Prospects of tensor-based numerical modeling of the collective electrostatics in many-particle systems. (English) Zbl 1469.82010
Summary: Recently the rank-structured tensor approach suggested a progress in the numerical treatment of the long-range electrostatics in many-particle systems and the respective interaction energy and forces. In this paper, we outline the prospects for tensor-based numerical modeling of the collective electrostatic potential on lattices and in many-particle systems of general type. Our approach, initially introduced for the rank-structured grid-based calculation of the interaction potentials on 3D lattices is generalized here to the case of many-particle systems with variable charges placed on \({{L}^{{ \otimes d}}}\) lattices and discretized on fine \({{n}^{{ \otimes d}}}\) Cartesian grids for arbitrary dimension \(d\). As a result, the interaction potential is represented in a parametric low-rank canonical format in \(O(dLn)\) complexity. The total interaction energy can be then calculated in \(O(dL)\) operations. Electrostatics in large bio-molecular systems is discretized on a fine \({{n}^{{ \otimes 3}}}\) grid by using the novel range-separated (RS) tensor format, which maintains the long-range part of the 3D collective potential of a many-body system in a parametric low-rank form in \(O(n)\)-complexity. We show how the energy and force field can be easily recovered by using the already precomputed electric field in the low-rank RS format. The RS tensor representation of the discretized Dirac delta enables the construction of the efficient energy preserving (conservative) regularization scheme for solving the 3D elliptic partial differential equations with strongly singular right-hand side arising in scientific computing. We conclude that the rank-structured tensor-based approximation techniques provide the promising numerical tools for applications to many-body dynamics in bio-sciences, protein docking and classification problems, for low-parametric interpolation of scattered data in data science, as well as in machine learning in many dimensions.
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
| 82D60 | Statistical mechanics of polymers |
| 81V70 | Many-body theory; quantum Hall effect |
| 81V55 | Molecular physics |
| 15A69 | Multilinear algebra, tensor calculus |
| 78A30 | Electro- and magnetostatics |
| 78M16 | Multipole methods applied to problems in optics and electromagnetic theory |
| 82M22 | Spectral, collocation and related (meshless) methods applied to problems in statistical mechanics |
Keywords:
Coulomb potential; Slater potential; long-range many-particle interactions; low-rank tensor decomposition; range-separated tensor formats; summation of electrostatic potentials; energy and force calculationsSoftware:
ICALABReferences:
| [1] | Khoromskaia, V.; Khoromskij, B. N., Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation, Comput. Phys. Commun., 185, 3162-3174 (2014) · Zbl 1360.78016 · doi:10.1016/j.cpc.2014.08.015 |
| [2] | Khoromskaia, V.; Khoromskij, B. N., Fast tensor method for summation of long-range potentials on 3D lattices with defects, Numer. Linear Algebra Appl., 23, 249-271 (2016) · Zbl 1413.65141 · doi:10.1002/nla.2023 |
| [3] | Benner, P.; Khoromskaia, V.; Khoromskij, B. N., Range-separated tensor format for many-particle modeling, SIAM J. Sci. Comput. A, 40, 1034-1062 (2018) · Zbl 1446.65202 · doi:10.1137/16M1098930 |
