The eigenspace spectral regularization method for solving discrete ill-posed systems (Q6097682)

Summary: 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 \(\kappa(K) = \|K^{-1}K\| = 1\). Thus, the condition number of ESRM is bounded by unity, unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.