Abstract
This paper shows that discrete linear equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, TST matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. Gauss least square method (GLSM), QR factorization method (QRFM), Cholesky decomposition method (CDM), and singular value decomposition (SVDM) failed to regularize these ill-posed problems. This paper introduces the eigenspace spectral regularization method (ESRM), which solves ill-posed discrete equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, and banded and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularizes such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator κ K = K − 1 K = 1 . Thus, the condition number of ESRM is bounded by unity, unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.
Highlights
Many mathematical problems are classified as ill-posed problems
The condition number of eigenspace spectral regularization method (ESRM) is bounded by unity, unlike the other regularization methods such as singular value decomposition (SVDM), Gauss least square method (GLSM), Cholesky decomposition method (CDM), and QR factorization method (QRFM)
Ill-posed discrete systems of linear equations are numerically unstable as they come with a large condition number
Summary
Many mathematical problems are classified as ill-posed problems. For instance, the discretization of linear ill-posed problems like the Fredholm integral equations of the first kind with a smooth kernel is ill-posed and referred to as a linear discrete ill-posed problem [1]. Ill-posed discrete systems of linear equations are numerically unstable as they come with a large condition number The instability of these matrix operators woefully compel the solution vector of the system of linear equations to be highly sensitive to perturbations in either the matrix operator or the right hand side of the system. All these regularization methods in restoring the wellposedness of the ill-posed discrete equation (1) fail to restore the existence, uniqueness, and stability conditions of wellposedness of a discrete system of linear equation [(7)]. In the area of application, the GLSM, QRFM, SVDM, and CDM have not been much consistently used to regularize an ill-posed discrete equation with perturbed right-hand side b in equation (1) (see authors in [4]).
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