Inclusion Reconstruction with Monotonicity-based Methods for the Elasto-oscillatory Wave Equation

• Applicant: Dr. Sarah Eberle-Blick
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 499303971

The main motivation of this project is the development and investigation of mathematical methods for the non-destructive testing of 3D elastic structures for inclusions, as required for example in material examinations, in exploration geophysics, as well as for medical diagnostics (elastography). Mathematically, this constitutes an inverse problem: "Find and reconstruct the unknown inclusions only with displacement measurements on the boundary of the elastic body resulting from prescribed boundary loads." Unfortunately, this problem is "ill-posed", i.e. even the smallest measurement error can completely distort the result. However, with the monotonicity methods, inclusions etc. can be detected and reconstructed inside an elastic body with this type of data. Thus, our aim is to develop, analyze, apply and verfiy a new method for solving the inverse problem for the elasto-oscillatory wave equation with monotonicity-based methods. The monotonicity methods have become more and more important in recent years and have been successfully applied to other inverse problems, such as electrical impedance tomography (EIT). The strengths of these methods are: they allow fast and globally convergent implementations based on rigorously proven theory, work in any dimensions d ≥ 2 for both full and partial boundary data, can enhance standard residual-based methods, yield rigorous resolution guarantees for realistic settings (noisy data) and provide an exact and unique solution of the inverse problem for exact data. In this project, the rigorous monotonicity methods are to be further developed. Specifically, the transition from the elasto-static wave equation already considered by the applicant to the elasto-oscillatory wave equation (frequency domain) is to be transferred and analyzed. In particular, two different approaches are to be pursued: the construction of monotonicity tests and a monotonicity-based regularization. Furthermore, both methods are to be implemented in order to simulate the reconstruction of inclusions numerically. Finally, laboratory experiments will be carried out to further investigate the monotonicity methods for the elastic wave equation and to verify them in a concrete application. This project thus combines the rigorously proven theory of the monotonicity methods developed for linear elasticity with the explicit application of the methods, i.e. the implementation and simulation of the reconstruction of inclusions in elastic bodies for both artificial and experimental data. In addition, the monotonicity methods have not yet been applied to any temporal problem. Thus, the study of the time-harmonic problem is an indispensable intermediate step in order to be able to solve timedependent inverse problems with monotonicity methods in the future.

• DFG Programme: Research Grants