Mathematical background

This page summarizes the mathematics implemented in SIES. For complete statements and proofs, see Ammari & Kang, Polarization and Moment Tensors (Springer, 2007) and the papers listed on the home page.

The conductivity problem

Let \(D_1, \dots, D_L\) be small, disjoint, \(C^2\)-smooth inclusions in \(\mathbb{R}^2\) with conductivities \(k_l \neq 1\) in a background of conductivity \(1\). A point source at \(x_s\) generates the potential \(u\) solving

\[ \nabla \cdot \big( 1 + \textstyle\sum_l (k_l - 1) \chi_{D_l} \big) \nabla u = \delta_{x_s}, \qquad u(x) - G(x - x_s) = O(|x|^{-1}), \]

where \(G(x) = \frac{1}{2\pi}\log|x|\) is the fundamental solution of the Laplacian. The measured data are the perturbations \(u - G\) collected at receivers for every source: the multistatic response (MSR) matrix.

Layer potentials

The perturbation admits the representation

\[ u - G = \sum_{l} S_{D_l}[\phi_l], \]

where \(S_D\) is the single layer potential and the densities \(\phi_l\) solve a system of boundary integral equations driven by the adjoint Neumann–Poincaré operator \(K_D^*\):

\[ (\lambda_l I - K_{D_l}^*)[\phi_l] - \sum_{m \neq l} \frac{\partial S_{D_m}[\phi_m]}{\partial \nu_l} = \frac{\partial G(\cdot - x_s)}{\partial \nu_l}, \qquad \lambda_l = \frac{k_l + 1}{2(k_l - 1)}. \]

SIES discretizes these operators with P0 boundary elements (sies.operators), with analytic treatment of the singular diagonals.

Generalized polarization tensors

The far-field expansion of \(u - G\) is governed by the contracted generalized polarization tensors (CGPT). With harmonic polynomials \(\mathrm{Re}(z^m)\), \(\mathrm{Im}(z^m)\):

\[ M^{cc}_{mn} = \int_{\partial D} \mathrm{Re}(z^n)\, (\lambda I - K_D^*)^{-1}\!\left[ \frac{\partial\, \mathrm{Re}(z^m)}{\partial \nu} \right] ds, \]

and similarly for the \(cs\), \(sc\), \(ss\) blocks (sies.asymptotics). The MSR matrix is then, to leading order, the linear image of the CGPT:

\[ \mathrm{MSR} \approx A_s \, M \, A_r^T, \]

where \(A_s, A_r\) depend only on the acquisition geometry. Inverting this linear system reconstructs the CGPT from measurements (sies.pde); for equispaced circular acquisitions the least-squares inverse is known in closed form.

Invariant shape descriptors

Under a translation \(z_0\), rotation \(\theta\) and scaling \(s\) of the shape, the complex CGPT pair \((N_1, N_2)\) transforms explicitly through lower-triangular binomial matrices and diagonal phase matrices. Normalizing out these factors yields descriptors \((\mathcal{I}_1, \mathcal{I}_2)\) invariant under all three transformations (sies.dictionary). Identification then amounts to a nearest-neighbor search in descriptor space, which is robust to measurement noise.

Tracking

When the target's CGPT is known, the position and orientation \((x_t, \theta_t)\) become the unknowns of the observation model

\[ h(x_t, \theta_t) = A_s \, M(x_t, \theta_t)\, A_r^T, \]

which is differentiable in closed form. An Extended Kalman Filter with a constant-velocity motion model estimates the trajectory online (sies.tracking).