Geodesic
Random walk-on-spheres algorithms for solving mixed and Neumann boundary-value problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 2, pp. 209-220

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We propose a new approach to constructing Monte Carlo methods for solving mixed boundary value problems for elliptic equations with constant coefficients. We derived a mean-value relation for point values of the solution. As a consequence, the walk-on-spheres algorithm can still be used even after a trajectory hits the reflecting boundary. Such an approach is significantly more efficient than the standard one. 
@article{SJVM_2007_10_2_a7,
     author = {N. A. Simonov},
     title = {Random walk-on-spheres algorithms for solving mixed and {Neumann} boundary-value problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {209--220},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2007_10_2_a7/}
}
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N. A. Simonov. Random walk-on-spheres algorithms for solving mixed and Neumann boundary-value problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 2, pp. 209-220. http://geodesic.mathdoc.fr/item/SJVM_2007_10_2_a7/