  Many applications in science and engineering require the solution of large
linear discrete ill-posed problems that are obtained by the discretization of a
Fredholm integral equation of the first kind in several space-dimensions. The
matrix that defines these problems is very ill-conditioned and generally
numerically singular, and the right-hand side, which represents measured data,
typically is contaminated by measurement error. Straightforward solution of
these problems generally is not meaningful due to severe error propagation.
Tikhonov regularization seeks to alleviate this difficulty by replacing the
given linear discrete ill-posed problem by a penalized least-squares problem,
whose solution is less sensitive to the error in the right-hand side and to
round-off errors introduced during the computations. This paper discusses the
construction of penalty terms that are determined by solving a matrix-nearness
problem. These penalty terms allow partial transformation to standard form of
Tikhonov regularization problems that stem from the discretization of integral
equations on a cube in several space-dimensions.
