The problem of CT is probably the most simple inverse problem. In general one has to solve the inverse problem for a partial differential equation. The numerical analysis of such inverse problems is still in its infancy. Of course one always can use a Newton method for solving the inverse problem simply as a nonlinear problem [31]. A method for solving the inverse problem (2.12) of ultrasound CT by an ART type method (compare section 3) is as follows [40]. For each direction 
, 
, define the nonlinear (Radon-type) operator 
where u is the solution to (2.12a) for 
with boundary values 
. Then one has
to find an approximate solution f to
This can be done by a Kaczmarz-type iteration. Starting out from an initial approximation 
we put
with some positive definite operator 
and a relaxation parameter 
. After p steps
one repeats the whole process. It turns out that for practical purposes, one can take to be the identity. In this form each step of (5.1) requires the solution of (2.12a) with boundary
values as specified above (this yields 
), and the solution of an adjoint boundary
value problem of the same structure for the application of 
. In the engineering literature this is known as the adjoint field method [9]. Nothing is known about convergence, and the performance - even though it is much faster than Newton-type methods - leaves much to be desired. Presently, 2D problems with a resolution 
can be solved in a few minutes on a workstation, but 3D problems in a clinical environment are
totally out of question. The same applies to equations such as (2.8) and to the equations of optical tomography (2.18).