A polarization tensor approximation for the Hessian in iterative solvers for non-linear inverse problems

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.