DL_MG: A Parallel Multigrid Poisson and Poisson–Boltzmann Solver for Electronic Structure Calculations in Vacuum and Solution - Publication - Bridge of Knowledge

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DL_MG: A Parallel Multigrid Poisson and Poisson–Boltzmann Solver for Electronic Structure Calculations in Vacuum and Solution

Abstract

The solution of the Poisson equation is a crucial step in electronic structure calculations, yielding the electrostatic potential -- a key component of the quantum mechanical Hamiltonian. In recent decades, theoretical advances and increases in computer performance have made it possible to simulate the electronic structure of extended systems in complex environments. This requires the solution of more complicated variants of the Poisson equation, featuring nonhomogeneous dielectric permittivities, ionic concentrations with nonlinear dependencies, and diverse boundary conditions. The analytic solutions generally used to solve the Poisson equation in vacuum (or with homogeneous permittivity) are not applicable in these circumstances, and numerical methods must be used. In this work, we present DL_MG, a flexible, scalable, and accurate solver library, developed specifically to tackle the challenges of solving the Poisson equation in modern large-scale electronic structure calculations on parallel computers. Our solver is based on the multigrid approach and uses an iterative high-order defect correction method to improve the accuracy of solutions. Using two chemically relevant model systems, we tested the accuracy and computational performance of DL_MG when solving the generalized Poisson and Poisson−Boltzmann equations, demonstrating excellent agreement with analytic solutions and efficient scaling to ~10^9 unknowns and 100s of CPU cores. We also applied DL_MG in actual large-scale electronic structure calculations, using the ONETEP linear-scaling electronic structure package to study a 2615 atom protein−ligand complex with routinely available computational resources. In these calculations, the overall execution time with DL_MG was not significantly greater than the time required for calculations using a conventional FFT-based solver.

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Category:
Articles
Type:
artykuł w czasopiśmie wyróżnionym w JCR
Published in:
Journal of Chemical Theory and Computation no. 14, edition 3, pages 1412 - 1432,
ISSN: 1549-9618
Language:
English
Publication year:
2018
Bibliographic description:
Womack J., Anton L., Dziedzic J., Hasnip P., Probert M., Skylaris C.: DL_MG: A Parallel Multigrid Poisson and Poisson–Boltzmann Solver for Electronic Structure Calculations in Vacuum and Solution// Journal of Chemical Theory and Computation. -Vol. 14, iss. 3 (2018), s.1412-1432
DOI:
Digital Object Identifier (open in new tab) 10.1021/acs.jctc.7b01274
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  1. Skylaris, C.-K.; Haynes, P. D.; Mostofi, A. A.; Payne, M. C. Introducing ONETEP: Linear-scaling density functional simulations on parallel computers. J. Chem. Phys. 2005, 122, 084119. open in new tab
  2. Mohr, S.; Ratcliff, L. E.; Genovese, L.; Caliste, D.; Boulanger, P.; Goedecker, S.; Deutsch, T. Accurate and efficient linear scaling DFT calculations with universal ap- plicability. Phys. Chem. Chem. Phys. 2015, 17, 31360-31370. open in new tab
  3. Gillan, M. J.; Bowler, D. R.; Torralba, A. S.; Miyazaki, T. Order-N first-principles calculations with the conquest code. Comput. Phys. Commun. 2007, 177, 14-18. open in new tab
  4. Duy, T. V. T.; Ozaki, T. A three-dimensional domain decomposition method for large- scale DFT electronic structure calculations. Comput. Phys. Commun. 2014, 185, 777- 789. open in new tab
  5. VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comput. Phys. Commun. 2005, 167, 103-128. open in new tab
  6. Soler, J. M.; Artacho, E.; Gale, J. D.; García, A.; Junquera, J.; Ordejón, P.; Daniel Sánchez-Portal, The SIESTA method for ab initio order-N materials simulation. J. Phys.: Condens. Matter 2002, 14, 2745. open in new tab
  7. Lever, G.; Cole, D. J.; Hine, N. D. M.; Haynes, P. D.; Payne, M. C. Electrostatic considerations affecting the calculated HOMO-LUMO gap in protein molecules. J. Phys.: Condens. Matter 2013, 25, 152101. open in new tab
  8. Brandt, A. Multi-level adaptive solutions to boundary-value problems. Math. Comp. 1977, 31, 333-390. open in new tab
  9. Briggs, W.; Henson, V.; McCormick, S. A Multigrid Tutorial, Second Edition; Other Titles in Applied Mathematics; Society for Industrial and Applied Mathematics, 2000; open in new tab
