Variations on a Theme

The ART algorithm for solving the system A x = b can be modified to include relaxation and regularization. There are also block-iterative and simultaneous versions of the ART. We consider these variations in this chapter.

Relaxed ART

The ART can be slow to converge and can be accelerated by altering the order in which the individual equations are accessed [111]. Another way to accelerate the ART is through the use of relaxation. We can introduce relaxation in ART using the relaxed ART (REART):

for n = 1 , , N, k = 0 , 1, , and m( k) = k(mod M ) + 1 and ? _k positive scalars in ( , 2 ? ), where is a scalar in (0 , 1).

A Simultaneous Version of ART

A simultaneous version of the ART was introduced by Cimmino [70]. It is obtained by projecting orthogonally onto each hyperplane simultaneously, then averaging the result. In closed form the Cimmino method is the following: for k = 0 , 1, , let

The iterative step can be written in vector form as

The Cimmino method is a special case of the Landweber iterative method given in Equation (44.5). Cimmino s method can also employ relaxation: using positive relaxation parameters ? _k in place of 1/ M, we get the relaxed Cimmino method

The convergence proof...