Geodesic
Monte Carlo methods for solving the first boundary value problem for a polyharmonic equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 3, pp. 495-508

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Results of solving the first boundary value problem for a polyharmonic equation are presented. The technique is based on the probabilistic representation of the solution of this problem constructed by the authors. Such a solution is shown to be a parametric derivative of the solution of a special Dirichlet problem for the Helmholtz equation. Based on this fact, new “walk-by-spheres” algorithms for a polyharmonic equation are developed. This made it possible to construct an algorithm implementing the Monte Carlo method for estimating the covariance function of the solution of a biharmonic equation with random functional parameters. 
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     author = {V. L. Lukinov and G. A. Mikhailov},
     title = {Monte {Carlo} methods for solving the first boundary value problem for a~polyharmonic equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
     volume = {45},
     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_3_a13/}
}
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V. L. Lukinov; G. A. Mikhailov. Monte Carlo methods for solving the first boundary value problem for a polyharmonic equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 3, pp. 495-508. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_3_a13/