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We present in this report our recent results on stochastic methods for solving a 2D Lame equation with random right-hand side describing stochastic volume loads. These problems are handled in different application fields, in particular, when analyzing the interaction of the elastic energy of polymers with turbulent energy of flows with large Reynolds numbers, in problems of wave radiation of biological tissues in elastography - a new intensively developing field of the diagnostic medicine, in poroelasticity, and in many others, where the physical fields are extremely irregular. In many such problems stochastic finite element methods are successively applied, however generally they are quite expensive since one needs to calculate large ensembles of solution in the whole domain even if it is desired to have a compact information in the form of a correlation and structure tensors.
The basic idea of the Monte Carlo approach to these problems is exactly in that we arrange the methods so that the statistical characteristics are evaluated directly, without calculating the ensemble of solution in the whole domain. In this study we suggest Random Walk algorithms for solving the first boundary value problem for the Lame equation with loads in the form of isotropic vector random field with a given spectral tensor. The method is based on a generalization of the known global algorithm using the adjoint trajectories.
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