By measuring the distance between Tikhonov regularization and Landweber iteration with the weighted -norm of the difference between their corresponding singular value filters, we find the Tikhonov regularization parameter which minimizes this distance. Choosing the minimizing parameter then makes the Tikhonov and Landweber solutions as close as possible (in an sense) across a range of data and matrix models which characterize linear inverse problems. For large Landweber iteration number k, an asymptotic analysis of the weighted -norm determines the value of the first term C in the asymptotic approximation of the minimizing parameter. This approximation remains accurate even for smaller k.