| [4] | Khoromskij, B. N., Range-separated tensor decomposition of the discretized Dirac delta and elliptic operator inverse, J. Comput. Phys., 401, 108998 (2020) · Zbl 1453.65035 · doi:10.1016/j.jcp.2019.108998 |
| [5] | M. J. Holst, PhD Thesis (Numerical Computing Group, Univ. of Illinois, Urbana-Champaign, IL, USA, 1994). |
| [6] | Jackson, J. D., Classical Electrodynamics (1999), New York: Wiley, New York · Zbl 0920.00012 |
| [7] | Cances, E.; Mennucci, B.; Tomasi, J., A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectrics, J. Chem. Phys., 107, 3032-3041 (1997) · doi:10.1063/1.474659 |
| [8] | Hünenberger, P. H.; McCammon, J. A., Effect of artificial periodicity in simulations of biomolecules under Ewald boundary conditions: A continuum electrostatics study, Biophys. Chem., 78, 69-88 (1999) · doi:10.1016/S0301-4622(99)00007-1 |
| [9] | Kaxiras, E., Atomic and Electronic Structure of Solids (2003), Cambridge: Cambridge Univ. Press, Cambridge · doi:10.1017/CBO9780511755545 |
| [10] | Lipparini, F.; Stamm, B.; Cances, E.; Maday, Y.; Mennucci, B., Domain decomposition for implicit solvation models, J. Chem. Theor. Comput., 139, 3637-3648 (2013) · doi:10.1021/ct400280b |
| [11] | Lu, B. Z.; Zhou, Y. C.; Holst, M. J.; McCammon, J. A., Recent progress in numerical methods for the Poisson-Boltzmann equation in biophysical applications, Commun. Comput. Phys., 3, 973-1009 (2008) · Zbl 1186.92005 |
| [12] | Lindgren, E. B.; Stace, A. J.; Polack, E.; Maday, Y.; Stamm, B.; Besley, E., An integral equation approach to calculate electrostatic interactions in many-body dielectric systems, J. Comput. Phys., 371, 712-731 (2018) · Zbl 1415.78002 · doi:10.1016/j.jcp.2018.06.015 |
| [13] | Khoromskij, B. N., Tensor Numerical Methods in Scientific Computing (2018), Berlin: De Gruyter, Berlin · Zbl 1405.65002 · doi:10.1515/9783110365917 |
| [14] | Khoromskaia, V.; Khoromskij, B. N., Tensor Numerical Methods in Quantum Chemistry (2018), Berlin: De Gruyter, Berlin · Zbl 1503.65003 · doi:10.1515/9783110365832 |
| [15] | L. De Lathauwer, PhD Thesis (Katholeke Univ., Leuven, 1997). |
| [16] | Comon, P., Mathematics in Signal Processing (2001), Oxford: Oxford Univ. Press, Oxford |
| [17] | Cichocki, A.; Amari, Sh., Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications (2002), Chichester: Wiley, Chichester · doi:10.1002/0470845899 |
| [18] | Smilde, A.; Bro, R.; Geladi, P., Multi-Way Analysis with Applications in the Chemical Sciences (2004), Chichester: Wiley, Chichester · doi:10.1002/0470012110 |
| [19] | Hackbusch, W., Tensor Spaces and Numerical Tensor Calculus (2012), Berlin: Springer, Berlin · Zbl 1244.65061 · doi:10.1007/978-3-642-28027-6 |
| [20] | Grasedyck, L.; Kressner, D.; Tobler, C., A literature survey of low-rank tensor approximation techniques, GAMM-Mitteilungen., 36, 53-78 (2013) · Zbl 1279.65045 · doi:10.1002/gamm.201310004 |
| [21] | Hackbusch, W.; Khoromskij, B. N., Low-rank Kronecker product approximation to multi-dimensional nonlocal operators: Part I. Separable approximation of multi-variate functions, Computing, 76, 177-202 (2006) · Zbl 1087.65049 · doi:10.1007/s00607-005-0144-0 |
| [22] | Gavrilyuk, I. P.; Hackbusch, W.; Khoromskij, B. N., Hierarchical tensor-product approximation to the inverse and related operators in high-dimensional elliptic problems, Computing, 74, 131-157 (2005) · Zbl 1071.65032 · doi:10.1007/s00607-004-0086-y |
| [23] | Khoromskij, B. N., Structured rank-(r_1, …, r_d) decomposition of function-related tensors in ℝ^d, Comput. Methods Appl. Math., 6, 194-220 (2006) · Zbl 1120.65052 · doi:10.2478/cmam-2006-0010 |
