We propose COSMA: a parallel matrix-matrix multiplication algorithm that is near communication-optimal for all combinations of matrix dimensions, processor counts, and memory sizes. The key idea behind COSMA is to derive an optimal (up to a factor of 0.03% for 10MB of fast memory) sequential schedule and then parallelize it, preserving I/O optimality. To achieve this, we use the red-blue pebble game to precisely model MMM dependencies and derive a constructive and tight sequential and parallel I/O lower bound proofs. Compared to 2D or 3D algorithms, which fix processor decomposition upfront and then map it to the matrix dimensions, it reduces communication volume by up to √ times. COSMA outperforms the established ScaLAPACK, CARMA, and CTF algorithms in all scenarios up to 12.8x (2.2x on average), achieving up to 88% of Piz Daint's peak performance. Our work does not require any hand tuning and is maintained as an open source implementation.

We present a scalable implementation of the Linearized Augmented Plane Wave method for distributed memory systems, which relies on an efficient distributed, block-cyclic setup of the Hamiltonian and overlap matrices and allows us to turn around highly accurate 1000+ atom all-electron quantum materials simulations on clusters with a few hundred nodes. The implementation runs efficiently on standard multi-core CPU nodes, as well as hybrid CPU-GPU nodes. The key for the latter is a novel algorithm to solve the generalized eigenvalue problem for dense, complex Hermitian matrices on distributed hybrid CPU-GPU systems. Performance tests for Li-intercalated CoO2 supercells containing 1501 atoms demonstrate that high-accuracy, transferable quantum simulations can now be used in throughput materials search problems. While our application can benefit and get scalable performance through CPU-only libraries like ScaLAPACK or ELPA2, our new hybrid solver enables the efficient use of GPUs and shows that a hybrid CPU-GPU architecture scales to a desired performance using substantially fewer cluster nodes, and notably, is considerably more energy efficient than the traditional multi-core CPU only systems for such complex applications.

SUMMARY For software to fully exploit the computing power of emerging heterogeneous computers, not only must the required computational kernels be optimized for the specific hardware architectures but also an effective scheduling scheme is needed to utilize the available heterogeneous computational units and to hide the communication between them. As a case study, we develop a static scheduling scheme for the tridiagonalization of a symmetric dense matrix on multicore CPUs with multiple graphics processing units (GPUs) on a single compute node. We then parallelize and optimize the Basic Linear Algebra Subroutines (BLAS)‐2 symmetric matrix‐vector multiplication, and the BLAS‐3 low rank symmetric matrix updates on the GPUs. We demonstrate the good scalability of these multi‐GPU BLAS kernels and the effectiveness of our scheduling scheme on twelve Intel Xeon processors and three NVIDIA GPUs. We then integrate our hybrid CPU‐GPU kernel into computational kernels at higher‐levels of software stacks, that is, a shared‐memory dense eigensolver and a distributed‐memory sparse eigensolver. Our experimental results show that our kernels greatly improve the performance of these higher‐level kernels, not only reducing the solution time but also enabling the solution of larger‐scale problems. Because such symmetric eigenvalue problems arise in many scientific and engineering simulations, our kernels could potentially lead to new scientific discoveries. Furthermore, these dense linear algebra algorithms present algorithmic characteristics that can be found in other algorithms. Hence, they are not only important computational kernels on their own but also useful testbeds to study the performance of the emerging computers and the effects of the various optimization techniques. Copyright © 2013 John Wiley & Sons, Ltd.

We present a new quantum cluster algorithm to simulate models of high-Tc superconductors. This algorithm extends current methods with continuous lattice self-energies, thereby removing artificial long-range correlations. This cures the fermionic sign problem in the underlying quantum Monte Carlo solver for large clusters and realistic values of the Coulomb interaction in the entire temperature range of interest. We find that the new algorithm improves time-to-solution by nine orders of magnitude compared to current, state of the art quantum cluster simulations. An efficient implementation is given, which ports to multi-core as well as hybrid CPU-GPU systems. Running on 18,600 nodes on ORNL's Titan supercomputer enables us to compute a converged value of Tc/t = 0.053 ± 0.0014 for a 28 site cluster in the 2D Hubbard model with U/t = 7 at 10% hole doping. Typical simulations on Titan sustain between 9.2 and 15.4 petaflops (double precision measured over full run), depending on configuration and parameters used.