Sensitivity to Initial Guess

The inversion of matrices arising from the ART family of algorithms from limited projection data is a mathematically ill-posed problem. As a rule, the number of equations here is much smaller than the number of unknowns. This makes the solution- set infinite in the sense that a unique solution is not guaranteed. Different initial guesses, may in principle, lead to different solution of this infinite set. In the absence of any knowledge about the field being studied, it is a difficult task to prescribe the initial guess. The sensitivity of the algorithms to the initial guess has been studied with reference to three different fields, namely:

1. a constant temperature field (= )
2. temperature distribution corresponding to two-longitudinal rolls, and
3. random field between and with an RMS value of

The guesses 1and 2 were seen to qualitatively reproduce the thermal field of figure 59 quite well(the reconstructed thermal field have not been shown as they are very close to the original). The noise present in the third guess was seen to be present in the reconstructed data. But the noise could be filtered in the frequency domain using a band-pass filter function. The reconstructed field after noise-removal was seen to be similar to the original in Figure 59. The errors, number of iterations and the CPU time for the three initial guesses are presented in Table 12.The fractional distribution of errors are reported in Table 13. With initial guesses 1and 2, the RMS and fractional of errors can be seen to be small for all the three algorithms. The maximum error is larger, but with reference to Table 13, it can be seen that large errors are restricted to small areas and are hence localized. Thus, in effect the initial guesses 1 and 2 may be considered to be equivalent. The errors corresponding to the third guess are uniformly higher for the proposed AVMART1 and AVMART2 algorithms, but small for AVMART3. The number of iterations for AVMART3are also smaller. Hence, AVMART3emertes as the best algorithm among those proposed in terms of error and CPU time for a noisy initial guess. For an unbiased and regular initial guess, computations over a wider range of parameters show AVMART2 to be the best (see the section on Sensitivity to Noise in Projection Data).

Table 13: Fractional Distribution of the Error over the fluid layer

The insensitivity of AVMART3algorithm to noise can be explained as follows , In the other two algorithms, correction is applied by finding the th root of the product of all corrections arising from ray with the cell under question. This improves the estimate of the path integral.