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

TL;DR

This paper shows how 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 and proposes to use it as an initial Hessian for quasi-Newton schemes.

Abstract

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.

Figures (11)

• Figure 1: Two types of non-linearity observed in inverse boundary-value problems for PDEs, in the case of EIT. (a) Saturation of a datum $d$ with material contrast of a single inclusion. (b) Lack of linear superposition, $\|\mathbf{d}'_{\mathrm{both}} - (\mathbf{d}'_1 + \mathbf{d}'_2)\|_2/\|\mathbf{d}'_{\mathrm{both}}\|_2$ against the separation distance between the boundary of two inclusions. $\mathbf{d}'_{\mathrm{both}}$ is the change in data from a homogeneous domain with two inclusions, $\mathbf{d}'_1$ and $\mathbf{d}'_2$ the change in data with the first and second inclusion alone, respectively.
• Figure 2: Derivative of components of the polarization tensor for an ellipsoid $B$ with $a=1$, $b=2$, $|B|=1$.
• Figure 3: Comparison of true and approximate diagonals of the Hessian matrix calculated on a homogeneous disc, for simulated data with a single inclusion with conductivity $\sigma=2.3$. The true Hessian is shown top-left (with elements of the matrix mapped to their corresponding element in the domain) and in blue below. The approximate Hessian using the freespace Green's function is shown top-centre and in red below. The approximate Hessian using the Neumann function for the disc is shown top-right, and in yellow below. Higher element indices correspond to elements closer to the boundary.
• Figure 4: $H_{ii}$ for selected elements $i$ away from the boundary, as a function of inclusion conductivity $\sigma$. Blue shows the true Hessian, and red and yellow the approximations using the free space Green's function and Neumann function on a unit disc, respectively.
• Figure 5: Relative error of the BFGS approximate Hessian to the true Hessian in Frobenius norm, initialised by the polarization tensor approximation (blue) and $\mathop{\mathrm{diag}}\limits(\mathrm{J}^T\mathrm{J})$ (red).

Theorems & Definitions (3)

• Remark 3.1
• Remark 4.1
• Remark 4.2