By Tokunaga Sh.

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Thus the square discrepancy of a sequence can also be measured in finite time. The Hammersley sequence and the Sobol sequence are quasi-random sequences with low square discrepancy. They are generated using algorithms from number theory, rather than randomly. We shall map them onto D and then measure the corresponding circular discrepancy. To obtain a sequence in D from a sequence in I2, we first need a suitable mapping between these two areas. 1 M a p p i n g the square to a disc We shall use two different ways of obtaining a sequence in D from one on 12.

7 Numerical results 2 Here, we look at the behavior of circular discrepancy with N . 3 shows the circular discrepancy for the regular vortex sequence, the Sobol sequence and the Hammersley sequence for N up to 3400. The data points for the ‘regular vortex’ are obtained using p = 2000 and p = 1000. 1 to the respective sequence on the square. The general trend is that the discrepancy decreases with more points. However this is not always the case. For almost all N , the ‘regular vortex’ distribution has the lowest circular discrepancy among the three sequences, with the Hammersley sequence coming closest to it.

4 (a) Uniform distribution of 400 points in 1 2 . 1, the distribution becomes rays on a disc. Fig. 5 (a) 400 points from the Hammersley sequence. 1. A Monte Carlo Algorithm for Generating a Low Circular Discrepancy Sequence 17 Fig. 6 (a) 400 points from the Antonov-Saleev variant of the Sobol sequence. 1. Fig. 7 (a) A regular vortex distribution of 400 points from a particle Monte Carlo simulations at p = 1000 and p = 2000. The boundary of the sequence is set to be d m . 1. 18 Vortex Dominated Flows Fig.