Listing 1. Contrast-source inversion–electrical-property tomography (CSI–EPT).

Given initial guesses ${\tilde{\chi }}^{[0]}$ and ${\tilde{\mathbf{w}}}^{[0]}$ for the contrast function and contrast source, respectively, For $k=1,2\dots$ Fix the contrast to ${\tilde{\chi }}^{[k-1]}$ and update the contrast source according to the update formula $${\tilde{\mathbf{w}}}^{[k]}={\tilde{\mathbf{w}}}^{[k-1]}+{\alpha }^{[k]}{\mathbf{v}}^{[k]}.$$ Compute the corresponding electric-field strength ${\widehat{\mathbf{E}}}^{[k]}$ according to (cf. Equation (5)) $${\tilde{\mathbf{E}}}^{[k]}(\mathbf{x})={\widehat{\mathbf{E}}}^{\mathrm{b}}(\mathbf{x})+{\mathcal{G}}_{\mathrm{body}}\{{\tilde{\mathbf{w}}}^{[k]}\}(\mathbf{x}).$$ Knowing contrast source ${\tilde{\mathbf{w}}}^{[k]}$ and corresponding electric-field strength ${\tilde{\mathbf{E}}}^{[k]}$, determine contrast function ${\tilde{\chi }}^{[k]}$ from constitutive relation ${\tilde{\mathbf{w}}}^{[k]}={\tilde{\chi }}^{[k]}{\tilde{\mathbf{E}}}^{[k]}$ by solving least-squares problem $\parallel \tilde{\chi }{\tilde{\mathbf{E}}}^{[k]}-{\tilde{\mathbf{w}}}^{[k]}{\parallel }_{\mathrm{body}}^2$ for minimum norm contrast function $\tilde{\chi }$. Stop if objective function is smaller than user-specified tolerance level, or if maximum number of iterations have been reached. End.