@article {IOPORT.02139524,
    author       = {Lukinov, V.L. and Mikhailov, G.A.},
    title        = {Probabilistic representation and Monte Carlo methods for the first boundary value problem for a polyharmonic equation.},
    year         = {2004},
    journal      = {Russian Journal of Numerical Analysis and Mathematical Modelling},
    volume       = {19},
    number       = {5},
    issn         = {0927-6467},
    pages        = {433-448},
    publisher    = {Walter de Gruyter, Berlin},
    doi          = {10.1163/1569398042395989},
    abstract     = {Consider the first boundary value problem for a polyharmonic equation of the form: $$(\Delta +c)^{p+1}u=-g, (\Delta+c)^k u\left| \right._\Gamma=\phi_k, (k=0,\dots,p).\tag1$$ The solution to the problem (1) is shown to be represented by the parameter derivative of the $p$th order of the solution to the Dirichret problem for the Helmholtz equation. Using the probabilistic representation of the solution to the Dirichret problem for the Helmholtz equation, new Monte Carlo estimation algorithm, `walk on the grid' and `random walk by spheres' algorithms, are proposed to estimate the solution to the problem (1). The properties of the bias and the variance of `$\varepsilon$- estimate', in which the `random walk by spheres' process terminates when it reaches the $\varepsilon'$-neighborhood, are also investigated based on the martingale property. The computational cost of the algorithm is provided.},
    reviewer     = {Katsuji Uosaki (Osaka)},
    identifier   = {02139524},
}