## Journal of Applied Mathematics

### An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems

Chein-Shan Liu

#### Abstract

It is known that the steepest-descent method converges normally at the first few iterations, and then it slows down. We modify the original steplength and descent direction by an optimization argument with the new steplength as being a merit function to be maximized. An optimal iterative algorithm with $m$-vector descent direction in a Krylov subspace is constructed, of which the $m$ optimal weighting parameters are solved in closed-form to accelerate the convergence speed in solving ill-posed linear problems. The optimally generalized steepest-descent algorithm (OGSDA) is proven to be convergent with very fast convergence speed, accurate and robust against noisy disturbance, which is confirmed by numerical tests of some well-known ill-posed linear problems and linear inverse problems.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 154358, 15 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808302

Digital Object Identifier
doi:10.1155/2013/154358

Mathematical Reviews number (MathSciNet)
MR3145010

Zentralblatt MATH identifier
06950536

#### Citation

Liu, Chein-Shan. An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems. J. Appl. Math. 2013 (2013), Article ID 154358, 15 pages. doi:10.1155/2013/154358. https://projecteuclid.org/euclid.jam/1394808302

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