| [24] | De Lathauwer, L.; De Moor, B.; Vandewalle, J., A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21, 1253-1278 (2000) · Zbl 0962.15005 · doi:10.1137/S0895479896305696 |
| [25] | Oseledets, I. V.; Tyrtyshnikov, E. E., Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM J. Sci. Comput., 31, 3744-3759 (2009) · Zbl 1200.65028 · doi:10.1137/090748330 |
| [26] | Grasedyck, L., Hierarchical singular value decomposition of tensors, SIAM. J. Matrix Anal. Appl., 31, 2029-2054 (2010) · Zbl 1210.65090 · doi:10.1137/090764189 |
| [27] | Oseledets, I. V., Tensor-train decomposition, SIAM J. Sci. Comput., 33, 2295-2317 (2011) · Zbl 1232.15018 · doi:10.1137/090752286 |
| [28] | Khoromskij, B. N.; Khoromskaia, V., Multigrid tensor approximation of function related arrays, SIAM J. Sci. Comput., 31, 3002-3026 (2009) · Zbl 1197.65215 · doi:10.1137/080730408 |
| [29] | G. Heidel, V. Khoromskaia, B. Khoromskij, and V. Schulz, “Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints,” J. Comput. Phys. 424, 109865 (2021). · Zbl 07508470 |
| [30] | B. Schmitt, B. Khoromskij, V. Khoromskaia, and V. Schulz, “Tensor method for optimal control problems constrained by fractional 3D elliptic operator with variable coefficients” (2020). arXiv:2006.09314. |
| [31] | Litvinenko, A.; Keyes, D.; Khoromskaia, V.; Khoromskij, B. N.; Matthies, H. G., Tucker tensor analysis of Matern functions in spatial statistics, Comput. Methods Appl. Math., 19, 101-122 (2019) · Zbl 1420.60084 · doi:10.1515/cmam-2018-0022 |
| [32] | P. Benner, V. Khoromskaia, B. N. Khoromskij, C. Kweyu, and M. Stein, “Computing biomolecular electrostatics using a range-separated regularization for the Poisson-Boltzmann equation” (2019). arXiv:1901.09864v1. · Zbl 1466.65180 |
| [33] | Stenger, F., Numerical Methods Based on Sinc and Analytic Functions (1993), New York: Springer-Verlag, New York · Zbl 0803.65141 · doi:10.1007/978-1-4612-2706-9 |
| [34] | Bertoglio, C.; Khoromskij, B. N., Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels, Comput. Phys. Commun., 183, 904-912 (2012) · Zbl 1308.65213 · doi:10.1016/j.cpc.2011.12.016 |
| [35] | Braess, D., Nonlinear Approximation Theory (1986), Berlin: Springer-Verlag, Berlin · Zbl 0656.41001 · doi:10.1007/978-3-642-61609-9 |
| [36] | Khoromskij, B. N.; Khoromskaia, V.; Flad, H.-J., Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format, SIAM J. Sci. Comput., 33, 45-65 (2011) · Zbl 1227.65113 · doi:10.1137/090777372 |
| [37] | Ewald, P. P., Die Berechnung optische und elektrostatischer Gitterpotentiale, Ann. Phys., 369, 253-287 (1921) · JFM 48.0566.02 · doi:10.1002/andp.19213690304 |
| [38] | Darten, T.; York, D.; Pedersen, L., Particle mesh Ewald: An O(N logN) method for Ewald sums in large systems, J. Chem. Phys., 98, 10089-10092 (1993) · doi:10.1063/1.464397 |
| [39] | Pollock, E. L.; Glosli, J., Comments on P^3M, FMM, and the Ewald method for large periodic Coulombic systems, Comput. Phys. Commun., 95, 93-110 (1996) · Zbl 0921.65081 · doi:10.1016/0010-4655(96)00043-4 |
| [40] | Hunenberger, P. H., Lattice-sum methods for computing electrostatic interactions in molecular simulations, AIP Conf. Proc., 492, 17-83 (1999) |
| [41] | Greengard, L.; Rochlin, V., A fast algorithm for particle simulations, J. Comput. Phys., 73, 325-348 (1987) · Zbl 0629.65005 · doi:10.1016/0021-9991(87)90140-9 |
