A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function

Hwang, Chi‐Ok; Mascagni, Michael; Given, James A.

doi:10.1016/s0378-4754(02)00224-0

Cited by 28 publications

(19 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case of a non-constant source term, the modified walk on spheres could be used to compute this contribution [23]. If the walk hits ∂D it stops.…”

Section: 33mentioning

confidence: 99%

Simulating diffusions with piecewise constant coefficients using a kinetic approximation

Lejay

Maire²

2010

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

Abstract. Using a kinetic approximation of a linear diffusion operator, we propose an algorithm that allows one to deal with the simulation of a multi-dimensional stochastic process in a media which is locally isotropic except on some surface where the diffusion coefficient presents some discontinuities. Basic numerical examples are given in dimensions one to three on PDEs or stochastic PDEs with or without source terms. Finally, we compute the hydrodynamic load in a porous media in the nuclear waste context.

show abstract

“…In the case of a non-constant source term, the modified walk on spheres could be used to compute this contribution [23]. If the walk hits ∂D it stops.…”

Section: 33mentioning

confidence: 99%

Simulating diffusions with piecewise constant coefficients using a kinetic approximation

Lejay

Maire²

2010

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

show abstract

“…We propose to check it when we use the so-called discrete Euler scheme [Gob00], which is the simplest procedure that can be used for general stopped diffusions. An alternative is the WOS scheme, which is especially efficient when we are dealing with the Brownian motion (see [HMG03] and the references therein). Some refinements to the discrete Euler scheme are also possible, using Brownian bridge simulations [Bal95,Gob00,Gob01].…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

“…We use an interpolation at the bidimensional Tchebychef grid, two types of discretization schemes, and Monte Carlo simulations. The first one is the modified WOS method [HMG03,GM04] which can take into account the source term f . This walk goes from one sphere to another until the motion reaches the ε-absorption layer.…”

Section: The Bidimensional Casementioning

confidence: 99%

Sequential Control Variates for Functionals of Markov Processes

Gobet¹,

Maire²

2005

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Using a sequential control variates algorithm, we compute Monte Carlo approximations of solutions of linear partial differential equations connected to linear Markov processes by the Feynman-Kac formula. It includes diffusion processes with or without absorbing/reflecting boundary and jump processes. We prove that the bias and the variance decrease geometrically with the number of steps of our algorithm. Numerical examples show the efficiency of the method on elliptic and parabolic problems. Introduction.We are concerned with the numerical evaluation of E(Ψ(X s : s ≥ t)|X t = x), where (X t ) t is a Markov process (with linear dynamics) and where Ψ belongs to a class of functionals related to Feynman-Kac representations. These issues arise, for example, in physics in the computations of the solution of diffusion equations (see [CDL + 89]), or in finance in the pricing of European options (see [DG95] and the references therein). Monte Carlo methods are usually used to evaluate these expectations for high-dimensional problems or when the functionals are complex. They give a rather poor approximation because of a slow convergence as σ/ √ M , M being the number of simulations and σ 2 the relative variance. A better accuracy can nevertheless be reached by using relevant variance-reduction tools like, for instance, the control variates method or importance sampling [Hal70, New94]. One of the most performing tools is the sequential Monte Carlo approach which consists in using iteratively these variance-reduction ideas [Hal62, Hal70, Boo89]. Using, respectively, importance sampling and control variates, this approach has been recently developed in [BCP00] for Markov chains and in [Mai03] for the numerical integration of multivariate smooth functions. We have introduced in [GM04] a sequential Monte Carlo method to solve the Poisson equation with Dirichlet boundary conditions over square domains. This method was based on Feynman-Kac computations of pointwise solutions combined with a global approximation on Tchebychef polynomials [BM97]. Pointwise solutions were computed using walk-on-spheres (WOS) simulations of stopped Brownian motion, which induces a simulation error due to the absorption layer thickness. We have nevertheless observed a geometric reduction up to a limit of both the simulation error and the variance with the number of steps of the algorithm. The global error was comparable to standard deterministic spectral methods [BM97] while avoiding the resolution of a linear system. Our goal here is twofold:• to extend the scope of the approach to general Markov processes connected to linear elliptic and parabolic Dirichlet problems;

show abstract

“…Under exponential timestepping, the corresponding improvement is from an error proportional to δt 1/2 , i.e., to the square root of the mean duration of a timestep, to an error proportional to δt [2]. Alternatively, under the walk-on-spheres method, a path is constructed as a sequence of jumps from ball centers to ball surfaces and an absorption layer is defined close to the boundary [16,17]. Realizations are brought to an end when they enter the layer.…”

Section: P[δt > T] = Exp(−λt) (11)mentioning

confidence: 99%

Multidimensional Exponential Timestepping with Boundary Test

Jansons¹,

Lythe²

2005

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. Exponential timestepping algorithms are efficient for exit-time problems because a boundary test can be performed at the end of each timestep, giving high-order convergence in numerical evaluation of mean exit times. Successive time increments are independent random variables with an exponential distribution. We show how to perform exact timestepping for Brownian motion in more than one dimension and consider hitting times of curved surfaces. Taking as examples the exit times from a circle, from a ball, from an ellipse, and from the region bounded by the two lines of a hyperbola, we report the results of numerical experiments performed with the algorithms developed in this work.

show abstract

A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function

Cited by 28 publications

References 13 publications

Simulating diffusions with piecewise constant coefficients using a kinetic approximation

Simulating diffusions with piecewise constant coefficients using a kinetic approximation

Sequential Control Variates for Functionals of Markov Processes

Multidimensional Exponential Timestepping with Boundary Test

Contact Info

Product

Resources

About