TY - JOUR
T1 - Direct Implementation of Tikhonov Regularization for the First Kind Integral Equation
AU - Jozi , Meisam
AU - Karimi , Saeed
JO - Journal of Computational Mathematics
VL - 3
SP - 335
EP - 353
PY - 2022
DA - 2022/02
SN - 40
DO - http://doi.org/10.4208/jcm.2010-m2020-0132
UR - https://global-sci.org/intro/article_detail/jcm/20240.html
KW - First kind integral equation, Golub-Kahan bidiagonalization, Tikhonov regularization, Quadrature Discretization.
AB - <p style="text-align: justify;">A common way to handle the Tikhonov regularization method for the first kind Fredholm integral equations, is first to discretize and then to work with the final linear system.
This unavoidably inflicts discretization errors which may lead to disastrous results, especially when a quadrature rule is used. We propose to regularize directly the integral
equation resulting in a continuous Tikhonov problem. The Tikhonov problem is reduced
to a simple least squares problem by applying the Golub-Kahan bidiagonalization (GKB)
directly to the integral operator. The regularization parameter and the iteration index are
determined by the discrepancy principle approach. Moreover, we study the discrete version
of the proposed method resulted from numerical evaluating the needed integrals. Focusing
on the nodal values of the solution results in a weighted version of GKB-Tikhonov method
for linear systems arisen from the Nyström discretization. Finally, we use numerical experiments on a few test problems to illustrate the performance of our algorithms.</p>