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###

The final test we present in this paper considers a case with a
much larger scattering contribution:
. The results
for a grid with voxels and angular points are shown in
Fig. 10. Fig. 11 shows the results
for slices along the coordinate axes (the two coordinates being centered, respectively).
Even though the dynamic range of the mean intensities is huge, nearly
12 dex, the
results are quite accurate.
For this case the lack of spatial resolution in the inner
parts of the voxel grid can be seen. Here is huge and cannot
be fully resolved by the 3D RT code (the 1D code naturally has much higher
resolution). However, only a few voxels away from the center
the results agree very well.

The convergence
plots in Fig 12 show the results for a very difficult
test case with
for a fixed voxel
and solid angle grid with voxels and angular points.
For this test, the iteration fails completely. The diagonal
operator provides significant speed-up, but still requires
more than 1600 iterations to reach the required convergence limit.
For this test, the Ng acceleration does not work with the diagonal operator,
the iteration process failed immediately after it was started. It is likely that
a better result could be obtained if Ng acceleration is started in the
steep part of the diagonal operator's convergence, e.g., after about 500
regular iterations. The nearest neighbor operator leads to much faster
convergence, even without Ng acceleration the solution converges in about
450 iterations. Here, Ng acceleration works very well with the nearest
neighbor operator, convergence is reached after 177 iterations. This is
still about a factor of 2 more than for the 1D code, but much better
than with the diagonal operator. The variation of the convergence
rate for the nearest neighbor operator and Ng acceleration with
the size of the solid angle grid is shown
in Fig. 13. The case with the smallest
angular grid ( points) actually converges *more slowly* than the
and grids. The higher resolution grids convergence rate compares well with the
1D code. This highlights the importance of the non-local
operator
and a large enough solid angle grid for rapid convergence and accuracy.

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Peter Hauschildt
2006-01-09