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Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics. (English) Zbl 1204.82020
This paper firstly derives a new probabilistic interpretation for a class of divergence-form operators, and then uses it to construct Monte Carlo methods for the linearized Poisson-Boltzmann equation in molecular dynamics. The Feynman-Kac formula is extended to the solution of a class of PDE involving divergence-form operators, and the proposed Monte Carlo algorithms are based on random walks on spheres.

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
92C40 Biochemistry, molecular biology
60J60 Diffusion processes
65C05 Monte Carlo methods
65C20 Probabilistic models, generic numerical methods in probability and statistics
68U20 Simulation (MSC2010)
82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
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[1] D.G. Aronson, Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc.73 (1967) 890-896. Zbl0153.42002 · Zbl 0153.42002 · doi:10.1090/S0002-9904-1967-11830-5
[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 1012.60003
[5] H. Brezis, Analyse fonctionnelle : Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983). Zbl0511.46001 · Zbl 0511.46001
[6] R. Dautray and J.-L. Lions, Evolution problems II, Mathematical analysis and numerical methods for science and technology6. Springer-Verlag, Berlin (1993).
[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). Zbl0592.60049 · Zbl 0592.60049
[8] M. Fukushima, Y. Ōshima and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics19. Walter de Gruyter & Co., Berlin (1994).
[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 1042.35002
[10] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library24. Second edition, North-Holland Publishing Co., Amsterdam (1989). · Zbl 0684.60040
[11] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics113. Second edition, Springer-Verlag, New York (1991). Zbl0734.60060 · Zbl 0734.60060
[12] O.A. Ladyzhenskaya and N.N. Ural’tseva, Linear and quasilinear elliptic equations. Academic Press, New York (1968). Zbl0164.13002 · Zbl 0164.13002
[13] B. Lapeyre, É. Pardoux and R. Sentis, Introduction to Monte-Carlo methods for transport and diffusion equations, Oxford Texts in Applied and Engineering Mathematics6. Oxford University Press, Oxford (2003). · Zbl 1136.65133
[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 0551.60059
[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. · Zbl 1231.76241 · doi:10.1016/j.cma.2010.03.002
[17] A. Lejay and M. Martinez, A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Ann. Appl. Probab.16 (2006) 107-139. Zbl1094.60056 · Zbl 1094.60056 · doi:10.1214/105051605000000656
[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 1113.82046 · doi:10.1093/imanum/drl008
[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. Paris342 (2006) 51-56. Zbl1082.60514 · Zbl 1082.60514 · doi:10.1016/j.crma.2005.10.025
[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. Zbl1075.65003 · Zbl 1075.65003 · doi:10.1137/S1064827503422221
[24] N.I. Portenko, Diffusion processes with a generalized drift coefficient. Teor. Veroyatnost. i Primenen.24 (1979) 62-77. · Zbl 0396.60071
[25] N.I. Portenko, Stochastic differential equations with a generalized drift vector. Teor. Veroyatnost. i Primenen.24 (1979) 332-347. · Zbl 0415.60055
[26] P.E. Protter, Stochastic integration and differential equations - Second edition, Version 2.1, Stochastic Modelling and Applied Probability21. Corrected third printing, Springer-Verlag, Berlin (2005).
[27] D. Revuz and M. Yor, Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften293. Springer-Verlag, Berlin (1991). · Zbl 0731.60002
[28] L.C.G. Rogers and D. Williams, Foundations, Diffusions, Markov processes, and martingales1. Reprint of the second edition (1994), Cambridge Mathematical Library, Cambridge University Press, Cambridge (2000).
[29] L.C.G. Rogers and D. Williams, Itô calculus, Diffusions, Markov processes, and martingales2. Reprint of the second edition (1994), Cambridge Mathematical Library, Cambridge University Press, Cambridge (2000).
[30] A. Rozkosz and L. Słomiński, Extended convergence of Dirichlet processes. Stochastics Stochastics Rep.65 (1998) 79-109. · Zbl 0917.60076
[31] K.K. Sabelfeld, Monte Carlo methods in boundary value problems. Springer Series in Computational Physics, Springer-Verlag, Berlin (1991).
[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. Zbl0824.65127 · Zbl 0824.65127 · doi:10.1515/mcma.1995.1.1.1
[33] N.A. Simonov, Walk-on-spheres algorithm for solving boundary-value problems with continuity flux conditions, in Monte Carlo and quasi-Monte Carlo methods2006, Springer, Berlin (2008) 633-643. Zbl1141.65315 · Zbl 1141.65315
[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. Zbl0651.47031 · Zbl 0651.47031 · numdam:SPS_1988__22__316_0 · eudml:113641
[36] D.W. Stroock and S.R.S. Varadhan, Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften233. Springer-Verlag, Berlin (1979). · Zbl 0426.60069
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