Институт вычислительной математики и математической геофизики СОРАН

Abstracts

Statistical simulation and Monte Carlo method

We consider the problem of reflected light intensity calculation for narrowly collimated detector. It is well known [1] that this problem may be solved with use of the double local estimate of Monte Carlo method. Usually for this estimate the additional point is constructed in the "visible area" of detector. However this estimate has an infinite variance because it contains the distance between the point of Markov chain and the additional point in the denominator. This distance may be very short.

Also, there is a so-called biased modification of the double local estimate with a finite variance. In this case, we exclude the points of chain which are closer than R from the additional point. In the report we find the bias of this estimate and obtain the expression that relates the minimum distance R, number of trajectories, and required error. This expression allow us to calulate the optimal value of the distance R.

Moreover, we propose an unbiased modification of the double local estimate (the combined estimate). In this estimate, the additional point is constructed differently in two areas. These areas are chosen in such a way that the variance is finite. If the point of Markov chain falls into the area that contains the "visibity area" of the detector, then the additional point is constructed according to the transition density of Markov chain. In another area, the additional point is constructed according to the ordinary double local estimate.

We calculate the radiation intensity for a semiinfinite space and the detector separated from the space. Obtained results are in agreement with the theory. Time dependence of the radiation intensity shows that the local estimate underestimates the result in the case of a narrow viewing angle of the detector and small number of trajectories. In this case, the biased double local estimate is close to the combined one and shows satisfactory results.

This work is supported by RFBR grant N 09-01-00035-a.

[1] Marchuk, G.I., Mikhailov, G.A., Nazaraliev, M.A., et al.: The Monte Carlo Method in Atmospheric Optics. Springer, Berlin Heidelberg (1980)

Note. Abstracts are published in author's edition

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