Variational optimization approach for reconstruction of dielectric permittivity and conductivity functions using partial boundary measurements

Abstract

We present a variational optimization approach for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and conductivity functions in time-dependent Maxwell's system using limited boundary observations of the electric field. The variational optimization approach is based on constructing a weak form of a Lagrangian which allows to use finite element based reconstruction algorithms. The optimality conditions for the Lagrangian and stability estimate for the adjoint problem are derived, as well as Frechét differentiability of it and of the regularized Tikhonov functional are also presented. Two- and three-dimensional numerical studies confirm our theoretical investigations.

Figures (12)

• Figure 1: 2D tests, test 1. Results from CGA, Algorithm 1. Figures a) and b) are connected to $\varepsilon$: a) exact $\varepsilon$, b) reconstructed $\varepsilon^{100}$ after 100 iterations in CGA. Figures c) and d) are connected to $\sigma$: c) exact $\sigma$, d) reconstructed $\sigma^{100}$ after 100 iterations in CGA.
• Figure 2: 2D tests, test 1. Convergence plots of errors over CGA iterations: a), b) errors $e^m_\varepsilon$ concerning $\varepsilon$; c), d) errors $e^m_\sigma$ concerning $\sigma$; e), f) errors $e^m_E$ concerning the term $E- \tilde{E}_{\rm obs}$. We note that figures a), c) and e) show the relative $L^2$-errors, and figures b), d) and f) show the supremum norms.
• Figure 3: 2D tests, test 1. Convergence plots of $L^2$ norms over CGA iterations: a) $\| \lambda \|$; b) $\|g_\varepsilon \|$; c) $\| g_\sigma\|$.
• Figure 4: 2D tests, test 2. Results from CGA, Algorithm 1. Figures a) and b) are connected to $\varepsilon$: a) exact $\varepsilon$, b) reconstructed $\varepsilon^{100}$ after 100 iterations in CGA. Figures c) and d) are connected to $\sigma$: c) exact $\sigma$, d) reconstructed $\sigma^{100}$ after 100 iterations in CGA.
• Figure 5: 2D tests, test 2. Convergence plots of errors over CGA iterations: a), b) errors $e^m_\varepsilon$ concerning $\varepsilon$; c), d) errors $e^m_\sigma$ concerning $\sigma$; e), f) errors $e^m_E$ concerning the term $E- \tilde{E}_{\rm obs}$. We note that figures a), c) and e) show the relative $L^2$-errors, and figures b), d) and f) show the supremum norms.

Theorems & Definitions (23)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 4.1
• Theorem 4.2
• proof
• Lemma 5.1
• proof
• Lemma 5.2