  10. DOI: 10.1137/1.9780898719505 DOI: 10.1137/1.9780898719505. open in new tab
  11. Trottenberg, U.; Oosterlee, C. W.; Schüller, A. Multigrid ; Academic Press, 2001.
  12. Merrick, M. P.; Iyer, K. A.; Beck, T. L. Multigrid Method for Electrostatic Computa- tions in Numerical Density Functional Theory. J. Phys. Chem. 1995, 99, 12478-12482. open in new tab
  13. Fattebert, J.-L.; Gygi, F. Density functional theory for efficient ab initio molecular dynamics simulations in solution. J. Comput. Chem. 2002, 23, 662-666. open in new tab
  14. Dabo, I.; Kozinsky, B.; Singh-Miller, N. E.; Marzari, N. Electrostatics in periodic boundary conditions and real-space corrections. Phys. Rev. B 2008, 77, 115139. open in new tab
  15. Sánchez, V. M.; Sued, M.; Scherlis, D. A. First-principles molecular dynamics simula- tions at solid-liquid interfaces with a continuum solvent. J. Chem. Phys. 2009, 131, 174108. open in new tab
  16. Dziedzic, J.; Helal, H. H.; Skylaris, C.-K.; Mostofi, A. A.; Payne, M. C. Minimal parameter implicit solvent model for ab initio electronic-structure calculations. EPL 2011, 95, 43001. open in new tab
  17. Dziedzic, J.; Fox, S. J.; Fox, T.; Tautermann, C. S.; Skylaris, C.-K. Large-scale DFT calculations in implicit solvent-A case study on the T4 lysozyme L99A/M102Q pro- tein. Int. J. Quantum Chem. 2013, 113, 771-785. open in new tab
  18. Garcia-Ratés, M.; López, N. Multigrid-Based Methodology for Implicit Solvation Mod- els in Periodic DFT. J. Chem. Theory Comput. 2016, 12, 1331-1341. open in new tab
  19. Briggs, E. L.; Sullivan, D. J.; Bernholc, J. Large-scale electronic-structure calculations with multigrid acceleration. Phys. Rev. B 1995, 52, R5471-R5474. open in new tab
  20. Briggs, E. L.; Sullivan, D. J.; Bernholc, J. Real-space multigrid-based approach to large-scale electronic structure calculations. Phys. Rev. B 1996, 54, 14362-14375. open in new tab
  21. Beck, T. L.; Iyer, K. A.; Merrick, M. P. Multigrid methods in density functional theory. Int. J. Quantum Chem. 1997, 61, 341-348. open in new tab
  22. Bernholc, J.; Hodak, M.; Lu, W. Recent developments and applications of the real-space multigrid method. J. Phys.: Condens. Matter 2008, 20, 294205. open in new tab
  23. Fattebert, J.-L.; Gygi, F. First-principles molecular dynamics simulations in a contin- uum solvent. Int. J. Quantum Chem. 2003, 93, 139-147. open in new tab
  24. Scherlis, D. A.; Fattebert, J.-L.; Gygi, F.; Cococcioni, M.; Marzari, N. A unified elec- trostatic and cavitation model for first-principles molecular dynamics in solution. J. Chem. Phys. 2006, 124, 074103. open in new tab
  25. Andreussi, O.; Dabo, I.; Marzari, N. Revised self-consistent continuum solvation in electronic-structure calculations. J. Chem. Phys. 2012, 136, 064102. open in new tab
  26. Fisicaro, G.; Genovese, L.; Andreussi, O.; Marzari, N.; Goedecker, S. A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments. J. Chem. Phys. 2016, 144, 014103. open in new tab
  27. Arfken, G. B.; Weber, H. J.; Harris, F. E. Mathematical Methods for Physicists: A Comprehensive Guide; Academic Press, 2011. open in new tab
