Hierarchical Interpolative Factorization for Self Green’s Function in 3D Modified Poisson-Boltzmann Equations

doi:10.1007/s42967-023-00352-z

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (2): 536-561.doi: 10.1007/s42967-023-00352-z

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Hierarchical Interpolative Factorization for Self Green’s Function in 3D Modified Poisson-Boltzmann Equations

Yihui Tu¹, Zhenli Xu², Haizhao Yang³

1 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China;
2 School of Mathematical Sciences, MOE-LSC and CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, China;
3 Department of Mathematics and Department of Computer Science, University of Maryland, College Park, MD 20742, USA

Received:2023-07-15 Revised:2023-11-14 Accepted:2023-11-20 Online:2025-06-20 Published:2025-04-21
Supported by:
Y. T. and Z. X. acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 12071288 and 12325113), the Science and Technology Commission of Shanghai Municipality of China (Grant No. 21JC1403700), and Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010403). H. Y. thanks the support of the US National Science Foundation under awards DMS-2244988 and DMS-2206333.

Abstract

Abstract: The modified Poisson-Boltzmann (MPB) equations are often used to describe the equilibrium particle distribution of ionic systems. In this paper, we propose a fast algorithm to solve the MPB equations with the self Green’s function as the self-energy in three dimensions, where the solution of the self Green’s function poses a computational bottleneck due to the requirement of solving a high-dimensional partial differential equation. Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green’s function, building upon our previous result of two dimensions. This approach yields an algorithm with a complexity of O(N log N) by strategically leveraging the locality and low-rank characteristics of the corresponding operators. Additionally, the theoretical O(N) complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction. Extensive numerical results are conducted to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.

Key words: Selected inversion, Hierarchical interpolative factorization, Linear scaling, Self Green’s function, Modified Poisson-Boltzmann (MPB) equations

CLC Number:

Yihui Tu, Zhenli Xu, Haizhao Yang. Hierarchical Interpolative Factorization for Self Green’s Function in 3D Modified Poisson-Boltzmann Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 536-561.

