Regularization algorithms based on total least squares

Discretizations of inverse problems lead to systems of linear equations with a highly ill-conditioned coeecient matrix, and in order to compute stable solutions to these systems it is necessary to apply regularization methods. Classical regularization methods, such as Tikhonov's method or truncated SVD, are not designed for problems in which both the coeecient matrix and the right-hand side are known only approximately. For this reason, we develop TLS-based regularization methods that take this situation into account. Here, we survey two diierent approaches to incorporation of regularization, or stabilization, into the TLS setting. The two methods are similar in spirit to Tikhonov regularization and truncated SVD, respectively. We analyze the regularizing properties of the methods and demonstrate by numerical examples that in certain cases with large perturbations, these new methods are able to yield more accurate regularized solutions than those produced by the standard methods. In this paper we study linear, and possibly overdetermined, systems of equations Ax b whose m n coeecient matrix A (with m n) is very ill conditioned. We restrict our attention to the important case where all the singular values of A decay gradually to zero, i.e., with no particular gap in the spectrum. Such ill-conditioned linear systems arise frequently in connection with discretizations of ill-posed problems, such as Fredholm integral equations of the rst kind, and the term discrete ill-posed problem is sometimes used to characterize these systems. For more details about the underlying theory see, e.g., 2], 3], 6], 8] and the references therein. Suuce it here to say that the gradual decay of the singular values of A is an intrinsic property of discretizations of many ill-posed problems. For discrete ill-posed problems, the ordinary least squares solution x LS , as well as the ordinary total least squares solution x TLS , are hopelessly contaminated by noise in the directions corresponding to the small singular values of A or (A ; b). Because of this, it is necessary to compute a regularized solution in which the eeects of the noise are ltered out. Surveys of regularization methods for discrete ill-posed problems are given in 6] and 8].