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. 2017 Jul 11;13(7):3378-3387.
doi: 10.1021/acs.jctc.7b00336. Epub 2017 Jun 7.

Acceleration of Linear Finite-Difference Poisson-Boltzmann Methods on Graphics Processing Units

Ruxi Qi  1 , Wesley M Botello-Smith  1 , Ray Luo  1
Affiliations

Acceleration of Linear Finite-Difference Poisson-Boltzmann Methods on Graphics Processing Units

Ruxi Qi et al. J Chem Theory Comput. .

Abstract

Electrostatic interactions play crucial roles in biophysical processes such as protein folding and molecular recognition. Poisson-Boltzmann equation (PBE)-based models have emerged as widely used in modeling these important processes. Though great efforts have been put into developing efficient PBE numerical models, challenges still remain due to the high dimensionality of typical biomolecular systems. In this study, we implemented and analyzed commonly used linear PBE solvers for the ever-improving graphics processing units (GPU) for biomolecular simulations, including both standard and preconditioned conjugate gradient (CG) solvers with several alternative preconditioners. Our implementation utilizes the standard Nvidia CUDA libraries cuSPARSE, cuBLAS, and CUSP. Extensive tests show that good numerical accuracy can be achieved given that the single precision is often used for numerical applications on GPU platforms. The optimal GPU performance was observed with the Jacobi-preconditioned CG solver, with a significant speedup over standard CG solver on CPU in our diversified test cases. Our analysis further shows that different matrix storage formats also considerably affect the efficiency of different linear PBE solvers on GPU, with the diagonal format best suited for our standard finite-difference linear systems. Further efficiency may be possible with matrix-free operations and integrated grid stencil setup specifically tailored for the banded matrices in PBE-specific linear systems.

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Figures

Figure 1
Figure 1
Correlations (a), (c) and differences (b), (d) of electrostatic solvation energies on GPU (Jacobi with CUSP library and DIA matrix format) and on CPU (CG) for the protein test set. Free space boundary condition was used. The convergence criterion was set to 10−3 (a), (b) and 10−6 (c), (d). The linear regression slopes are 0.999931 and 0.999996 for 10−3 and 10−6 criterion respectively, and the correlation coefficients are 1.0 for both.
Figure 2
Figure 2
Correlations (a), (c) and differences (b), (d) of electrostatic solvation energies on GPU (Jacobi with CUSP library and DIA matrix format) and on CPU (CG) for the protein test set. The periodic boundary condition (PBC) was used. The convergence criterion was set to 10−3 (a), (b) and 10−6 (c), (d). The linear regression slopes are 0.999639 and 1.0 for 10−3 and 10−6 criterion respectively, and the correlation coefficients are 1.0 for both.
Figure 3
Figure 3
Comparison between PB solvers on GPU (Jacobi with CUSP library and DIA matrix format) and on CPU with free space boundary condition for the protein test set, as functions of number of atoms and grids respectively. The convergence criterion was set to 10−3 (a), (b) and 10−6 (c), (d).
Figure 4
Figure 4
Comparison between PB solvers on GPU (Jacobi with CUSP library and DIA matrix format) and on CPU (ICCG) with PBC for the protein test set, as functions of number of atoms and grids respectively. The convergence criterion was set to 10−3 (a), (b) and 10−6 (c), (d).
Figure 5
Figure 5
Average time used by different CPU and GPU solvers for selected test proteins. The detail of each solver combination is elaborated in Table 1. The GPU/ICCG solver is not listed due to their extremely long execution times. The convergence criterion was set to 10−3 (top) and 10−6 (bottom).
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