Issue |
ESAIM: M2AN
Volume 44, Number 5, September-October 2010
Special Issue on Probabilistic methods and their applications
|
|
---|---|---|
Page(s) | 997 - 1048 | |
DOI | https://doi.org/10.1051/m2an/2010050 | |
Published online | 26 August 2010 |
Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics
1
TOSCA project-team, INRIA Sophia Antipolis –
Méditerranée, France. Mireille.Bossy@sophia.inria.fr; Nicolas.Champagnat@sophia.inria.fr; Denis.Talay@sophia.inria.fr
2
IMATH, Université du sud
Toulon-Var, France. maire@univ-tln.fr
Received:
31
July
2009
Revised:
23
February
2010
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 . 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.
Mathematics Subject Classification: 35Q60 / 92C40 / 60J60 / 65C05 / 65C20 / 68U20
Key words: Divergence form operator / Poisson-Boltzmann equation / Feynman-Kac formula / random walk on sphere algorithm
© EDP Sciences, SMAI, 2010
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