Computationally Efficient Technique for Nonlinear Poisson-Boltzmann Equation

  • Sanjay Kumar Khattri
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3991)

Abstract

Discretization of non-linear Poisson-Boltzmann Equation equations results in a system of non-linear equations with symmetric Jacobian. The Newton algorithm is the most useful tool for solving non-linear equations. It consists of solving a series of linear system of equations (Jacobian system). In this article, we adaptively define the tolerance of the Jacobian systems. Numerical experiment shows that compared to the traditional method our approach can save a substantial amount of computational work. The presented algorithm can be easily incorporated in existing simulators.

References

  1. 1.
    Khattri, S.K.: Analyzing Finite Volume for Single Phase Flow in Porous Media. Journal of Porous Media. Accepted for Publication (2006)Google Scholar
  2. 2.
    Aksoyw, B.: Adaptive Multilevel Numerical Methods with Applications in Diffusive Biomolecular Reactions. PhD Thesis, The University of California, San Diego (2001)Google Scholar
  3. 3.
    Khattri, S.K.: Newton-Krylov Algorithm with Adaptive Error Correction for the Poisson-Boltzmann Equation. MATCH Commun. Math. Comput. Chem., 56 (2006)Google Scholar
  4. 4.
    Chow, S.-S.: Finite element error estimates for nonlinear elliptic equations of monotone type. Numer. Math. 54, 373–393 (1989)Google Scholar
  5. 5.
    Eymard, R., Gallouët, T., Hilhorst, D., Naït Slimane, Y.: Finite volumes and nonlinear diffusion equations. RAIRO Math. Model. Numer. Anal. 32, 747–761 (1998)Google Scholar
  6. 6.
    Lui, S.H.: On Schwarz Alternating Methods For Non Linear Elliptic PDEs. SIAM Journal on Scientific Computing 21, 1506–1523 (2000)Google Scholar
  7. 7.
    Fogolari, F., Brigo, A., Molinari, H.: The Poisson Boltzmann equation for Biomolecular electrostatics: A Tool for Structural Biology. Journal of Molecular Recognition 15, 377–392 (2002)CrossRefGoogle Scholar
  8. 8.
    Kuo, S.S., Altman, M.D., Bardhan, J.P., Tidor, B., White, J.K.: Fast Methods for Simulation of Biomolecule Electrostatics. In: International Conference on Computer Aided Design (2002)Google Scholar
  9. 9.
    Host, M., Kozack, R.E., Saied, F., Subramaniam, S.: Treatment of Electrostatic Effects in Proteins: Multigrid-based Newton Iterative Method for Solution of the Full Nonlinear Poisson-Boltzmann Equation. Proteins: Structure, Function, and Genetics 18, 231–245 (1994)CrossRefGoogle Scholar
  10. 10.
    Host, M., Kozack, R.E., Saied, F., Subramaniam, S.: Protein electrostatics: Rapid multigrid-based Newton algorithm for solution of the full nonlinear Poisson-Boltzmann equation. J. of Biomol. Struct. & Dyn. 11, 1437–1445 (1994)Google Scholar
  11. 11.
    Host, M., Kozack, R.E., Saied, F., Subramaniam, S.: Multigrid-based Newton iterative method for solving the full Nonlinear Poisson-Boltzmann equation. Biophysical Journal 66, A130–A130 (1994)Google Scholar
  12. 12.
    Holst, M., Saied, F.: Numerical solution of the nonlinear Poisson-Boltzmann equation: Developing more robust and efficient methods. J. Comput. Chem. 16, 337–364 (1995)Google Scholar
  13. 13.
    Baker, N., Sept, D., Holst, M., McCammon, J.A.: The adaptive multilevel finite element solution of the Poisson-Boltzmann equation on massively parallel computers. IBM J. Research and Development 45, 427–438 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sanjay Kumar Khattri
    • 1
  1. 1.Department of MathematicsUniversity of BergenNorway

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