  For many inverse parameter problems for partial differential equations in
which the domain contains only well-separated objects, an asymptotic solution
to the forward problem involving 'polarization tensors' exists. These are
functions of the size and material contrast of inclusions, thereby describing
the saturation component of the non-linearity. As such, these asymptotic
expansions can allow fast and stable reconstruction of small isolated objects.
In this paper, we show how such an asymptotic series can be applied to
non-linear least-squares reconstruction problems, by deriving an approximate
diagonal Hessian matrix for the data misfit term.
  Often, the Hessian matrix can play a vital role in dealing with the
non-linearity, generating good update directions which accelerate the solution
towards a global minimum which may lie in a long curved valley, but
computational cost can make direct calculation infeasible. Since the
polarization tensor approximation assumes sufficient separation between
inclusions, our approximate Hessian does not account for non-linearity in the
form of lack of superposition in the inverse problem. It does however account
for the non-linear saturation of the change in the data with increasing
material contrast. We therefore propose to use it as an initial Hessian for
quasi-Newton schemes.
  This is demonstrated for the case of electrical impedance tomography in
numerical experimentation, but could be applied to any other problem which has
an equivalent asymptotic expansion. We present numerical experimentation into
the accuracy and reconstruction performance of the approximate Hessian,
providing a proof of principle of the reconstruction scheme.
