Commun. Comput. Phys., 13 (2013), pp. 107-128.

Mathematical and Numerical Aspects of the Adaptive Fast Multipole Poisson-Boltzmann Solver

Bo Zhang 1, Benzhuo Lu 2, Xiaolin Cheng 3, Jingfang Huang 4*, Nikos P. Pitsianis 5, Xiaobai Sun 1, J. Andrew McCammon 6

1 Department of Computer Science, Duke University, NC 27708, USA.
2 Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100910, China.
3 Center for Molecular Biophysics, Oak Ridge National Laboratory, TN 37831, USA.
4 Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA.
5 Department of Computer Science, Duke University, NC 27708, USA; and Department of Electrical and Computer Engineering, Aristotle University, Thessaloniki, 54124, Greece.
6 Department of Chemistry & Biochemistry, Center for Theoretical Biological Physics, Department of Pharmacology, Howard Hughes Medical Institute, University of California, San Diego, CA 92093, USA.

Received 21 July 2011; Accepted (in revised version) 11 November 2011
Available online 12 June 2012


This paper summarizes the mathematical and numerical theories and computational elements of the adaptive fast multipole Poisson-Boltzmann (AFMPB) solver. We introduce and discuss the following components in order: the Poisson-Boltzmann model, boundary integral equation reformulation, surface mesh generation, the node-patch discretization approach, Krylov iterative methods, the new version of fast multipole methods (FMMs), and a dynamic prioritization technique for scheduling parallel operations. For each component, we also remark on feasible approaches for further improvements in efficiency, accuracy and applicability of the AFMPB solver to large-scale long-time molecular dynamics simulations. The potential of the solver is demonstrated with preliminary numerical results.

AMS subject classifications: 45B05, 65Y05, 68W10, 90B10, 92C05, 92C40

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Key words: Biomolecular system, electrostatics, Poisson-Boltzmann equation, fast multipole methods, mesh generation, directed acyclic graph, dynamic prioritization, parallelization.

*Corresponding author.
Email: (B. Zhang), (B. Lu), (X. Cheng), (J. Huang), (N. P. Pitsianis), (X. Sun), (J. A. McCammon)

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