  This paper is concerned with the solution of large-scale linear discrete
ill-posed problems with error-contaminated data. Tikhonov regularization is a
popular approach to determine meaningful approximate solutions of such
problems. The choice of regularization matrix in Tikhonov regularization may
significantly affect the quality of the computed approximate solution. This
matrix should be chosen to promote the recovery of known important features of
the desired solution, such as smoothness and monotonicity. We describe a novel
approach to determine regularization matrices with desired properties by
solving a matrix nearness problem. The constructed regularization matrix is the
closest matrix in the Frobenius norm with a prescribed null space to a given
matrix. Numerical examples illustrate the performance of the regularization
matrices so obtained.