  28. We have adopted the nomenclature used in Ref. 25 for the variants of the Poisson equation. The standard (Eq. 1) and generalized (Eq. 7) forms of the equation are also sometimes referred to as the "homogeneous" and "nonhomogeneous" Poisson equations, respectively, where the (non-)homogeneity of the equation refers to the structure of the dielectric permittivity, ε(r). We eschew this terminology because it is potentially ambiguous-the descriptor "homogeneous" has a specific and different meaning when describing differential equations. Note that the generalized Poisson equation is also sometimes referred to as the "variable-coefficient Poisson equation" (see, for example, Ref. 81). open in new tab
  29. Fogolari, F.; Brigo, A.; Molinari, H. The Poisson-Boltzmann equation for biomolecular electrostatics: a tool for structural biology. J. Mol. Recognit. 2002, 15, 377-392. open in new tab
  30. Lu, B.; Zhou, Y.; Holst, M.; McCammon, J. Recent progress in numerical methods for the Poisson-Boltzmann equation in biophysical applications. Commun. Comput. Phys. 2008, 3, 973-1009.
  31. Grochowski, P.; Trylska, J. Continuum molecular electrostatics, salt effects, and counte- rion binding-A review of the Poisson-Boltzmann theory and its modifications. Biopoly- mers 2008, 89, 93-113. open in new tab
  32. Ringe, S.; Oberhofer, H.; Hille, C.; Matera, S.; Reuter, K. Function-Space-Based Solu- tion Scheme for the Size-Modified Poisson-Boltzmann Equation in Full-Potential DFT. open in new tab
  33. J. Chem. Theory Comput. 2016, 12, 4052-4066. open in new tab
  34. Ringe, S.; Oberhofer, H.; Reuter, K. Transferable ionic parameters for first-principles Poisson-Boltzmann solvation calculations: Neutral solutes in aqueous monovalent salt solutions. J. Chem. Phys. 2017, 146, 134103. open in new tab
  35. Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, UK, 1992.
  36. Payne, M. C.; Teter, M. P.; Allan, D. C.; Arias, T. A.; Joannopoulos, J. D. Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients. Rev. Mod. Phys. 1992, 64, 1045-1097. open in new tab
  37. Hine, N. D. M.; Dziedzic, J.; Haynes, P. D.; Skylaris, C.-K. Electrostatic interactions in finite systems treated with periodic boundary conditions: Application to linear-scaling density functional theory. J. Chem. Phys. 2011, 135, 204103. open in new tab
  38. Tomasi, J.; Mennucci, B.; Cammi, R. Quantum Mechanical Continuum Solvation Mod- els. Chem. Rev. 2005, 105, 2999-3094. open in new tab
  39. York, D. M.; Karplus, M. A Smooth Solvation Potential Based on the Conductor-Like Screening Model. J. Phys. Chem. A 1999, 103, 11060-11079. open in new tab
  40. Fisicaro, G.; Genovese, L.; Andreussi, O.; Mandal, S.; Nair, N. N.; Marzari, N.; Goedecker, S. Soft-Sphere Continuum Solvation in Electronic-Structure Calculations. open in new tab
  41. J. Chem. Theory Comput. 2017, 13, 3829-3845. open in new tab
  42. Skylaris, C.-K.; Haynes, P. D.; Mostofi, A. A.; Payne, M. C. Implementation of linear- scaling plane wave density functional theory on parallel computers. Phys. Status Solidi B 2006, 243, 973-988. open in new tab
  43. Hine, N. D. M.; Haynes, P. D.; Mostofi, A. A.; Skylaris, C. K.; Payne, M. C. Linear- scaling density-functional theory with tens of thousands of atoms: Expanding the scope and scale of calculations with ONETEP. Comput. Phys. Commun. 2009, 180, 1041- 1053. open in new tab
  44. Wilkinson, K. A.; Hine, N. D. M.; Skylaris, C.-K. Hybrid MPI-OpenMP Parallelism in the ONETEP Linear-Scaling Electronic Structure Code: Application to the Delamina- tion of Cellulose Nanofibrils. J. Chem. Theory Comput. 2014, 10, 4782-4794. open in new tab
