Numerical solution of the nonlinear Poisson–Boltzmann equation: Developing more robust and efficient methods

TLDR: A robust and efficient numerical method for solution of the nonlinear Poisson‐Boltzmann equation arising in molecular biophysics that will converge in the case of molecules for which many of the existing methods will not.

Abstract: We present a robust and efficient numerical method for solution of the nonlinear Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the box method, and solution of the discrete equations is accomplished with a global inexact-Newton method, combined with linear multilevel techniques we have described in an article appearing previously in this journal. A detailed analysis of the resulting method is presented, with comparisons to other methods that have been proposed in the literature, including the classical nonlinear multigrid method, the nonlinear conjugate gradient method, and nonlinear relaxation methods such as successive overrelaxation. Both theoretical and numerical evidence suggests that this method will converge in the case of molecules for which many of the existing methods will not. In addition, for problems which the other methods are able to solve, numerical experiments show that the new method is substantially more efficient, and the superiority of this method grows with the problem size. The method is easy to implement once a linear multilevel solver is available and can also easily be used in conjunction with linear methods other than multigrid. © 1995 by John Wiley & Sons, Inc.

Figures

Table 3: Storage required by various nonlinear (and some linear) elliptic solvers.

Figure 5: Comparison of various methods for the nonlinear crambin problem.

Figure 6: Comparison of various methods for the nonlinear acetamide problem.

Figure 8: Comparison of various methods for the nonlinear SOD problem.

Figure 2: Banded matrix produced by the box-method.

Figure 7: Comparison of various methods for the nonlinear tRNA problem.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Numerical solution of the nonlinear poisson-boltzmann equation: developing more robust and efficient methods" ?

The authors present a robust and efficient numerical method for solution of the nonlinear Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the box method, and solution of the discrete equations is accomplished with a global inexact-Newton method, combined with linear multilevel techniques the authors have described in a paper appearing previously in this journal. A detailed analysis of the resulting method is presented, with comparisons to other methods that have been proposed in the literature, including the classical nonlinear multigrid method, the nonlinear conjugate gradient method, and nonlinear relaxation methods such as successive over-relaxation. Both theoretical and numerical evidence suggests that this method will converge in the case of molecules for which many of the existing methods will not. 

Q2. How many meshes are required to represent a vector having length n?

With regard to multilevel methods, since the number of unknowns drops by a factor of eight as one moves to a coarser mesh in three dimensions if standard successively refined non-uniform Cartesian meshes are used, the authors seethat the storage required to represent on all meshes a vector having length n on the finest mesh is:n + n8 +n64 + · · · = n ·(1 8 + 1 64 + · · ·)≤ 

Q3. What is the coarse problem solved with the nonlinear conjugate gradient method?

The coarse problem is solved with the nonlinear conjugate gradient method, and a damping parameter, as described earlier is required; otherwise, the method does not converge for rapid nonlinearities such as those present in the nonlinear Poisson-Boltzmann equation. 

Q4. What did the experiments with the simplest molecules show divergence?

Their experiments with such a “vanilla” nonlinear multigrid method showed divergence except for the simplest possible molecules (acetamide) at very low ionic strengths (Is < 0.0001) and small mesh sizes (31 × 31 × 31). 

Q5. Why is the Jacobian matrix of the inexact-Newton approach of interest?

A second reason for their interest in the inexactNewton approach is that the efficient multilevel methods for the linearized Poisson-Boltzmann equation [5, 6, 14] can be used effectively for the Jacobian systems; this is because the Jacobian F ′(u) is essentially the linearized Poisson-Boltzmann operator, where only the diagonal Helmholtz-like term N ′(·) changes from one Newton iteration to the next. 

Q6. How much improvement is the nonlinear conjugate gradient method?

The method DINMH is extremely efficient, representing an improvement of more than a factor of fifty over the nonlinear SOR and nonlinear conjugate gradient methods, and a factor of ten over the nonlinear multigrid method NMH. 

Q7. What did the authors use to obtain convergence even for the crambin case?

To obtain convergence even for the crambin case (Figure 5) required both the harmonic coefficient averaging approach developed in Holst and Saied [14] and the damping parameter discussed earlier in this paper. 

Q8. What is the effectiveness of coefficient averaging techniques?

The effectiveness of coefficient averaging techniques, applied to the linearized Poisson-Boltzmann equation, is discussed in detail in Holst and Saied [14], and also in references [5, 6]. 

Q9. What is the eigenvalue of the Jacobi iteration matrix?

In the paper of Nicholls and Honig [13], an adaptive SOR procedure is developed for the linearized PoissonBoltzmann equation, employing a power method to estimate the largest eigenvalue of the Jacobi iteration matrix, which enables estimation of the optimal relaxation parameter for SOR using Young’s formula (page 110 in Varga [7]). 

Q10. How can the authors find a by minimizing Jk() along the descent?

The authors can find such a λ by minimizing Jk(·) along the descent direction wk, which is equivalent to solving the following onedimensional problem:dJ(uk + λwk)dλ = 0.As in the discussion of the nonlinear conjugate gradient method, the one-dimensional problem can be solved with Newton’s method:λm+1 = λm − XY ,where (exactly as for the nonlinear CG method)X = λm(Akwk, wk)k − (rk , wk)k + (Nk(uk + λ mwk)−Nk(uk), wk)k,Y = (Akwk, wk)k + (N ′ k(uk + λ mwk)wk, wk)k. 

Q11. What is the common way to extend classical linear methods?

The classical linear methods discussed earlier, such as Gauss-Seidel and SOR, can be extended in the obvious way to nonlinear algebraic equations of the form (3).