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Appendix A: Iterative maximization approach

We assume that the optimal model point position is shifted by some small distance, δφ_i, relative to the grid point location found using RMCLEAN, (A.1)The ML solution is obtained by minimizing H in Eq. (4) with respect to δφ_i, which gives (A.2)where is the same as , Eq. (2), except that the sum is over k ≠ i. The operator ℜ selects the real part of its argument. This expression cannot be solved analytically for δφ_i, but we can use it to search for the solutions iteratively. To do so we Taylor expand the exponential term in Eq. (A.2) to first order, thereby making H second order in δφ_i. This gives (A.3)which we can rewrite in the form (A.4)where (A.5)We must also solve for an updated flux. We can again extremize Eq. (4), this time with respect to m_i. We find (A.6)and thus (A.7)In our example implementation, we solve for δφ and m iteratively until convergence is achieved. We also attempt to merge nearby model components to reduce the degrees of freedom in the model according to the prescription described in Sect. 2.

We tried other iterative schemes, e.g. solving for position and flux changes by inverting the Hessian matrix of H, but found that the approach given here is the most stable.

© ESO, 2013