Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics

Bossy, Mireille; Champagnat, Nicolas; Maire, Sylvain; Talay, Denis

ESAIM: Modélisation mathématique et analyse numérique, Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 997-1048

Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of ${ℝ}^d$. This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator belongs to this family, as the solution of a SDE including a non standard local time term related to the interface of discontinuity. We then prove an extended Feynman-Kac formula for the Poisson-Boltzmann equation. This formula allows us to justify various probabilistic numerical methods to approximate the free energy of a molecule. We analyse the convergence rate of these simulation procedures and numerically compare them on idealized molecules models.

Classification : 35Q60, 92C40, 60J60, 65C05, 65C20, 68U20
Keywords: divergence form operator, Poisson-Boltzmann equation, Feynman-Kac formula, random walk on sphere algorithm

[1] D.G. Aronson, Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 (1967) 890-896. | Zbl

[2] N.A. Baker, D. Sept, M.J. Holst and J.A. Mccammon, The adaptive multilevel finite element solution of the Poisson-Boltzmann equation on massively parallel computers. IBM J. Res. Dev. 45 (2001) 427-437.

[3] N.A. Baker, D. Bashford and D.A. Case, Implicit solvent electrostatics in biomolecular simulation, in New algorithms for macromolecular simulation, Lect. Notes Comput. Sci. Eng. 49, Springer, Berlin (2005) 263-295.

[4] A.N. Borodin and P. Salminen, Handbook of Brownian motion-facts and formulae. Probability and its Applications, 2nd edition, Birkhäuser Verlag, Basel (2002). | Zbl

[5] H. Brezis, Analyse fonctionnelle : Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983). | Zbl

[6] R. Dautray and J.-L. Lions, Evolution problems II, Mathematical analysis and numerical methods for science and technology 6. Springer-Verlag, Berlin (1993). | Zbl

[7] S.N. Ethier and T.G. Kurtz, Markov processes - Characterization and convergence. Wiley Series in Probability and Mathematical Statistics, Probability and Mathematical Statistics, John Wiley & Sons Inc., New York (1986). | Zbl

[8] M. Fukushima, Y. Ōshima and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics 19. Walter de Gruyter & Co., Berlin (1994). | Zbl

[9] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics, Reprint of the 1998 edition, Springer-Verlag, Berlin (2001). | Zbl

[10] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library 24. Second edition, North-Holland Publishing Co., Amsterdam (1989). | Zbl

[11] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics 113. Second edition, Springer-Verlag, New York (1991). | Zbl

[12] O.A. Ladyzhenskaya and N.N. Ural'Tseva, Linear and quasilinear elliptic equations. Academic Press, New York (1968). | Zbl

[13] B. Lapeyre, É. Pardoux and R. Sentis, Introduction to Monte-Carlo methods for transport and diffusion equations, Oxford Texts in Applied and Engineering Mathematics 6. Oxford University Press, Oxford (2003). | Zbl

[14] J.-F. Le Gall, One-dimensional stochastic differential equations involving the local times of the unknown process, in Stochastic analysis and applications (Swansea, 1983), Lecture Notes Math. 1095, Springer, Berlin (1984) 51-82. | Zbl

[15] A. Lejay, Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme divergence : Cas linéaires et semi-linéaires. Ph.D. Thesis, Université de Provence, Marseille, France (2000).

[16] A. Lejay and S. Maire, Simulating diffusions with piecewise constant coefficients using a kinetic approximation. Comput. Meth. Appl. Mech. Eng. 199 (2010) 2014-2023.

[17] A. Lejay and M. Martinez, A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Ann. Appl. Probab. 16 (2006) 107-139. | Zbl

[18] N. Limić, Markov jump processes approximating a nonsymmetric generalized diffusion. Preprint, arXiv:0804.0848v4 (2008).

[19] S. Maire, Réduction de variance pour l'intégration numérique et pour le calcul critique en transport neutronique. Ph.D. Thesis, Université de Toulon et du Var, France (2001).

[20] S. Maire and D. Talay, On a Monte Carlo method for neutron transport criticality computations. IMA J. Numer. Anal. 26 (2006) 657-685. | Zbl

[21] M. Martinez, Interprétations probabilistes d'opérateurs sous forme divergence et analyse des méthodes numériques probabilistes associées. Ph.D. Thesis, Université de Provence, Marseille, France (2004).

[22] M. Martinez and D. Talay, Discrétisation d'équations différentielles stochastiques unidimensionnelles à générateur sous forme divergence avec coefficient discontinu. C. R. Math. Acad. Sci. Paris 342 (2006) 51-56. | Zbl

[23] M. Mascagni and N.A. Simonov, Monte Carlo methods for calculating some physical properties of large molecules. SIAM J. Sci. Comput. 26 (2004) 339-357. | Zbl

[24] N.I. Portenko, Diffusion processes with a generalized drift coefficient. Teor. Veroyatnost. i Primenen. 24 (1979) 62-77. | Zbl

[25] N.I. Portenko, Stochastic differential equations with a generalized drift vector. Teor. Veroyatnost. i Primenen. 24 (1979) 332-347. | Zbl

[26] P.E. Protter, Stochastic integration and differential equations - Second edition, Version 2.1, Stochastic Modelling and Applied Probability 21. Corrected third printing, Springer-Verlag, Berlin (2005). | Zbl

[27] D. Revuz and M. Yor, Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften 293. Springer-Verlag, Berlin (1991). | Zbl

[28] L.C.G. Rogers and D. Williams, Foundations, Diffusions, Markov processes, and martingales 1. Reprint of the second edition (1994), Cambridge Mathematical Library, Cambridge University Press, Cambridge (2000). | Zbl

[29] L.C.G. Rogers and D. Williams, Itô calculus, Diffusions, Markov processes, and martingales 2. Reprint of the second edition (1994), Cambridge Mathematical Library, Cambridge University Press, Cambridge (2000). | Zbl

[30] A. Rozkosz and L. Słomiński, Extended convergence of Dirichlet processes. Stochastics Stochastics Rep. 65 (1998) 79-109. | Zbl

[31] K.K. Sabelfeld, Monte Carlo methods in boundary value problems. Springer Series in Computational Physics, Springer-Verlag, Berlin (1991). | Zbl

[32] K.K. Sabelfeld and D. Talay, Integral formulation of the boundary value problems and the method of random walk on spheres. Monte Carlo Meth. Appl. 1 (1995) 1-34. | Zbl

[33] N.A. Simonov, Walk-on-spheres algorithm for solving boundary-value problems with continuity flux conditions, in Monte Carlo and quasi-Monte Carlo methods 2006, Springer, Berlin (2008) 633-643. | Zbl

[34] N.A. Simonov, M. Mascagni and M.O. Fenley, Monte Carlo-based linear Poisson-Boltzmann approach makes accurate salt-dependent solvation free energy predictions possible. J. Chem. Phys. 127 (2007) 185105.

[35] D.W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, in Séminaire de Probabilités, XXII, Lecture Notes in Math. 1321, Springer, Berlin (1988) 316-347. | Zbl | Numdam

[36] D.W. Stroock and S.R.S. Varadhan, Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften 233. Springer-Verlag, Berlin (1979). | Zbl