  45. Collatz, L. The Numerical Treatment of Differential Equations; open in new tab
  46. Springer Berlin Heidel- berg: Berlin, Heidelberg, 1960; DOI: 10.1007/978-3-642-88434-4. open in new tab
  47. Schaffer, S. Higher order multigrid methods. Math. Comp. 1984, 43, 89-115, S1. open in new tab
  48. Wesseling, P.; Oosterlee, C. W. Geometric multigrid with applications to computational fluid dynamics. Journal of Computational and Applied Mathematics 2001, 128, 311- 334. open in new tab
  49. Stüben, K. A review of algebraic multigrid. Journal of Computational and Applied Mathematics 2001, 128, 281-309. open in new tab
  50. Chow, E.; Falgout, R. D.; Hu, J. J.; Tuminaro, R. S.; Yang, U. M. In Parallel Processing for Scientific Computing; open in new tab
  51. Heroux, M. A., Raghavan, P., Simon, H. D., Eds.; SIAM series on Software, Environments and Tools; SIAM, 2006. open in new tab
  52. Zhang, J. Acceleration of five-point red-black Gauss-Seidel in multigrid for Poisson equation. Appl. Math. Comput. 1996, 80, 73-93.
  53. Terboven, C.; an Mey, D.; Schmidl, D.; Jin, H.; Reichstein, T. Data and Thread Affinity in OpenMP Programs. Proceedings of the 2008 Workshop on Memory Access on Future Processors: A Solved Problem? New York, NY, USA, 2008; pp 377-384. open in new tab
  54. Gropp, W.; Torsten, H.; Thakur, R.; Lusk, E. Using Advanced MPI: Modern Features of the Message-Passing Interface; MIT Press, 2014. open in new tab
  55. Chapman, B.; Jost, G.; van der Pas, R. Using OpenMP: Portable Shared Memory Parallel Programming; 2014. open in new tab
  56. Holst, M. J.; Saied, F. Numerical solution of the nonlinear Poisson-Boltzmann equation: Developing more robust and efficient methods. J. Comput. Chem. 1995, 16, 337-364. open in new tab
  57. Anton, L.; Womack, J. C.; Dziedzic, J. DL MG multigrid solver. 2017; https: //ccpforge.cse.rl.ac.uk/gf/project/dl-mg/.
  58. Wolfram Research Inc, Mathematica, Version 11.2 ; 2017. open in new tab
  59. Anton, L.; Dziedzic, J.; Skylaris, C.-K.; Probert, M. Multi- grid solver module for ONETEP, CASTEP and other codes; 2013;
  60. Clark, S. J.; Segall, M. D.; Pickard, C. J.; Hasnip, P. J.; Probert, M. I. J.; Refson, K.; Payne, M. C. First principles methods using CASTEP. Z. Kristallogr. 2005, 220, 567- 570. open in new tab
  61. Parrish, R. M.; Burns, L. A.; Smith, D. G. A.; Simmonett, A. C.; DePrince, A. E.; Hohenstein, E. G.; Bozkaya, U.; Sokolov, A. Y.; Di Remigio, R.; Richard, R. M.; Gonthier, J. F.; James, A. M.; McAlexander, H. R.; Kumar, A.; Saitow, M.; Wang, X.; Pritchard, B. P.; Verma, P.; Schaefer, H. F.; Patkowski, K.; King, R. A.; Valeev, E. F.; Evangelista, F. A.; Turney, J. M.; Crawford, T. D.; Sherrill, C. D. Psi4 1.1: An Open- Source Electronic Structure Program Emphasizing Automation, Advanced Libraries, and Interoperability. J. Chem. Theory Comput. 2017, 13, 3185-3197. open in new tab
  62. Womack, J. C.; Anton, L.; Dziedzic, J.; Hasnip, P. J.; Probert, M. I. J.; Skylaris, C.- K. Implementation and optimisation of advanced solvent modelling functionality in CASTEP and ONETEP ; 2017; http://www.archer.ac.uk/community/eCSE/eCSE07- 06/eCSE07-06.php. open in new tab
  63. Howard, J. C.; Womack, J. C.; Dziedzic, J.; Skylaris, C.-K.; Pritchard, B. P.; Craw- ford, T. D. Electronically Excited States in Solution via a Smooth Dielectric Model Combined with Equation-of-Motion Coupled Cluster Theory. J. Chem. Theory Com- put. 2017, 13, 5572-5581. open in new tab