| [42] | Bernstein, F. C.; Koetzle, T. F.; Williams, G. J. B.; Meyer, E. F.; Brice, M. D.; Rodgers, J. R.; Kennard, O.; Shimanouchi, T.; Tasumi, M., The protein data bank: A computer-based archival file for macromolecular structures, J. Mol. Biol., 112, 535-542 (1977) · doi:10.1016/S0022-2836(77)80200-3 |
| [43] | Deserno, M.; Holm, C., How to mesh up Ewald sums: II. A theoretical and numerical comparison of various particle mesh routines, J. Chem. Phys., 109, 7694-7701 (1998) · doi:10.1063/1.477415 |
| [44] | Hockney, R. W.; Eastwood, J. W., Computer Simulation Using Particles (1988), New York: Taylor and Francis, New York · Zbl 0662.76002 · doi:10.1201/9781439822050 |
| [45] | Cances, E.; Maday, Y.; Stamm, B., Domain decomposition for implicit solvation models, J. Chem. Phys., 139, 054111 (2013) · doi:10.1063/1.4816767 |
| [46] | Xie, D.; Ying, J., A new box iterative method for a class of nonlinear interface problems with application in solving Poisson-Boltzmann equation, J. Comput. Appl. Math., 307, 3319-3334 (2016) · Zbl 1348.65170 · doi:10.1016/j.cam.2016.01.005 |
| [47] | Maz’ya, V.; Schmidt, G., Approximate Approximations (2007), Providence, R.I.: Am. Math. Soc., Providence, R.I. · Zbl 1120.41013 · doi:10.1090/surv/141 |
| [48] | Hsiao, G. C.; Wendland, W. L., Boundary Integral Equations (2008), Berlin: Springer, Berlin · Zbl 1157.65066 · doi:10.1007/978-3-540-68545-6 |
| [49] | Sauter, S. A.; Schwab, Ch., Boundary Element Methods (2011), Berlin: Springer, Berlin · Zbl 1215.65183 · doi:10.1007/978-3-540-68093-2 |
| [50] | Hitchcock, F. L., The expression of a tensor or a polyadic as a sum of products, J. Math. Phys., 6, 164-189 (1927) · JFM 53.0095.01 · doi:10.1002/sapm192761164 |
| [51] | Tucker, L. R., Some mathematical notes on three-mode factor analysis, Psychometrika, 31, 279-311 (1966) · doi:10.1007/BF02289464 |
| [52] | Khoromskij, B. N.; Khoromskaia, V., Low rank Tucker-type tensor approximation to classical potentials, Central Eur. J. Math., 5, 523-550 (2007) · Zbl 1130.65060 · doi:10.2478/s11533-007-0018-0 |
| [53] | Gavrilyuk, I. P.; Hackbusch, W.; Khoromskij, B. N., Data-sparse approximation to operator-valued functions of elliptic operator, Math. Comput., 73, 1297-1324 (2003) · Zbl 1065.47009 · doi:10.1090/S0025-5718-03-01590-4 |
| [54] | White, S. R., Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B, 48, 10345-10356 (1993) · doi:10.1103/PhysRevB.48.10345 |
| [55] | Schollwöck, U., The density-matrix renormalization group in the age of matrix product states, Ann. Phys., 326, 96-192 (2011) · Zbl 1213.81178 · doi:10.1016/j.aop.2010.09.012 |
| [56] | Oseledets, I. V.; Tyrtyshnikov, E. E., TT-cross approximation for multidimensional arrays, Linear Algebra Appl., 432, 70-88 (2010) · Zbl 1183.65040 · doi:10.1016/j.laa.2009.07.024 |
| [57] | Dolgov, S. V.; Khoromskij, B. N.; Oseledets, I. V., Fast solution of multi-dimensional parabolic problems in the TT/QTT formats with initial application to the Fokker-Planck equation, SIAM J. Sci. Comput. A, 34, 3016-3038 (2012) · Zbl 1259.82075 · doi:10.1137/120864210 |
| [58] | Dolgov, S. V.; Oseledets, I. V., Solution of linear systems and matrix inversion in the TT-format, SIAM J. Sci. Comput. A, 34, 2718-2739 (2012) · Zbl 1259.65071 · doi:10.1137/110833142 |
| [59] | Lubich, Ch.; Rohwedder, T.; Schneider, R.; Vandereycken, B., Dynamical approximation of hierarchical tucker and tensor-train tensors, SIAM J. Matrix Anal. Appl., 34, 470-494 (2013) · Zbl 1391.15087 · doi:10.1137/120885723 |
| [60] | Lubich, Ch.; Oseledets, I. V.; Vandereycken, B., Time integration of tensor trains, SIAM J. Numer. Anal., 53, 917-941 (2015) · Zbl 1312.65114 · doi:10.1137/140976546 |
