Noniterative solution of inverse problems by the linear least square method

Abstract

In this paper, a noniterative linear least-squares error method developed by Yang and Chen for solving the inverse problems is re-examined. For the method, condition for the existence of a unique solution and the error bound of the resulting inverse solution considering the measurement errors are derived. Though the method was shown to be able to give the unique inverse solution at only one iteration in the literature, however, it is pointed out with two examples that for some inverse problems the method is practically not applicable, once the unavoidable measurement errors are included. The reason behind this is that the so-called reverse matrix for these inverse problems has a huge number of 1-norm, thus, magnifying a small measurement error to an extent that is unacceptable for the resulting inverse solution in a practical sense. In other words, the method fails to yield a reasonable solution whenever applied to an ill-conditioned inverse problem. In such a case, two approaches are recommended for decreasing the very high condition number: (i) by increasing the number of measurements or taking measurements as close as possible to the location at which the to-be-estimated unknown condition is applied, and (ii) by using the singular value decomposition (SVD). © 2001 Elsevier Science Inc. All rights reserved.