Memory-Efficient Recursive Evaluation of 3-Center Gaussian Integrals

Abstract

To improve the efficiency of Gaussian integral evaluation on modern accelerated architectures, FLOP-efficient Obara-Saika-based recursive evaluation schemes are optimized for the memory footprint. For the 3-center 2-particle integrals that are key for the evaluation of Coulomb and other 2-particle interactions in the density-fitting approximation, the use of multiquantal recurrences (in which multiple quanta are created or transferred at once) is shown to produce significant memory savings. Other innovations include leveraging register memory for reduced memory footprint and direct compile-time generation of optimized kernels (instead of custom code generation) with compile-time features of modern C++/CUDA. Performance of conventional and CUDA-based implementations of the proposed schemes is illustrated for both the individual batches of integrals involving up to Gaussians with low and high angular momenta (up to L = 6) and contraction degrees, as well as for the density-fitting-based evaluation of the Coulomb potential. The computer implementation is available in the open-source library.

High-Performance Evaluation of High Angular Momentum 4-Center Gaussian Integrals on Modern Accelerated Processors

“…To optimize the bandwidth, it is necessary to maximize the occupancy, which means minimizing the fast memory footprint. Our approach is to evaluate the 1-index integrals using eq for monotonically decreasing auxiliary indices m , reusing the memory occupied by false[ boldr̃ false] false( m + 1 false) to store false[ boldr̃ false] false( m false) ; this is akin to the in-place evaluation techniques we explored in ref . The prefactors in eq and metadata (maps from index triplets boldr̃ to their ordinals) are independent of m .…”

“…Each thread computes a 2-day round-robin range of 2-index integrals, one at a time, to approximately balance the load between threads. By minimizing the memory footprint of false[ boldr̃ false] false( m false) integrals using the in-place evaluation technique of ref , it is possible to evaluate 1-index integrals even for the [ii|ii] integrals using only 23 kB of shared memory. This allows us to assign 4 thread blocks to each SM even on the V100 GPU with a relatively modest amount of shared memory per SM and make the performance of the 2-index integral evaluation less dependent on the hardware details to ensure efficient execution on current and future generations of accelerators.…”

“…To make the performance analysis easier and more meaningful, we focus here on microbenchmarking the integral kernels, i.e., we analyze their performance for specific integral classes rather than computing, e.g., the entire Fock operator matrix and/or the entire set of integrals for a given problem. While microbenchmarking is less common, ,, it provides a more detailed model of performance by removing extra details (e.g., screening) that can greatly influence performance of integration benchmarks. We strongly encourage others to follow suit.…”

“…An even more serious issue is the high memory footprint of such kernels that exceeds the size of the lowest levels of memory hierarchy (registers and scratchpad memory) even for relatively low angular momenta, thereby reducing the performance. Although detailed performance can be difficult to extract from the numerous publications dedicated to Gaussian AO integral evaluation on GPUs, − the performance of the Head-Gordon–Pople , refinement of the Obara-Saika recurrence scheme implemented by Barca et al is in our experience representative: whereas the performance for 4-center integrals of low total angular momenta (up to (pp|pp), with s, p, d, f, g, h, i, k... denoting Gaussian AOs with angular momenta l = 0, 1, 2, 3, 4, 5, 6, 7..., respectively), was found to reach a substantial (20–50%) fraction of the peak FP64 FLOP rate, the performance for higher angular momenta dropped rapidly to 2% of the peak rate for the (dd|dd) integrals. Another, albeit a less direct, datapoint comes from a study by Johnson et al who observed significant loss of efficiency of the GPU code for the Coulomb matrix evaluation (using McMurchie-Davidson recurrence-based formalism) vs the CPU counterpart as the basis set is enlarged to include higher angular momenta.…”

“…Recently, we reconsidered the design of Gaussian AO integral algorithms in order to optimize their memory footprint . For the specific case of 3-center Gaussian AO integrals, we argued that even for high angular momenta the Obara-Saika recurrence-based schemes , would outperform the Rys quadrature , commonly thought to lead to optimally compact memory footprints; however, even with several algorithmic and programming innovations the performance was reasonable for integrals up to (ff|f) but dropped for higher angular momenta to only a few percent of the hardware peak.…”