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

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.