A spectral Monte Carlo method for the Poisson equation

Gobet, Emmanuel; Maire, Sylvain

doi:10.1515/mcma.2004.10.3-4.275

Cited by 9 publications

(19 citation statements)

References 11 publications

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“…We can mention Fredholm integral equations or spectral methods for partial differential equations. In particular, coupling this approximation and the sequential Monte Carlo algorithm developed in [7] seems really promising for the numerical solution of partial differential equations. These ideas are under development.…”

Section: Resultsmentioning

confidence: 99%

Quasi-Monte Carlo quadratures for multivariate smooth functions

Maire¹,

Luigi²

2006

Applied Numerical Mathematics

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Section: Resultsmentioning

confidence: 99%

Quasi-Monte Carlo quadratures for multivariate smooth functions

Maire¹,

Luigi²

2006

Applied Numerical Mathematics

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“…Even if our algorithm is based on independent random drawings, we have observed in [GM04] that one could use lowdiscrepancy sequences to speed up the convergence of the algorithm. We hence use here a version of the algorithm based on Halton sequences, which is twice as fast as the Monte Carlo version.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Note that the numerical computations of Au n−1 in (3.2) (and analogously those of Bu n−1 ) are performed using the evaluations of (AC j ) j through the equality Au n−1 = N j=1 u n−1 (x j )AC j (see [GM04]). …”

Section: Initializationmentioning

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%

“…We do not notice any difference on the final accuracy for the two values ε 1 and ε 2 . CPU times are of course smaller for ε 1 and also about eight times smaller than in [GM04] by using a recursive computation of the Tchebychef polynomials instead of their trigonometric expression. We now use the corrected Euler scheme with discretization parameters t 1 = 0.005 and t 2 = 0.002.…”

Section: The Bidimensional Casementioning

confidence: 99%

See 2 more Smart Citations

Sequential Control Variates for Functionals of Markov Processes

Gobet¹,

Maire²

2005

SIAM J. Numer. Anal.

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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;

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Stochastic Spectral Formulations for Elliptic Problems

Maire

Tanré²

2009

Monte Carlo and Quasi-Monte Carlo Methods 2008

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We describe new stochastic spectral formulations with very good properties in terms of conditioning. These formulations are built by combining Monte Carlo approximations of the Feynman-Kac formula and standard deterministic approximations on basis functions. We give error bounds on the solutions obtained using these formulations in the case of linear approximations. Some numerical tests are made on an anisotropic diffusion equation using a tensor product Tchebychef polynomial basis and one random point schemes quantified or not.

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A spectral Monte Carlo method for the Poisson equation

Cited by 9 publications

References 11 publications

Quasi-Monte Carlo quadratures for multivariate smooth functions

Quasi-Monte Carlo quadratures for multivariate smooth functions

Sequential Control Variates for Functionals of Markov Processes

Stochastic Spectral Formulations for Elliptic Problems

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