# GPU-Accelerated Fock Matrix Computation with Efficient Reduction
## 1 Introduction
### Contributions  
To address the memory contention problem in current Fock matrix construction frameworks, we propose an efficient GPU acceleration scheme that efficiently and suitably distributes the destination addresses of the atomic operations from multiple GPU threads. Our approach capitalizes on a distributed atomic reduction scheme and differs from existing approaches in accelerating Fock matrix updates without reducing the number of atomic operations. Concretely speaking, our primary contributions are as follows:  
**Distributed atomic reduction across replicated Fock matrices:**  
We propose a parallelization method that accelerates the Fock matrix computation using an efficient reduction scheme. Parallel computation of the Fock matrix often suffers from overhead due to memory contention arising from multiple threads simultaneously attempting to write to the same address using atomic operations. To address the arising memory contention problem, we distribute the targets of the atomic addition operations in Equation (1). As such, we decompose the Fock matrix into multiple replicas located at distinct memory addresses, allowing each thread to contribute to a different replica. Then, each replica serves as an intermediate buffer of the same size as the original matrix, and the desired Fock matrix is constructed by summing all the corresponding replicas. Each thread determines the target replica based on its own index, significantly reducing the frequency of the memory contention at the same address compared to concentrating the atomic additions on a single Fock matrix.  
**Hybrid approach with thread-local reduction:**  
Furthermore, we demonstrate that the proposed reduction scheme (above-mentioned) is able to accelerate Fock matrix construction of existing thread-local reduction techniques, enabling a hybridized Fock matrix construction scheme. The memory contention that could not be avoided even with the existing approaches that minimize the number of atomic operations can be further reduced by applying our replicated Fock matrices approach. The experimental results suggest that our proposal does not conflict with existing implementations of the Fock matrix construction, and can enhance computational performance by alleviating their bottlenecks.  
**Computational experiments with relevant molecules:**  
We implemented the above two reduction methods on an NVIDIA A100 GPU hardware and evaluated the performance landscape using a relevant set/configuration of molecules. Our proposed replicated Fock matrix approach achieved up to a 3.75\(\times\) speedup compared to the case where atomic additions concentrate on a single Fock matrix. Furthermore, the hybrid approach combining our proposed reduction scheme with the existing thread-local reduction method rendered an additional performance improvement of up to 1.98 times.  
This paper is organized as follows: Section 2 briefly summarizes the Hartree-Fock method, outlining the key concepts and major computational bottlenecks behind the Schrodinger equation and the Fock matrix computation. Section 3 reviews the previous studies addressing the memory contention problem in the Fock matrix construction, outlines the contemporary challenges, and highlights the key aspects/motivations behind our proposal. Section 4 describes the main proposal of this paper, the distributed atomic reduction method using the replicated Fock matrices. Section 5 evaluates the performance landscape of our proposed approach, as well as of the hybridized Fock matrix construction scheme combining with the existing thread-local reduction technique. Finally, Section 6 concludes and summarizes the major findings in this study.