The method of Tikhonov regularization is commonly used to obtain regularized solutions of ill-posed linear inverse problems. We use its natural connection to optimal Bayes estimators to determine optimal experimental designs that can be used with Tikhonov regularization; they are designed to control a measure of total relative efficiency. We present an iterative/semidefinite programming hybrid method to explore the configuration space efficiently. Two examples from geophysics are used to illustrate the type of applications to which the methodology can be applied. © 2013 SAGE Publications.