The Calderon problem [1] concerns the recovery of an unknown coefficient of a partial differential equation from the boundary values of its solution. These measurements induce a highly nonlinear forward operator, posing challenges for the development of reconstruction methods, which usually suffer from the problem of local convergence. To circumvent this issue, we propose an alternative approach based on lifting and convex relaxation techniques, that has been successfully developed for finite-dimensional quadratic inverse problems [2, 4, 5]. This leads to a convex optimization problem whose solution coincides with the sought-after coefficient, provided that a non-degenerate source condition holds [3, 6]. While we leave the analysis of the source condition in the Calder´on setting to future works, we demonstrate the validity of our approach in a toy model where the solution of the PDE is known everywhere in the domain (instead of only voltage-to-current measurements on the boundary). In this simplified setting, we verify that the non-degenerate source condition holds under certain assumptions on the unknown coefficient.

References

[1] Calderon, A., ´ On an inverse boundary value problem. Seminar On Numerical Analysis And Its Applications To Continuum Physics. 25 pp. 65-73 (1980).

[2] Candes, E. & Li, X. ` Solving quadratic equations via PhaseLift when there are about as many equations as unknowns. Foundations Of Computational Mathematics. 14 pp. 1017- 1026 (2014).

[3] Candes, E. & Recht, B.,Exact matrix completion via convex optimization. Communications Of The ACM. 55, 111-119 (2012).

[4] Candes, E., Strohmer, T. & Voroninski, V., Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming. Communications On Pure And Applied Mathematics. 66, 1241-1274 (2013).

[5] Li, X. & Voroninski, V., Sparse signal recovery from quadratic measurements via convex programming. SIAM Journal On Mathematical Analysis. 45, 3019-3033 (2013).

[6] Vaiter, S., Peyre, G. & Fadili, J., ´ Low complexity regularization of linear inverse problems. Sampling Theory, A Renaissance: Compressive Sensing And Other Developments. pp. 103-153 (2015).

Key words : Calderon problem; Lifting; Convex relaxation.

Field : Inverse Problems

Sedimentary rocks contain traces of the Earth's ancient magnetic field, forming a valuable archive for reconstructing the geodynamic history of the Earth: continental drift, magnetic reversals, or internal movements of the core. To read this fossil memory, geophysicists rely on measurement devices, such as superconducting quantum interference device (SQUID) or quantum diamond microscopes (QDM), which locally measure components of the magnetic field at a fixed distance above the sample. However, the measured field strength is particularly weak, and extremely sensitive to noise, which leads to an ill-posed inverse problem for reconstructing the internal magnetization of the sample.

The magnetic field measurements recorded by these microscopes are modeled as the projection, along a direction $\vec{l} \in \mathbb{R}^3$, of the field generated by an unknown magnetization, supported in a planar domain $S \in \mathbb{R}^2$ containing the rock sample. This magnetization is a vector-valued function $m$ assumed to belong $[L^2(S)]^3$ , while the measurements are performed on a set $Q \in \mathbb{R}^2$ located at a height $h > 0$ above the sample (see Figure 1).

The link between the magnetization $\vec{m}$ and the measured field is based on an integral operator, derived from Maxwell's equations in the static regime, and relies on the Poisson−Laplace equation $\Delta \phi = \nabla \cdot \vec{m}$, satisfied by the scalar magnetic potential $\phi$. This potential is then expressed using the Poisson kernel and the Riesz transforms. Within this framework, we define the measurement operator $b_{\vec{l}}$.

The objective of this work is to develop a mathematically rigorous and numerically stable approach to estimate the magnetic moment $<\vec{m}> = \displaystyle\int \int_S \vec{m}(t)\, dt \in \mathbb{R}^3$ of a sample from directional magnetic field measurements $b_{\vec{l}}$ on $Q$.

References

[1] L. Baratchart, D. P. Hardin, E. A. Lima, E. B. Saff, and B. P. Weiss, Characterizing Kernels of Operators Related to Thin-Plate Magnetizations via Generalizations of Hodge Decompositions, Inverse Problems, vol. 29, no. 1, 2013.

[2] CL. Baratchart, S. Chevillard, D. P. Hardin, J. Leblond, and E. Andrade Lima, Magnetic Moment Estimation and Bounded Extremal Problems, Inverse Problems and Imaging, vol. 13, no. 1, 2019.

Key words : Inverse source problems, elliptic partial differential equations, harmonic functions, geosciences and planetary sciences, paleomagnetism.

Field : Inverse Problems

The Kohn-Vogelius method is an effective approach for solving inverse parameter identification problems. We present an abstract framework that generalizes the work of [1, 2, 3] to non-symmetric, second-order, linear PDE operators. This framework allows us to solve inverse problems involving the identification of singular parameters from boundary measurements. The method transforms the inverse problem into an optimization problem by constructing an energy-like cost functional $J$ defined over an admissible set $Q_{(ad)}$ , such that the inverse problem’s solution q is the unique minimizer of $J$. The difficulty for non symmetric PDE is that the functional $J$ cannot coincide with the bilinear form associated with the forward problem. We study the derivatives of $J$ with respect to various perturbations of $q$, either in terms of $L^\infty$ perturbations or in terms of perturbations of $q$’s discontinuity points, to enable the use of gradient descent algorithms. Finally, we present several applications related to reconstructing the discontinuity points of $q$, especially in impedance or generalized-impedance models, accompanied by two-dimensional numerical illustrations.

References

[1] S. Chaabane, I. Feki, N. Mars, Numerical reconstruction of a piecewise constant Robin parameter in the two-or three-dimensional case, Inverse Problems, 28 (6), (2012).

[2] S. Chaabane, B. Charfi, H. Haddar. Reconstruction of discontinuous parameters in a second order impedance boundary operator, Inverse Problems, 32 (10), (2016).

[3] B. Charfi, Identification du support de singularit´e d’une imp´edance g´en´eralis´ee, PHD Thesis, Facult´e des Sciences de Sfax, (2019).

Key words : Identification of discontinuity sets, Kohn-Vogelius method.

Field : Inverse Problems

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