Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems

The aim of this paper is to extend the applicability of an algorithm for solving inconsistent linear systems to the rank-deficient case, by employing incomplete projections onto the set of solutions of the augmented system Ax-r = b. The extended algorithm converges to the unique minimal norm solution of the least squares solutions. For that purpose, incomplete oblique projections are used, defined by means of matrices that penalize the norm of the residuals. The theoretical properties of the new algorithm are analyzed, and numerical experiences are presented comparing its performance with some well-known projection methods.Facultad de Ciencias ExactasFacultad de Ingenierí

Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems

The aim of this paper is to extend the applicability of an algorithm for solving inconsistent linear systems to the rank-deficient case, by employing incomplete projections onto the set of solutions of the augmented system Ax-r = b. The extended algorithm converges to the unique minimal norm solution of the least squares solutions. For that purpose, incomplete oblique projections are used, defined by means of matrices that penalize the norm of the residuals. The theoretical properties of the new algorithm are analyzed, and numerical experiences are presented comparing its performance with some well-known projection methods.Facultad de Ciencias ExactasFacultad de Ingenierí

Globally Convergent Image Reconstruction for Emission Tomography Using Relaxed Ordered Subsets Algorithms

We present two types of globally convergent relaxed ordered subsets (OS) algorithms for penalized-likelihood image reconstruction in emission tomography: modified block sequential regularized expectation-maximization (BSREM) and relaxed OS separable paraboloidal surrogates (OS-SPS). The global convergence proof of the existing BSREM (De Pierro and Yamagishi, 2001) required a few a posteriori assumptions. By modifying the scaling functions of BSREM, we are able to prove the convergence of the modified BSREM under realistic assumptions. Our modification also makes stepsize selection more convenient. In addition, we introduce relaxation into the OS-SPS algorithm (Erdogan and Fessler, 1999) that otherwise would converge to a limit cycle. We prove the global convergence of diagonally scaled incremental gradient methods of which the relaxed OS-SPS is a special case; main results of the proofs are from (Nedic and Bertsekas, 2001) and (Correa and Lemarechal, 1993). Simulation results showed that both new algorithms achieve global convergence yet retain the fast initial convergence speed of conventional unrelaxed ordered subsets algorithms.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86017/1/Fessler67.pd