| [61] | Sulimov, A. V.; Zheltkov, D. A.; Oferkin, I. V.; Kutov, D. C.; Katkova, E. V.; Tyrtyshnikov, E. E.; Sulimov, V. B., Russian Supercomputing Days (2017), Cham: Springer, Cham |
| [62] | Benner, P.; Onwunta, A.; Stoll, M., A low-rank inexact Newton-Krylov method for stochastic eigenvalue problems, Comput. Methods Appl. Math., 19, 5-22 (2019) · Zbl 1420.65043 · doi:10.1515/cmam-2018-0030 |
| [63] | Hackbusch, W.; Kressner, D.; Uschmajew, A., Perturbation of higher-order singular values, SIAM J. Appl. Algebra Geom., 1, 374-387 (2017) · Zbl 1371.15011 · doi:10.1137/16M1089873 |
| [64] | Khoromskij, B. N., O(d logN)-quantics approximation of N-d tensors in high-dimensional numerical modeling, Constr. Approximation, 34, 257-289 (2011) · Zbl 1228.65069 · doi:10.1007/s00365-011-9131-1 |
| [65] | Oseledets, I. V., Approximation of 2^d × 2^d matrices by using tensor decomposition, SIAM J. Matrix Anal. Appl., 31, 2130-2145 (2010) · Zbl 1200.15005 · doi:10.1137/090757861 |
| [66] | Rakhuba, M.; Oseledets, I., Fast multidimensional convolution in low-rank tensor formats via cross approximation, SIAM J. Sci. Comput. A, 37, 565-582 (2015) · Zbl 1320.65197 · doi:10.1137/140958529 |
| [67] | Kazeev, V.; Khoromskij, B. N.; Tyrtyshnikov, E. E., Multilevel Toeplitz matrices generated by tensor-structured vectors and convolution with logarithmic complexity, SIAM J. Sci. Comput. A, 35, 1511-1536 (2013) · Zbl 1275.15018 · doi:10.1137/110844830 |
| [68] | Kazeev, V.; Khammash, M.; Nip, M.; Schwab, Ch., Direct solution of the chemical master equation using quantized tensor trains, PLoS Comput. Biol., 10, e1003359 (2014) · doi:10.1371/journal.pcbi.1003359 |
| [69] | Kazeev, V.; Reichmann, O.; Schwab, C., Low-rank tensor structure of linear diffusion operators in the TT and QTT formats, Linear Algebra Appl., 438, 4204-4221 (2013) · Zbl 1281.15024 · doi:10.1016/j.laa.2013.01.009 |
| [70] | Cichocki, A.; Lee, N.; Oseledets, I.; Pan, A. H.; Zhao, Q.; Mandic, D. P., Tensor networks for dimensionality reduction and large-scale optimization: Part 1. Low-rank tensor decompositions, Found. Trends Mach. Learn., 9, 249-429 (2016) · Zbl 1364.68320 · doi:10.1561/2200000059 |
| [71] | Kressner, D.; Steinlechner, M.; Uschmajew, A., Low-Rank tensor methods with subspace correction for symmetric eigenvalue problems, SIAM J. Sci. Comput. A, 36, 2346-2368 (2014) · Zbl 1307.65040 · doi:10.1137/130949919 |
| [72] | Dolgov, S.; Khoromskij, B. N.; Litvinenko, A.; Matthies, H. G., Computation of the response surface in the tensor train data format, SIAM J. Uncert. Quantif., 3, 1109-1135 (2015) · Zbl 1329.65271 · doi:10.1137/140972536 |
| [73] | Rakhuba, M. V.; Oseledets, I. V., Grid-based electronic structure calculations: The tensor decomposition approach, J. Comput. Phys., 312, 19-30 (2016) · Zbl 1351.82040 · doi:10.1016/j.jcp.2016.02.023 |
| [74] | M. Espig, W. Hackbusch, A. Litvinenko, H. G. Matthies, and E. Zander, “Post-processing of high-dimensional data” (2019). arXiv:1906.05669. · Zbl 1436.65054 |
| [75] | Bachmayr, M.; Schneider, R.; Uschmajew, A., Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations, Found. Comput. Math., 16, 1423-1472 (2016) · Zbl 1357.65153 · doi:10.1007/s10208-016-9317-9 |
| [76] | C. Marcati, M. Rakhuba, and C. Schwab, “Tensor rank bounds for point singularities in ℝ^3,” SAM Res. Rep. 2019-68 (Seminar for Applied Mathematics, ETH Zurich, Zurich, 2019), p. 47. |
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