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References

1. Bazant, M.Z., Storey, B.D., Kornyshev, A.A.: Double layer in ionic liquids: overscreening versus crowding. Phys. Rev. Lett. 106(4), 046102 (2011)
2. Boroudjerdi, H., Kim, Y.-W., Naji, A., Netz, R.R., Schlagberger, X., Serr, A.: Statics and dynamics of strongly charged soft matter. Phys. Rep. 416, 129–199 (2005)
3. Borukhov, I., Andelman, D., Orland, H.: Steric effects in electrolytes: a modified Poisson-Boltzmann equation. Phys. Rev. Lett. 79(3), 435–438 (1998)
4. Brandt, D.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 138, 333– 390 (1977)
5. Chapman, D.L.: A contribution to the theory of electrocapillarity. Phil. Mag. 25, 475–481 (1913)
6. Cheng, H., Gimbutas, Z., Martinsson, P., Rokhlin, V.: On the compression of low rank matrices. SIAM J. Sci. Comput. 26(4), 1389–1404 (2005)
7. Corry, B., Kuyucak, S., Chung, S.H.: Dielectric self-energy in Poisson-Boltzmann and PoissonNernst-Planck models of ion channels. Biophys. J. 84(6), 3594–3606 (2003)
8. Daiguji, H., Yang, P., Majumdar, A.: Ion transport in nanofluidic channels. Nano Lett. 4(1), 137– 142 (2004)
9. Duff, I.S., Reid, J.K.: The multifrontal solution of indefinite sparse symmetric linear. ACM Trans. Math. Softw. 9(3), 302–325 (1983)
10. George, A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10(2), 345– 363 (1973)
11. Gillman, A., Martinsson, P.G.: A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method. SIAM J. Sci. Comput. 36(4), 2023–2046 (2013)
12. Gillman, A., Martinsson, P.G.: An O(N) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads. Adv. Comput. Math. 40(4), 773–796 (2014)
13. Gouy, G.: Constitution of the electric charge at the surface of an electrolyte. J. Phys. 9, 457–468 (1910)
14. Grasedyck, L., Kriemann, R., Borne, S.L.: Domain-decomposition based H-LU preconditioners. Numer. Math. 112(4), 565–600 (2009)
15. Ho, K.L., Ying, L.: Hierarchical interpolative factorization for elliptic operators: differential equations. Comm. Pure Appl. Math. 69(8), 1415–1451 (2016)
16. Ho, K.L., Ying, L.: Hierarchical interpolative factorization for elliptic operators: integral equations. Comm. Pure Appl. Math. 69(7), 1314–1353 (2016)
17. Ji, L., Liu, P., Xu, Z., Zhou, S.: Asymptotic analysis on dielectric boundary effects of modified Poisson-Nernst-Planck equations. SIAM J. Appl. Math. 78, 1802–1822 (2018)
18. Kenneth, L.H.: FLAM: fast linear algebra in MATLAB—algorithms for hierarchical matrices. J. Open Source Softw. 5, 1906 (2020)
19. Liljeström, V., Seitsonen, J., Kostiainen, M.: Electrostatic self-assembly of soft matter nanoparticle cocrystals with tunable lattice parameters. ACS Nano 9(11), 11278–11285 (2015)
20. Lin, L., Lu, J., Ying, L., Car, R., E, W.: Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems. Commun. Math. Sci. 7(3), 755–777 (2009)
21. Lin, L., Yang, C., Lu, J., Ying, L., E, W.: A fast parallel algorithm for selected inversion of structured sparse matrices with application to 2D electronic structure calculations. SIAM J. Sci. Comput. 33(3), 1329–1351 (2011)
22. Lin, L., Yang, C., Meza, J.C., Lu, J., Y, L., E, W.: SelInv–an algorithm for selected inversion of a sparse symmetric matrix. ACM Trans. Math. Softw. 37(4), 1–19 (2011)
23. Liu, C., Wang, C., Wise, S., Yue, X., Zhou, S.: A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system. Math. Comput. 90, 2071–2106 (2021)
24. Liu, H., Wang, Z.: A free energy satisfying finite difference method for Poisson-Nernst-Planck equations. J. Comput. Phys. 268(2), 363–376 (2014)
25. Liu, J.-L., Eisenberg, R.S.: Molecular mean-field theory of ionic solutions: a Poisson-NernstPlanck-Bikerman model. Entropy 22(5), 550 (2020)
26. Liu, J.W.H.: The multifrontal method for sparse matrix solution: theory and practice. SIAM Rev. 34(1), 82–109 (1992)
27. Liu, P., Ji, X., Xu, Z.: Modified Poisson-Nernst-Planck model with accurate Coulomb correlation in variable media. SIAM J. Appl. Math. 78, 226–245 (2018)
28. Ma, M., Xu, Z.: Self-consistent field model for strong electrostatic correlations and inhomogeneous dielectric media. J. Chem. Phys. 141(24), 244903 (2014)
29. Ma, M., Xu, Z., Zhang, L.: Modified Poisson-Nernst-Planck model with Coulomb and hard-sphere correlations. SIAM J. Appl. Math. 81, 1645–1667 (2021)
30. Netz, R.R., Orland, H.: Beyond Poisson-Boltzmann: fluctuation effects and correlation functions. Eur. Phys. J. E 1(2), 203–214 (2000)
31. Netz, R.R., Orland, H.: Variational charge renormalization in charged systems. Eur. Phys. J. E 11(3), 301–311 (2003)
32. Podgornik, R.: Electrostatic correlation forces between surfaces with surface specific ionic interactions. J. Chem. Phys. 91, 5840–5849 (1989)
33. Schmitz, P.G., Ying, L.: A fast direct solver for elliptic problems on general meshes in 2D. J. Comput. Phys. 231(4), 1314–1338 (2012)
34. Schoch, R.B., Han, J., Renaud, P.: Transport phenomena in nanofluidics. Rev. Mod. Phys. 80, 839– 883 (2008)
35. Tu, Y., Pang, Q., Yang, H., Xu, Z.: Linear-scaling selected inversion based on hierarchical interpolative factorization for self Green’s function for modified Poisson-Boltzmann equation in two dimensions. J. Comput. Phys. 461, 110893 (2022)
36. Wang, Z.-G.: Fluctuation in electrolyte solutions: the self energy. Phys. Rev. E 81, 021501 (2010)
37. Xia, J., Chandrasekaran, S., Gu, M., Li, X.: Superfast multifrontal method for large structured linear systems of equations. SIAM J. Matrix Anal. Appl. 31(3), 1382–1411 (2009)
38. Xia, J., Xi, Y., Cauley, S., Balakrishnan, V.: Fast sparse selected inversion. SIAM J. Matrix Anal. Appl. 36(3), 1283–1314 (2015)
39. Xu, Z., Maggs, A.C.: Solving fluctuation-enhanced Poisson-Boltzmann equations. J. Comput. Phys. 36(3), 310–322 (2014)
40. Xu, Z., Ma, M., Liu, P.: Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches. Phys. Rev. E 90(1), 013307 (2014)

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Hierarchical Interpolative Factorization for Self Green’s Function in 3D Modified Poisson-Boltzmann Equations

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