Abstract

Quantum Monte Carlo (QMC) is among the most accurate methods for solving the time independent Schrödinger equation. Unfortunately, the method is very expensive and requires a vast array of computing resources in order to obtain results of a reasonable convergence level. On the other hand, the method is not only easily parallelizable across CPU clusters, but as we report here, it also has a high degree of data parallelism. This facilitates the use of recent technological advances in Graphical Processing Units (GPUs), a powerful type of processor well known to computer gamers. In this paper we report on an end-to-end QMC application with core elements of the algorithm running on a GPU. With individual kernels achieving as much as 30× speed up, the overall application performs at up to 6× faster relative to an optimized CPU implementation, yet requires only a modest increase in hardware cost. This demonstrates the speedup improvements possible for QMC in running on advanced hardware, thus exploring a path toward providing QMC level accuracy as a more standard tool. The major current challenge in running codes of this type on the GPU arises from the lack of fully compliant IEEE floating point implementations. To achieve better accuracy we propose the use of the Kahan summation formula in matrix multiplications. While this drops overall performance, we demonstrate that the proposed new algorithm can match CPU single precision.

Keywords: Graphical processing units; Quantum Monte Carlo; Matrix multiplication; Floating point error; Kahan summation formula; De-normals

PACS classification codes: 07.05.Bx; 02.70.Ss; 02.60.Dc; 89.20.Ff

Article Outline

1. Introduction

2. Intro to Graphical Processing Units

3. Intro to Quantum Monte Carlo

4. Implementation on the GPU

4.1. Walker batch scheme

4.2. Basis function evaluation

4.2.1. Kernel 1: Data generation

4.2.2. Kernel 2: Layout conversion

4.3. Matrix multiplication

4.4. Jastrow functions

5. GPU floating point error

5.1. Underflow corrections

5.2. Kahan method

6. Results

7. Conclusion

Acknowledgements

References