Geophysical mapping and imaging are applications where we seek an approximate pseudo inverse of a matrix of very high order. Say, constructs theoretical data from model parameters using a linear operator .Experience shows that the transpose of the simulation operator provides a useful image. Thus the suggested image is where is the observed data, and is the transpose (Hilbert adjoint) of .Mathematically, this means that when the dimensionality is very high, we often approximate an inverse by a transpose .
Experience shows that a ``better'' image is often created by a unitary operator. Practitioners often discover improvements in the form of scaling diagonal matrices, say where and are the scaling diagonal matrices. We seek an operator that is as unitary as possible, i.e., .We believe that is also in some sense ideally preconditioned for linear solvers. A basic, recurring problem is a lack of straightforward theory for finding and .