Monte Carlo Methods for the Linearized Poisson-Boltzmann Equation
نویسندگان
چکیده
We review efficient grid-free random walk methods for solving boundary value problems for the linearized Poisson-Boltzmann equation (LPBE). First we introduce the “Walk On Spheres” (WOS) algorithm [1] for the LPBE. Based on this WOS algorithm, another, related, Monte Carlo algorithm is presented. This modified Monte Carlo method reinterprets the weights used in the original WOS algorithm as survival probabilities for the random walker used in the computation [2]. In addition, a Feynman-Kac path-integral implementation for solving the LPBE is given [3]. This Feynman-Kac approach uses the WOS method to provide a technique for estimating certain Gaussian path integrals without the need for simulating Brownian trajectories in detail. We then similarly interpret the exponential weight in the Feynman-Kac formula as a survival probability. It is then shown that this method is mathematically equivalent to the previous modified WOS method for the LPBE. The effectiveness of these methods is illustrated by computing four analytically solvable problems. In all four cases, excellent agreement is shown. In particular, for the problem of calculating the electrostatic potential in an electrolyte between two infinite parallel flat plates, our modified WOS method is compared with the old WOS method and with our Feynman-Kac WOS (FK WOS) method. Our modified WOS method is the most efficient one, but FK WOS method holds the promise of extension to more complicated equations such as the time-independent Schrödinger equation. Finally, we illustrate the use of a Monte Carlo approach for the LPBE in a more complicated setting related to the computation of the electrostatic free energy of a large molecule. Here, we couple the LPBE solution in the exterior of a compact domain (molecule) with the solution of the Poisson equation inside, and with continuity boundary conditions linking these two solutions. The Monte Carlo method performs quite well in this complicated situation. Preprint submitted to Elsevier Science 13 May 2003
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