The Eigenspace Spectral Regularization Method for solving Discrete
Ill-Posed Systems
Abstract
In this paper, it is shown that discrete equations with Hilb ert matrix
operator, circulant matrix operator, conference matrix operator, banded
matrix operator, and sparse matrix operator are ill-posed in the sense
of Hadamard. These ill-posed problems cannot be regularized by Gauss
Least Square Method (GLSM), QR Factorization Method (QRFM), Cholesky
Decomposition Method (CDM) and Singular Value Decomposition (SVDM). To
overcome the limitations of these methods of regularization, an
Eigenspace Spectral Regularization Method (ESRM) is introduced which
solves ill-p os ed discrete equations with Hilb ert matrix operator,
circulant matrix operator, conference matrix operator, banded matrix
operator, and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM,
the ESRM regularize such a system. In addition, the ESRM has a unique
property, the norm of the eigenspace spectral matrix operator κ (K) =
||K − 1K|| = 1. Thus, the condition
number of ESRM is bounded by unity unlike the other regularization
methods such as SVDM, GLSM, CDM, and QRFM.