## 35R30 Inverse problems

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We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from Eberle and Harrach (Inverse Probl 37(4):045006, 2021), but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.

The Calderón problem with finitely many unknowns is equivalent to convex semidefinite optimization
(2023)

We consider the inverse boundary value problem of determining a coefficient function in an elliptic partial differential equation from knowledge of the associated Neumann-Dirichlet-operator. The unknown coefficient function is assumed to be piecewise constant with respect to a given pixel partition, and upper and lower bounds are assumed to be known a-priori.
We will show that this Calderón problem with finitely many unknowns can be equivalently formulated as a minimization problem for a linear cost functional with a convex non-linear semidefinite constraint. We also prove error estimates for noisy data, and extend the result to the practically relevant case of finitely many measurements, where the coefficient is to be reconstructed from a finite-dimensional Galerkin projection of the Neumann-Dirichlet-operator.
Our result is based on previous works on Loewner monotonicity and convexity of the Neumann-Dirichlet-operator, and the technique of localized potentials. It connects the emerging fields of inverse coefficient problems and semidefinite optimization.

We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton’s method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and for finitely many directions. We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of determining an unknown inverse Robin transmission coefficient in an elliptic PDE. Using a relation to monotonicity and localized potentials techniques, we show that a piecewise-constant coefficient on an a-priori known partition with a-priori known bounds is uniquely determined by finitely many boundary measurements and that it can be uniquely and stably reconstructed by a globally convergent Newton iteration. We derive a constructive method to identify these boundary measurements, calculate the stability constant and give a numerical example.

Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming
(2021)

Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.