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.

Reduction of communication and efficient partitioning are key issues for achieving scalability in hierarchical N -Body algorithms like FMM. In the present work, we propose four independent strategies to improve partitioning and reduce communication. First of all, we show that the conventional wisdom of using space-filling curve partitioning may not work well for boundary integral problems, which constitute about 50% of FMM's application user base. We propose an alternative method which modifies orthogonal recursive bisection to solve the cell-partition misalignment that has kept it from scaling previously. Secondly, we optimize the granularity of communication to find the optimal balance between a bulk-synchronous collective communication of the local essential tree and an RDMA per task per cell. Finally, we take the dynamic sparse data exchange proposed by Hoefler et al. [1] and extend it to a hierarchical sparse data exchange, which is demonstrated at scale to be faster than the MPI library's MPI Alltoallv that is commonly used.

The LEGaTO project leverages task-based programming models to provide a software ecosystem for Made in-Europe heterogeneous hardware composed of CPUs, GPUs, FPGAs and dataflow engines. The aim is to attain one order of magnitude energy savings from the edge to the converged cloud/HPC, balanced with the security and resilience challenges. LEGaTO is an ongoing three-year EU H2020 project started in December 2017.

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