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6. Mean RCS computed using four different sampling methods. Fig. 7. Mean RCS and sensitivity for a two-parameter random problem computed using the adaptive sparse grid method. and the variance of RCS as Q var(RCS) = ωi RCSi − RCS 2 , (27) i =1 where the number of terms, Q , and the integration weights, ωi , depend on the speciﬁc integration technique used. We ﬁrst consider the problem with one random parameter, taken to be a cylinder with one hole of random size. The size π , π ) in the polar angle.
Liu et al. / Applied Numerical Mathematics 61 (2011) 24–37 35 Fig. 8. RCS computed for cylinders illustrated in Fig. 4, with 2 (left), 3 (middle), and 4 (right) holes with uniformly distributed hole size and angle of incidence of the illuminating wave. Fig. 9. Poisson density associated with the number of holes in the cylinder. 3. Example of a higher-dimensional problem To simulate a more realistic random wave problem, we need to consider higher-dimensional problems. We achieve this in two different ways.
This is vastly advantageous over previously used methods which simply compare solutions at the two levels, often resulting in having to compute an additional level at substantial cost – typically the cost of a new level is comparable to the combined cost of all previous levels. Instead of this, we use the Richardson extrapolation to estimate the result at the next level, hence dramatically reducing the overall cost without impacting the accuracy. To test the validity of this approach as a way to accelerate the adaptive sparse grid algorithm and reduce the computational cost we consider a set of high-dimensional test functions proposed in [7,8].