Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space

Abstract

The conjugate gradient method for nonsymmetric linear operators in Hilbert space is investigated. Conditions on the coincidence of the full and truncated versions, known from the finite-dimensional case, are extended to the Hilbert space setting. The focus is on preconditioning by the symmetric part of the operator, in which case estimates are given for the resulting condition number. An important motivation for this study is given by differential operators, for which the obtained estimates yield mesh independent conditioning properties of the full CG method, and are in fact achieved by the simpler truncated version.

^†Dedicated to the memory of Jean-Jacques Lions: a source of inspiration for rigour in Applied Mathematics.

Keywords:

• Conjugate gradient method in Hilbert space
• Nonsymmetric linear operators
• Mesh independent
• Preconditioning

acknowledgment

This research was done during J. K.'s Hungarian post-doc scholarship Magyary Zoltán.

Notes

^†Dedicated to the memory of Jean-Jacques Lions: a source of inspiration for rigour in Applied Mathematics.