A new monte carlo method for solving a stationary diffusion equation

Mikhaı̆lov, G. A.; Burmistrov, A. V.

doi:10.1007/bf02674746

Cited by 1 publication

(5 citation statements)

References 3 publications

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“…The same procedure can be used to estimate the second term. Thus, by Theorem 4 in [4] we obtain the following assertion. …”

Section: Estimation Of Solutions By 'Random Walk By Spheres and Balls'mentioning

confidence: 63%

“…The Green 'nonpivotal' function and its normal derivative are expressed in terms of the formulae (see [4]): …”

Section: Estimation Of Solutions By 'Random Walk By Spheres and Balls'mentioning

confidence: 99%

“…In order to study variance finiteness, by analogy with [4] we modify the estimate as follows. We calculate the function À´Ö µ only for the case of the steṕ Ö µ ´Ö ¼ ¼µ, i.e.…”

Section: Estimation Of Solutions By 'Random Walk By Spheres and Balls'mentioning

confidence: 99%

“…The procedure in [4] can be used to study the value obtained. The same procedure can be used to estimate the second term.…”

Section: Estimation Of Solutions By 'Random Walk By Spheres and Balls'mentioning

confidence: 99%

“…Suppose the order of the error in the Euler scheme for the estimate of both the solution and its derivative is Õ , ½ ¾ Õ ½.Proof. By analogy with[4] the assertions of the theorem are obtained by repeated averaging which replaces the 'Euler' estimate by its mathematical expectation.Remark 2.1.To estimate the derivative of the solution Ù with respect to any direction (not only with respect to the direction of the velocity Ú) we should use at the first step of the algorithm the representation for ´Öµ ¿ Ù Brought to you by | Florida International University Libraries…”

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confidence: 99%

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Estimation of the gradient of the solution of an adjoint diffusion equation by the Monte Carlo method

Burmistrov¹,

Mikhaı̆lov²

2002

Russian Journal of Numerical Analysis and Mathematical Modelling

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For the case of isotropic diffusion we consider the representation of the weighted concentration of trajectories and its space derivatives in the form of integrals (with some weights) of the solution to the corresponding boundary value problem and its directional derivative of a convective velocity. If the convective velocity at the domain boundary is degenerate and some other additional conditions are imposed, this representation allows us to construct an efficient 'random walk by spheres and balls' algorithm. When these conditions are violated, transition to modelling the diffusion trajectories by the Euler scheme is realized, and the directional derivative of velocity is estimated by the dependent testing method, using the parallel modelling of two closely-spaced diffusion trajectories. We succeeded in justifying this method by statistically equivalent transition to modelling a single trajectory after the first step in the Euler scheme, using a suitable weight. This weight also admits direct differentiation with respect to the initial coordinate along a given direction. The resulting weight algorithm for calculating concentration derivatives is especially efficient if the initial point is in the subdomain in which the coefficients of the diffusion equation are constant.¼ has the meaning of the weighted [with weight ´¡µ] concentration of diffusion trajectories starting at the point Ö. In this work, we construct the representation of the concentration Ù´¡µ and its space derivatives in £

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“…The same procedure can be used to estimate the second term. Thus, by Theorem 4 in [4] we obtain the following assertion. …”

Section: Estimation Of Solutions By 'Random Walk By Spheres and Balls'mentioning

confidence: 63%

“…The Green 'nonpivotal' function and its normal derivative are expressed in terms of the formulae (see [4]): …”

Section: Estimation Of Solutions By 'Random Walk By Spheres and Balls'mentioning

confidence: 99%

“…In order to study variance finiteness, by analogy with [4] we modify the estimate as follows. We calculate the function À´Ö µ only for the case of the steṕ Ö µ ´Ö ¼ ¼µ, i.e.…”

Section: Estimation Of Solutions By 'Random Walk By Spheres and Balls'mentioning

confidence: 99%

“…The procedure in [4] can be used to study the value obtained. The same procedure can be used to estimate the second term.…”

Section: Estimation Of Solutions By 'Random Walk By Spheres and Balls'mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Estimation of the gradient of the solution of an adjoint diffusion equation by the Monte Carlo method

Burmistrov¹,

Mikhaı̆lov²

2002

Russian Journal of Numerical Analysis and Mathematical Modelling

Self Cite

View full text Add to dashboard Cite

show abstract

A new monte carlo method for solving a stationary diffusion equation

Cited by 1 publication

References 3 publications

Estimation of the gradient of the solution of an adjoint diffusion equation by the Monte Carlo method

Estimation of the gradient of the solution of an adjoint diffusion equation by the Monte Carlo method

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