This article is the second in a series of two papers concerning the mathematical study of a boundary integral equation of the second kind that describes the interaction of N dielectric spherical particles undergoing mutual polarisation. The first article presented the numerical analysis of the Galerkin method used to solve this boundary integral equation and derived N-independent convergence rates for the induced surface charges and total electrostatic energy. The current article will focus on computational aspects of the algorithm. We provide a convergence analysis of the iterative method used to solve the underlying linear system and show that the number of liner solver iterations required to obtain a solution is independent of N. Additionally, we present two linear scaling solution strategies for the computation of the approximate induced surface charges. Finally, we consider a series of numerical experiments designed to validate our theoretical results and explore the dependence of the numerical errors and computational cost of solving the underlying linear system on different system parameters.
Keywords: Boundary integral equations, complexity analysis, linear scaling, $$-body problem, polarisation
@article{M2AN_2021__55_S1_S625_0,
author = {Bramas, B\'erenger and Hassan, Muhammad and Stamm, Benjamin},
title = {An integral equation formulation of the $N$-body dielectric spheres problem. {Part} {II:} complexity analysis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {S625--S651},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {Suppl\'ement},
doi = {10.1051/m2an/2020055},
mrnumber = {4221318},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020055/}
}
TY - JOUR AU - Bramas, Bérenger AU - Hassan, Muhammad AU - Stamm, Benjamin TI - An integral equation formulation of the $N$-body dielectric spheres problem. Part II: complexity analysis JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - S625 EP - S651 VL - 55 IS - Supplément PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020055/ DO - 10.1051/m2an/2020055 LA - en ID - M2AN_2021__55_S1_S625_0 ER -
%0 Journal Article %A Bramas, Bérenger %A Hassan, Muhammad %A Stamm, Benjamin %T An integral equation formulation of the $N$-body dielectric spheres problem. Part II: complexity analysis %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P S625-S651 %V 55 %N Supplément %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020055/ %R 10.1051/m2an/2020055 %G en %F M2AN_2021__55_S1_S625_0
Bramas, Bérenger; Hassan, Muhammad; Stamm, Benjamin. An integral equation formulation of the $N$-body dielectric spheres problem. Part II: complexity analysis. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S625-S651. doi: 10.1051/m2an/2020055
[1] , , , , and , Task-based FMM for multicore architectures. SIAM J. Sci. Comput. 36 (2014) C66–C93. | MR | Zbl | DOI
[2] , An efficient program for many-body simulation. SIAM J. Sci. Stat. Comput. 6 (1985) 85–103. | MR | DOI
[3] and , A hierarchical force-calculation algorithm . Nature 324 (1986) 446–449. | DOI
[4] and , Dielectric effects in the self-assembly of binary colloidal aggregates. Phys. Rev. Lett. 113 (2014) 017801. | DOI
[5] , and , Efficient and accurate simulation of dynamic dielectric objects. J. Chem. Phys. 140 (2014) 064903. | DOI
[6] , , , , , and , ScalFMM: a generic parallel fast multipole library. In: SIAM Conference on Computational Science and Engineering, 2015.
[7] and , Comparison of treecodes for computing electrostatic potentials in charged particle systems with disjoint targets and sources. J. Comput. Chem. 34 (2013) 2159–2167. | DOI
[8] , , and , Direct measurement of three-body interactions amongst charged colloids. Phys. Rev. Lett. 92 (2004) 078301. | DOI
[9] , and , A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155 (1999) 468–498. | MR | Zbl | DOI
[10] and , Many-body electrostatic interactions in electrorheological fluids. Phys. Rev. E 48 (1993) 2721. | DOI
[11] , A very fast and momentum-conserving tree code. Astrophys. J. Lett. 536 (2000) L39. | DOI
[12] , , and , Many-body interactions in colloidal suspensions. J. Phys.: Condens. Matter 15 (2002) S263.
[13] , , and , Numerical techniques for large cosmological -body simulations. Astrophys. J. Suppl. Ser. 57 (1985) 241–260. | DOI
[14] and , Geometric aspects of the theory of Krylov subspace methods. Acta Numer. 10 (2001) 251–312. | MR | Zbl | DOI
[15] , and , Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20 (1983) 345–357. | MR | Zbl | DOI
[16] , Iterative methods for large, sparse, nonsymmetric systems of linear equations, Ph.D. thesis, Yale University New Haven, CO (1982).
