Regular iterative algorithms

S. Rao and T. Kailath. Architecture design for regular iterative algorithms. In E. E. Swartzlander, editor, Systolic Signal Processing Systems, pages 209-297. Dekker Inc, New York, 1987. 

S. K. Rao. Regular iterative algorithms and their implementations on processor arrays. PhD thesis. Information System Lab, Stanford University, October 1985. 

S. Rao and T. Kailath. Regular iterative algorithms and their implementation on processor arrays. Proc. of IEEE, 76, number 3, pages 259-269, 1988. 

Notice that in this case, if we use regular iteration the result only converged if the equation was solved in a certain way. Using weighted iteration, it is possible to solve it either way and obtain a solution, but one way is clearly faster than the other. However, weighting will accelerate the algorithm in most cases and is relatively easy to implement, so it is a worthwhile method to use. 

Several numerical procedures for EADF evaluation have also been proposed. Morrison and Ross [19] developed the so-called CAEDMON (Computed Adsorption Energy Distribution in the Monolayer) method. Adamson and Ling [20] proposed an iterative approximation that needs no a priori assumptions. Later, House and Jaycock [21] improved that method and proposed the so-called HILDA (Heterogeneity Investigation at Loughborough by a Distribution Analysis) algorithm. Stanley et al. [22,23] presented two regularization methods as well as the method of expectation maximalization. 

When started with a smooth image, iterative maximum likelihood algorithms can achieve some level of regularization by early stopping of the iterations before convergence (see e.g. Lanteri et al., 1999). In this case, the regularized solution is not the maximum fikelihood one and it also depends on the initial solution and the number of performed iterations. A better solution is to explicitly account for additional regularization constraints in the penalty criterion. This is explained in the next section. 

Figure 3.3 The errors produced by an iterative reconstruction algorithm have patterns which appear, at each iteration, completely random (A), but if successive patterns are memorised together, it is possible to observe regular structures appearing in the memory matrix (B).

An alternative approach is based on the simple idea, which we discussed already above, that the regularization parameter a can be updated in the process of the iterative inversion. For example, one can use the following algorithm for the RCG method... 

In turn, the ARMS algorithm makes use of MH as a way to guarantee (under regularity conditions) that in the long run the marginal distribution estimate of X converges to its target function it. In summary, ARMS iterates as follows ... 

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