  64. Skylaris, C.-K.; Mostofi, A. A.; Haynes, P. D.; Diéguez, O.; Payne, M. C. Nonorthogonal generalized Wannier function pseudopotential plane-wave method. Phys. Rev. B 2002, 66, 035119. open in new tab
  65. Mostofi, A. A.; Haynes, P. D.; Skylaris, C.-K.; Payne, M. C. Preconditioned iterative (68) Although the grid dimensions used for pbez and erf eps differ by a single grid point in the x and y directions (section 4.1.1), the resulting difference in overall number of grid points is negligible, allowing direct comparison of the timings for the grid sizes used in each test case. open in new tab
  66. Amdahl, G. M. Validity of the Single Processor Approach to Achieving Large Scale Computing Capabilities. Proceedings of the April 18-20, 1967, Spring Joint Computer Conference. New York, NY, USA, 1967; pp 483-485. open in new tab
  67. Hill, M. D.; Marty, M. R. Amdahl's Law in the Multicore Era. Computer 2008, 41, 33-38. open in new tab
  68. Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865-3868. open in new tab
  69. Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple [Phys. Rev. Lett. 77, 3865 (1996)]. open in new tab
  70. Phys. Rev. Lett. 1997, 78, 1396-1396. open in new tab
  71. Bennett, J. W. Discovery and Design of Functional Materials: Integration of Database Searching and First Principles Calculations. Physics Procedia 2012, 34, 14-23. open in new tab
  72. A sulfur pseudopotential from a suite of pseudopotentials generated by K. Refson to supplement the Rappe-Bennett library was also used in calculations on the T4 lysozyme complex reported in section 4.2. ONETEP-and CASTEP-compatible versions of the Rappe-Bennett library and K. Refson's supplementary set of pseudopotentials are avail- able to download from the CASTEP project page on CCPForge. 82 open in new tab
  73. ARCHER is the UK's national supercomputing service, based on a Cray XC30 super- computer. At the time of writing, ARCHER consists of 4920 nodes connected with an Aries interconnect. Each node contains 2 × 12 core Intel Ivy Bridge processors, with standard nodes sharing 64 GiB between the two processors. For further information, see https://www.archer.ac.uk/. open in new tab
  74. Frigo, M.; Johnson, S. G. The Design and Implementation of FFTW3. Proc. IEEE 2005, 93, 216-231. open in new tab
  75. Cray Inc., XC TM Series Programming Environment User Guide, s-2529-17.05 ed.; 2017; https://pubs.cray.com/content/S-2529/17.05/xctm-series-programming- environment-user-guide-1705-s-2529. open in new tab
  76. Gholami, A.; Malhotra, D.; Sundar, H.; Biros, G. FFT, FMM, or Multigrid? A com- parative Study of State-Of-the-Art Poisson Solvers for Uniform and Nonuniform Grids in the Unit Cube. SIAM J. Sci. Comput. 2016, 38, C280-C306. open in new tab
  77. Verga, L. G.; Aarons, J.; Sarwar, M.; Thompsett, D.; Russell, A. E.; Skylaris, C.-K. Effect of graphene support on large Pt nanoparticles. Phys. Chem. Chem. Phys. 2016, 18, 32713-32722. open in new tab
  78. Nagel, J. R. Numerical Solutions to Poisson Equations Using the Finite-Difference Method [Education Column]. open in new tab
  79. IEEE Antenn. Propag. M. 2014, 56, 209-224. open in new tab
  80. CASTEP project page on CCPForge. https://ccpforge.cse.rl.ac.uk/gf/ project/castep/.
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