[17] , Polynomial Based Iteration Methods for Symmetric Linear Systems. In: Vol. 68 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA (2011). | MR | Zbl
[18] , Perturbative many-body expansion for electrostatic energy and field for system of polarizable charged spherical ions in a dielectric medium. J. Chem. Phys. 141 (2014) 034115. | DOI
[19] , , and , A hybrid method for systems of closely spaced dielectric spheres and ions. SIAM J. Sci. Comput. 38 (2016) B375–B395. | MR | DOI
[20] and , A treecode-accelerated boundary integral Poisson-Boltzmann solver for electrostatics of solvated biomolecules. J. Comput. Phys. 247 (2013) 62–78. | MR | DOI
[21] , Iterative Methods for Solving Linear Systems. In: Vol. 17 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA (1997). | MR | Zbl
[22] , The rapid evaluation of potential fields in particle systems, ACM Distinguished Dissertations. MIT Press, Cambridge, MA (1988). | MR | Zbl
[23] , The numerical solution of the -body problem. Comput. Phys. 4 (1990) 142–152. | DOI
[24] and , A fast algorithm for particle simulations. J. Comput. Phys. 73 (1987) 325–348. | MR | Zbl | DOI
[25] , , , and , Electrostatic self-assembly of macroscopic crystals using contact electrification. Nat. Mater. 2 (2003) 241–245. | DOI
[26] , Mathematical analysis of boundary integral equations and domain decomposition methods with applications in polarisable electrostatics. Ph.D. thesis. RWTH Aachen University (2020).
[27] and , An integral equation formulation of the -body dielectric spheres problem. Part I: numerical analysis. ESAIM:M2AN (2020). https://doi.org/10.1051/m2an/2020030. | MR | Numdam
[28] and , A linear scaling in accuracy numerical method for computing the electrostatic forces in the -body dielectric spheres problem. Commun. Comput. Phys. Preprint arXiv:2002.01579 (2020)). | MR
[29] and , Computer Simulation Using Particles. CRC Press (1988). | Zbl | DOI
[30] , , , , , , , , , and , Improvements to the APBS biomolecular solvation software suite. Protein Sci. 27 (2018) 112–128. | DOI
[31] , and , Multi-level adaptive particle mesh (MLAPM): a c code for cosmological simulations. Mon. Not. R. Astron. Soc. 325 (2001) 845–864. | DOI
[32] , and , A Cartesian treecode for screened coulomb interactions. J. Comput. Phys. 228 (2009) 3858–3868. | MR | Zbl | DOI
[33] , , and , Interaction forces between colloidal particles in liquid: theory and experiment. Adv. Colloid Interface Sci. 134 (2007) 151–166. | DOI
[34] and , Convergence analysis of Krylov subspace methods. GAMM-Mitt. 27 (2004) 153–173. | MR | Zbl | DOI
[35] , , , , and , The effect of like-charge attraction on aerosol growth in the atmosphere of Titan. Icarus 291 (2017) 245–253. | DOI
[36] , , , , and , An integral equation approach to calculate electrostatic interactions in many-body dielectric systems. J. Comput. Phys. 371 (2018) 712–731. | MR | DOI
[37] , , , and , Dynamic simulations of many-body electrostatic self-assembly. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 376 (2018) 20170143. | DOI
[38] , Electrostatics in the presence of spherical dielectric discontinuities. J. Chem. Phys. 128 (2008) 214505. | DOI
[39] and , An analytical electrostatic model for salt screened interactions between multiple proteins. J. Chem. Theory Comput. 2 (2006) 541–555. | DOI
[40] , and , Electrostatic self-assembly of polystyrene microspheres by using chemically directed contact electrification. Angew. Chem. Int. Ed. 46 (2007) 206–209. | DOI
[41] , and , Many-body electrostatic forces between colloidal particles at vanishing ionic strength. Phys. Rev. Lett. 103 (2009) 138301. | DOI
[42] , Image charges in spherical geometry: application to colloidal systems. J. Chem. Phys. 117 (2002) 11062–11074. | DOI
[43] , , and , Optimized M2L kernels for the Chebyshev interpolation based fast multipole method. Preprint arXiv:1210.7292(2012).
[44] , and , How fast are nonsymmetric matrix iterations? SIAM J. Matrix Anal. App. 13 (1992) 778–795. | MR | Zbl | DOI
[45] , Giant polarization in high polymers. J. Electron. Mater. 15 (1986) 201–203. | DOI
[46] , and , Image method for induced surface charge from many-body system of dielectric spheres. J. Chem. Phys. 145 (2016) 124903. | DOI
[47] , , , , and , A theory of interactions between polarizable dielectric spheres. J. Colloid Interface Sci. 469 (2016) 237–241. | DOI
[48] , Krylov subspace methods for solving large unsymmetric linear systems. Math. Comput. 37 (1981) 105–126. | MR | Zbl | DOI
[49] , Iterative Methods for Sparse Linear Systems, 2nd edition. Society for Industrial and Applied Mathematics, Philadelphia, PA (2003). | MR | Zbl
[50] and , GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7 (1986) 856–869. | MR | Zbl | DOI
[51] and , Boundary Element Methods. In: Vol. 39 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin-Heidelberg (2011). | MR | Zbl | DOI
[52] , , , and , Structural diversity in binary nanoparticle superlattices. Nature 439 (2006) 55–59. | DOI
[53] , Electrostatic interaction in the presence of dielectric interfaces and polarization-induced like-charge attraction. Phys. Rev. E 87 (2013) 013307. | DOI
[54] and , Calculating the bimolecular rate of protein–protein association with interacting crowders. J. Chem. Theory Comput. 9 (2013) 2481–2489. | DOI
Cité par Sources :
