the full-scale of the Cray XC40 supercomputer. The numerical results match the analytical solution with convergence at 1.0e-4 relative 2-norm residual accuracy. To the best of our knowledge, this work presents the fastest and the most scalable FMM-accelerated linear solver for oscillatory kernels.

On the Spurious Interior Resonance Modes of Time Domain Integral Equations for Analyzing Acoustic Scattering from Penetrable Objects

“…It should be noted here that TDCPIE is obtained by linearly combining TDPIE with its normal derivative. Coupling parameters of this combination are carefully selected to enable the computation of the singular integrals that appear in the expressions of the matrix elements resulting from the Nyström discretization in space [28][29][30][31][32][33][34][35][36][37][38][39][40].…”

A parallel fast multipole method for a space-time boundary element method for the heat equation

“…Many publications have been devoted to the efficient implementation and parallelization of the FMM (see, e.g., [14,15,16,17,18,19]) mainly for particle simulations, but also for the solution of classical spatial boundary integral equations [20,21], and, less frequently, space-time boundary integral equations [22]. A parallelization of a standard Galerkin space-time BEM for the heat equation in two spatial dimensions was considered in [23].…”

Solving Acoustic Boundary Integral Equations Using High Performance Tile Low-Rank LU Factorization

“…However, despite their enormous success, iterative methods may still encounter several major bottlenecks when compared to direct solvers. Indeed, iterative methods are often inadequate for ill-conditioned problems which arise when solving a scattering problem near resonant frequencies [2] or when the scatterer exhibits multi-scale geometric features. In contrast, direct methods are stable and are not as sensitive to ill-conditioning.…”