Increasing the efficiency of radiation shielding calculations by using antithetic variates

This paper considers the passage of a parallel, uniform stream of particles through a parallel-sided, homogeneous slab of material capable of scattering and absorption. Particle histories are followed until the particle escapes from the near or far surface of the slab. Each particle carries a weight (initially 1, at the near surface of the slab) and the total weight of particles escaping through the far surface is compared with the total number of particles initiated originally. To increase efficiency of the computation, (i) we disregard absorption, by multiplying the weight of each particle by its non-absorption probability at each collision, and (ii) we select N parallel geometric surfaces inside the slab (parallel to the sides of the slab) and, as a particle crosses any such surface – if it is crossing it in a backward direction – we play “Russian roulette” with probability 1/m (multiplying its weight by m if it survives), while – if it is crossing the surface in a forward direction – we “split” the particle into m identical particles (all initially moving with the same velocity vector), each with weight 1/m times that of the particle we split. In the case of splitting, the paper compares two alternative techniques. Either (1) we select m independent random numbers with which to compute the m free path lengths to the next (scattering) collision, or (2) we select one random number and form m “antithetically” balanced values with which to compute the m free path lengths (with appropriate weight-adjustments). The results obtained are very promising. The use of “antithetic splitting” increases the efficiency (in terms of the work required to obtain a given variance in the resulting estimate of the transmission coefficient) by a factor ranging from about 1 to about 30 (the improvement appears to be most marked for scattering probabilities around 0.5, rather than near 1.0).