  We propose a conceptually new method for solving nonlinear inverse scattering
problems (ISPs) such as are commonly encountered in tomographic ultrasound
imaging, seismology and other applications. The method is inspired by the
theory of nonlocality of physical interactions and utilizes the relevant
formalism. We formulate the ISP as a problem whose goal is to determine an
unknown interaction potential $V$ from external scattering data. Although we
seek a local (diagonally-dominated) $V$ as the solution to the posed problem,
we allow $V$ to be nonlocal at the intermediate stages of iterations. This
allows us to utilize the one-to-one correspondence between $V$ and the T-matrix
of the problem, $T$. Here it is important to realize that not every $T$
corresponds to a diagonal $V$ and we, therefore, relax the usual condition of
strict diagonality (locality) of $V$. An iterative algorithm is proposed in
which we seek $T$ that is (i) compatible with the measured scattering data and
(ii) corresponds to an interaction potential $V$ that is as
diagonally-dominated as possible. We refer to this algorithm as to the
data-compatible T-matrix completion (DCTMC). This paper is Part I in a two-part
series and contains theory only. Numerical examples of image reconstruction in
a strongly nonlinear regime are given in Part II. The method described in this
paper is particularly well suited for very large data sets that become
increasingly available with the use of modern measurement techniques and
instrumentation.
