Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations

Abstract

Abstract.Iterative methods are developed for computing the Moore Penrose pseudoinverse solution of a linear system Ax=b, where A is an m × n sparse matrix. The methods do not require the explicit formation of ArA or AA v and therefore are advantageous to use when these matrices are much less sparse than A itself. The methods are based on solving the two related systems (i) x= Ary, AAIy= b, and (ii) ArAx = ATb. First it is shown how the SORand SSOR-methods for these two systems can be implemented efficiently. Further, the acceleration of the SSOR-method by Chebyshev semi-iteration and the conjugate gradient method is discussed. In particular it is shown that the SSOR-cg method for (i) and (ii) can be implemented in such a way that each step requires only two sweeps through successive rows and columns of A respectively. In the general rank deficient and inconsistent case it is shown how the pseudoinverse solution can be computed by a two step procedure, Some possible applications are mentioned and numerical results are given for some problems from picture reconstruction.

A general extending and constraining procedure for linear iterative methods

“…In this respect have been designed extensions of the algorithm (3) for the inconsistent case of (1) which are based on relaxation parameters, column relaxations or supplementary steps introduced in the iteration (see [21,11,23,26,2,30] and references therein). Moreover, for problems related to image reconstruction in computerized tomography the iteration step (3) was combined with a constraining strategy, usually acting on the components of the successive approximations [20,23,28]).…”

“…The fact that it was used in this section as an application for our considerations in section 2 is only an historical poin of view. Regarding the extension procedure proposed in (35)-(37), it differs from the older "multi-steps" methods (see [30,2]) or methods that uses in the inconsistent case for (1), the associated augmented system (which is always consistent) by the following two aspects: firstly, the modification of the right-hand side in (36) is included in the iteration of the extended algorithm, thus in the global convergence of the algorithm (so do no more appear accumulation of errors due to approximate solutions in the different steps of "multi-steps" methods), and the second one, the fact that, acting on the initial problem, the extended method (35)-(37) is influenced by its condition number and not by the squared one, as in the case of augmented system or normal equation (see [3] and the numerical experiments in [22]). Moreover, we want to point out that the extending and constraining approach developed under the assumptions (8)- (12) is quite general and can include other algorithms, in image reconstruction or elsewhere (e.g.…”

The effect of the nonlinearity on GCV applied to conjugate gradients in computerized tomography

“…The elements of the matrix L are not explicitly known, but in [2] was shown an effi cient way to compute the products A t C −t ω r and C −1 ω Aw.…”

“…Another approach is to apply an iterative method directly to the least square problem minimize ||Ax − b|| 2 and take as as a 'solution' an early iterate of the method. In [10] and [23], it is proven that this approach is equivalent, in some sense, to (1.3).…”

Krylov subspace methods for solving large unsymmetric linear systems

“…Note that sm = hm + XmV*vm+x and that sm is the solution of the least squares problem (see [19]) (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) "iin||Fmj-/im+Imt;m+1||, for which many efficient algorithms are available; see [3], [13]. It should be added that only a moderate accuracy is needed in practice, so the bidiagonalization algorithm BIDIAG described in [13] is suitable for solving (3.13) with moderate accuracy.…”