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Solving the multidimensional difference biharmonic equation by the Monte Carlo method. (Russian, English) Zbl 0993.65006

Sib. Mat. Zh. 42, No. 5, 1125-1135 (2001); translation in Sib. Math. J. 42, No. 5, 942-951 (2001).
The authors construct and justify new weighted Monte Carlo methods for estimating solutions to the Dirichlet problem for the multidimensional difference biharmonic equation using the so-called random walk by lattice. Vector versions of the algorithms constructed are extended to metaharmonic equations, preserving conditions of unbiasedness of estimates and boundedness of their dispersions, which makes it possible to present a simple algorithm for estimating the first eigenvalue for the multidimensional difference Laplace operator.
Moreover, special algorithms based on random walk by lattice are presented. These algorithms make it possible to estimate solutions to the Dirichlet problem for the biharmonic equation with weak nonlinearity and mixed boundary conditions including the Neumann problem. The authors conclude that the weighted Monte Carlo algorithms are effective for estimating solutions to multidimensional problems under a small number of points and for estimating parametric derivatives.

MSC:

65C05 Monte Carlo methods
35J40 Boundary value problems for higher-order elliptic equations
65N06 Finite difference methods for boundary value problems involving PDEs
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