Convergence analysis of a solver for the linear Poisson--Boltzmann model

TL;DR

The authors address the convergence of a nonoverlapping interior–exterior domain decomposition method (ddLPB) for the linear Poisson--Boltzmann equation by recasting the problem as an interior–exterior transmission and introducing an interior–exterior Sobolev constant. They establish a spectral equivalence between the DtN operators governing the interior and exterior subproblems, using a specialized inner product on $H^{1/2}(\Gamma)$ to prove convergence of a preconditioned Richardson iteration with relaxation parameter $0<\alpha<2$. The key theoretical contributions include explicit constants $C_1$ and $C_2$ bounding the spectrum of the iteration operator and, consequently, an optimal choice $\alpha_{op}=2/(C_1+C_2)$ that accelerates convergence; for realistic solvent models with $\varepsilon_1\le\varepsilon_2$ and small $\kappa$, convergence is guaranteed and practically efficient. Numerical experiments on spheres, small molecules (benzene, caffeine), and proteins validate the theory, quantify optimal or near-optimal $\alpha$ values (e.g., around 4/3 to 3/2 in typical cases), and illustrate rapid convergence with energies that are robust to $\alpha$ variations. These results provide a solid theoretical justification and practical guidance for using ddLPB in solvation energy computations, particularly in large biomolecular systems where linearized PB physics applies.

Abstract

This work investigates the convergence of a domain decomposition method for the Poisson-Boltzmann model that can be formulated as an interior-exterior transmission problem. To study its convergence, we introduce an interior-exterior constant providing an upper bound of the $L^2$ norm of any harmonic function in the interior, and establish a spectral equivalence for related Dirichlet-to-Neumann operators to estimate the spectrum of interior-exterior iteration operator. This analysis is nontrivial due to the unboundedness of the exterior subdomain, which distinguishes it from the classical analysis of the Schwarz alternating method with nonoverlapping bounded subdomains. It is proved that for the linear Poisson-Boltzmann solvent model in reality, the convergence of interior-exterior iteration is ensured when the relaxation parameter lies between 0 and 2. This convergence result interprets the good performance of ddLPB method developed in [SIAM Journal on Scientific Computing, 41 (2019), pp. B320-B350] where the relaxation parameter is set to 1. Numerical simulations are conducted to verify our convergence analysis and to investigate the optimal relaxation parameter for the interior-exterior iteration.

Figures (5)

• Figure 1: Iteration number $N_{\rm ite}(\alpha)$ v.s. the relaxation parameter $\alpha$ for 1 sphere and 2 spheres ($\varepsilon_1=1,~R=1,~\kappa=1$Å$^{-1}$). The "optimal" parameter $\bar{\alpha}_{\rm op}$ computed by \ref{['eq:alphaop2']} is the abscissa of star with different colors corresponding different $\varepsilon_2$.
• Figure 2: $\tilde{\alpha}_{\rm op}$ defined in \ref{['eq:alphaop_tilde']} v.s. $\varepsilon_2$ for 1 sphere and 2 spheres ($\varepsilon_1=1,~R=1,~\kappa=1$Å$^{-1}$), and $\bar{\alpha}_{\rm op}$ in \ref{['eq:alphaop3']} is the black dash line.
• Figure 3: Left: $\tilde{\alpha}_{\rm op},~\tilde{\alpha}_{\rm max}$ v.s. $R$ for 2 spheres ($\varepsilon_1=1,~\varepsilon_2=2,~\kappa=1$Å$^{-1}$). Right: $\tilde{\alpha}_{\rm op},~\tilde{\alpha}_{\rm max}$ v.s. $\kappa$ for 2 spheres ($\varepsilon_1=1,~\varepsilon_2=2,~R=1$).
• Figure 4: Iteration number $N_{\rm ite}(\alpha)$ v.s. the relaxation parameter $\alpha$ for protein molecules ($\varepsilon_1=1,~\kappa=0.104$Å$^{-1}$). The optimal parameter $\bar{\alpha}_{\rm op}$ in \ref{['eq:alphaop2']} is the abscissa of star with different colors corresponding different $\varepsilon_2$.
• Figure 5: $\lambda_{\mathcal{A}}(\ell)$ v.s. $\ell$ for the unit sphere where $\varepsilon_1=1,~\varepsilon_2=2$.

Theorems & Definitions (13)

• Definition 3.1: Interior-exterior constant
• Lemma 3.1: Interior-exterior estimate
• proof
• Theorem 3.1: Spectral equivalence and convergence
• proof
• Remark 3.1
• Proposition 3.1: Properties on spectrum, reed1972methods
• proof
• Theorem 3.2
• proof