Simulating diffusions with piecewise constant coefficients using a kinetic approximation

Lejay, Antoine; Maire, Sylvain

doi:10.1016/j.cma.2010.03.002

Cited by 11 publications

(13 citation statements)

References 33 publications

(43 reference statements)

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“…This is typically the case in neutron transport approximations of diffusion processes, which naturally suggest a new way to move the particle from Γ. This idea has been introduced in [16]. Define …”

Section: Neutron Transport Jump (Ntj) Algorithmmentioning

confidence: 99%

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

Bossy¹,

Champagnat²,

Maire³

et al. 2010

ESAIM: M2AN

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Abstract.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.Mathematics Subject Classification. 35Q60, 92C40, 60J60, 65C05, 65C20, 68U20.

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Section: Neutron Transport Jump (Ntj) Algorithmmentioning

confidence: 99%

“…. 16) where r = μ ext R, r 0 = μ ext R 0 and γ = μ ext ε. Now, fix a > 0 and let ϕ(x) = x(a + x)/ sinh x for x > 0.…”

Section: Estimation Of the Computational Costmentioning

confidence: 99%

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

Bossy¹,

Champagnat²,

Maire³

et al. 2010

ESAIM: M2AN

Self Cite

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“…For the second method, the probabilities to go to the two subdomains are the same but there is only two replacement points chosen normally at a distance h to the interface [20]. The third method is the Kinetic approximation introduced in [15]. In this method, the motion goes equiprobably to D 1 and D 2 but it goes further inside the subdomain when the diffusion coefficient is larger.…”

Section: Three Dimensional Transmission Problemmentioning

confidence: 99%

“…Many algorithms essentially monodimensional have been proposed to deal with this interface conditions [7,12,19]. In recent works, some new approaches have been introduced based either on stochastic processes tools [14,15,25] or on finite differences techniques [2,20]. In particular, both methodologies have been improved and compared in [16] on elliptic, parabolic and eigenvalue problems.…”

Section: Introductionmentioning

confidence: 99%

Stochastic finite differences for elliptic diffusion equations in stratified domains

Maire

Nguyen

2016

Mathematics and Computers in Simulation

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International audienceWe describe Monte Carlo algorithms to solve elliptic partial differen- tial equations with piecewise constant diffusion coefficients and general boundary conditions including Robin and transmission conditions as well as a damping term. The treatment of the boundary conditions is done via stochastic finite differences techniques which possess an higher order than the usual methods. The simulation of Brownian paths inside the domain relies on variations around the walk on spheres method with or without killing. We check numerically the efficiency of our algorithms on various examples of diffusion equations illustrating each of the new techniques introduced here

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“…The Euler scheme can be used for the simulation of a wide class of stochastic processes linked to second order elliptic operators. If we deal with the Laplace operator or with divergence form operators with constant or piecewise constant diffusion coefficients [17], more efficient simulations of Brownian paths are available like walk on spheres (WOS) [24] or walk on rectangles algorithms [9]. In the case of Dirichlet boundary conditions, the stochastic representation of the solution implies a Brownian path up to the first time it hits the boundary ∂D of the domain.…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo approximations of the Neumann problem

Maire

Tanré

2013

Monte Carlo Methods and Applications

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We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes especially devised for local time approximation on the boundary of the domain. Global approximations are computed thanks to a stochastic spectral formulation taking into account the property of zero mean value of the solution. This stochastic formulation is asymptotically perfect in terms of conditioning. Numerical examples are given on the Laplace operator on a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary conditions. A more general convection-diffusion equation is also numerically studied.

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Simulating diffusions with piecewise constant coefficients using a kinetic approximation

Cited by 11 publications

References 33 publications

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

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

Stochastic finite differences for elliptic diffusion equations in stratified domains

Monte Carlo approximations of the Neumann problem

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