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Reduced basis method for the nonlinear Poisson–Boltzmann equation regularized by the range-separated canonical tensor format

  • Cleophas Kweyu EMAIL logo , Lihong Feng , Matthias Stein and Peter Benner
Published/Copyright: May 17, 2022
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 8

Abstract

The Poisson–Boltzmann equation (PBE) is a fundamental implicit solvent continuum model for calculating the electrostatic potential of large ionic solvated biomolecules. However, its numerical solution encounters severe challenges arising from its strong singularity and nonlinearity. In (P. Benner, V. Khoromskaia, B. Khoromskij, C. Kweyu, and M. Stein, “Regularization of Poisson-Boltzmann type equations with singular source terms using the range-separated tensor format,” SIAM J. Sci. Comput., vol. 43, no. 1, pp. A415–A445, 2021; C. Kweyu, V. Khoromskaia, B. Khoromskij, M. Stein, and P. Benner, “Solution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format,” arXiv:2109.14073, 2021), the effect of strong singularities was eliminated by applying the range-separated (RS) canonical tensor format (P. Benner, V. Khoromskaia, and B. N. Khoromskij, “Range-separated tensor format for many-particle modeling,” SIAM J. Sci. Comput., vol. 40, no. 2, pp. A1034–A1062, 2018; B. N. Khoromskij, “Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse,” J. Comput. Phys., vol. 401, p. 108998, 2020) to construct a solution decomposition scheme for the PBE. The RS tensor format allows deriving a smooth approximation to the Dirac delta distribution in order to obtain a regularized PBE (RPBE) model. However, solving the RPBE is still computationally demanding due to its high dimension N , where N is always in the millions. In this study, we propose to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the RPBE in order to construct a reduced order model (ROM) of low dimension N ≪ N , whose solution accurately approximates the nonlinear RPBE. The long-range potential can be obtained by lifting the ROM solution back to the N -space while the short-range potential is directly precomputed analytically, thanks to the RS tensor format. The sum of both provides the total electrostatic potential. The main computational benefit is the avoidance of computing the numerical approximation of the singular electrostatic potential. We demonstrate in the numerical experiments, the accuracy and efficacy of the reduced basis (RB) approximation to the nonlinear RPBE (NRPBE) solution and the corresponding computational savings over the classical nonlinear PBE (NPBE) as well as over the RBM being applied to the classical NPBE.

Keywords: canonical tensor representation; Dirac delta distribution; Newton kernel; range-separated tensor format; reduced basis method; singularity
AMS Subject Classification: 65F30; 65F50; 65N35; 65F10
Corresponding author: Cleophas Kweyu, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, D-39106 Magdeburg, Germany; and Department of Mathematics and Physics, Moi University, P.O. Box 3900-30100, Eldoret, Kenya, E-mail:

Acknowledgement

The authors thank the following organizations for financial and material support on this project: International Max Planck Research School (IMPRS) for Advanced Methods in Process and Systems Engineering and Max Planck Society for the Advancement of Science (MPG).

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2021-03-11
Revised: 2022-04-01
Accepted: 2022-04-26
Published Online: 2022-05-17

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Testing of logarithmic-law for the slip with friction boundary condition
  4. A new clique polynomial approach for fractional partial differential equations
  5. The modified Rusanov scheme for solving the phonon-Bose model
  6. Delta-shock for a class of strictly hyperbolic systems of conservation laws
  7. The Cădariu–Radu method for existence, uniqueness and Gauss Hypergeometric stability of a class of Ξ-Hilfer fractional differential equations
  8. Novel periodic and optical soliton solutions for Davey–Stewartson system by generalized Jacobi elliptic expansion method
  9. The simulation of two-dimensional plane problems using ordinary state-based peridynamics
  10. Reduced basis method for the nonlinear Poisson–Boltzmann equation regularized by the range-separated canonical tensor format
  11. Simulation of the crystallization processes by population balance model using a linear separation method
  12. PS and GW optimization of variable sliding gains mode control to stabilize a wind energy conversion system under the real wind in Adrar, Algeria
  13. Characteristics of internal flow of nozzle integrated with aircraft under transonic flow
  14. Magnetogasdynamic shock wave propagation using the method of group invariance in rotating medium with the flux of monochromatic radiation and azimuthal magnetic field
  15. The influence pulse-like near-field earthquakes on repairability index of reversible in mid-and short-rise buildings
  16. Intelligent controller for maximum power extraction of wind generation systems using ANN
  17. A new self-adaptive inertial CQ-algorithm for solving convex feasibility and monotone inclusion problems
  18. Existence and Hyers–Ulam stability of solutions for nonlinear three fractional sequential differential equations with nonlocal boundary conditions
  19. A study on solvability of the fourth-order nonlinear boundary value problems
  20. Adaptive control for position and force tracking of uncertain teleoperation with actuators saturation and asymmetric varying time delays
  21. Framing the hydrothermal significance of water-based hybrid nanofluid flow over a revolving disk
  22. Catalytic surface reaction on a vertical wavy surface placed in a non-Darcy porous medium
  23. Carleman framework filtering of nonlinear noisy phase-locked loop system
  24. Corrigendum
  25. Corrigendum to: numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
https://doi.org/10.1515/ijnsns-2021-0103
Keywords for this article
canonical tensor representation; Dirac delta distribution; Newton kernel; range-separated tensor format; reduced basis method; singularity

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Testing of logarithmic-law for the slip with friction boundary condition
  4. A new clique polynomial approach for fractional partial differential equations
  5. The modified Rusanov scheme for solving the phonon-Bose model
  6. Delta-shock for a class of strictly hyperbolic systems of conservation laws
  7. The Cădariu–Radu method for existence, uniqueness and Gauss Hypergeometric stability of a class of Ξ-Hilfer fractional differential equations
  8. Novel periodic and optical soliton solutions for Davey–Stewartson system by generalized Jacobi elliptic expansion method
  9. The simulation of two-dimensional plane problems using ordinary state-based peridynamics
  10. Reduced basis method for the nonlinear Poisson–Boltzmann equation regularized by the range-separated canonical tensor format
  11. Simulation of the crystallization processes by population balance model using a linear separation method
  12. PS and GW optimization of variable sliding gains mode control to stabilize a wind energy conversion system under the real wind in Adrar, Algeria
  13. Characteristics of internal flow of nozzle integrated with aircraft under transonic flow
  14. Magnetogasdynamic shock wave propagation using the method of group invariance in rotating medium with the flux of monochromatic radiation and azimuthal magnetic field
  15. The influence pulse-like near-field earthquakes on repairability index of reversible in mid-and short-rise buildings
  16. Intelligent controller for maximum power extraction of wind generation systems using ANN
  17. A new self-adaptive inertial CQ-algorithm for solving convex feasibility and monotone inclusion problems
  18. Existence and Hyers–Ulam stability of solutions for nonlinear three fractional sequential differential equations with nonlocal boundary conditions
  19. A study on solvability of the fourth-order nonlinear boundary value problems
  20. Adaptive control for position and force tracking of uncertain teleoperation with actuators saturation and asymmetric varying time delays
  21. Framing the hydrothermal significance of water-based hybrid nanofluid flow over a revolving disk
  22. Catalytic surface reaction on a vertical wavy surface placed in a non-Darcy porous medium
  23. Carleman framework filtering of nonlinear noisy phase-locked loop system
  24. Corrigendum
  25. Corrigendum to: numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
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