Inverse problems are encountered in several areas in applied science and technology. When one studies a physical phenomenon which is governed by the equations of mathematical physics, the application of the model to real life situations often requires the knowledge of constitutive and/or geometrical parameters which in practice may not be completely known or are inaccessible to direct measurements. Other applications involve the development of non-destructive methods for identifying damage or defects in mechanical systems or for the identification of unknown forcing terms. Generally speaking, one has to deal with a class of problems in which the roles of the unknowns and the data is reversed, at least in part, with respect to the direct problems.
The analysis of inverse problems has inherent mathematical interest since, when compared with the corresponding problems of the direct theory, they do not usually satisfy the Hadamard postulates of well-posedeness and are highly non-linear, even if the corresponding direct problem is linear. In most cases, in order to overcome these analytical obstacles, it is impossible to invoke all-purpose, ready-made, theoretical procedures. Instead, it is necessary to single out a suitable approach and trade-off with the intrinsic ill-posedeness by using original ideas and a deep use of mathematical methods from various areas. Another important aspect of the study of inverse problems concerns with their numerical treatment, and the need of figuring out ad-hoc strategies for the implementation of robust solving algorithms, considering the presence of noise, measurement errors and the probabilistic character required in some contexts. The above critical issues are further amplified when inverse methods are applied to the study of full-scale mechanical systems, as additional obstructions arise because of the complexity of mechanical modelling, the inadequacy of the analytical models used to describe the physical phenomena, and incompleteness of the field data.
In this course we aim to provide a review of recent developments of inverse problems for mechanical systems. The topics will range from advanced techniques for damage detection and source identification in structures, dynamic methods for model updating and active vibration control, photo-acoustic imaging using resonating nanoparticles, the use of nanostructures as mass resonant sensors, probabilistic and deep learning-based methods, with attention to the computational and experimental aspects relevant for practical applications.
With this lecture series, we mainly aim to attract doctoral students and researchers interested in working in inverse problems in the areas of civil and mechanic engineering, mathematical physics, and applied mathematics. The objective is to provide the audience with a solid theoretical framework, mathematical tools needed to tackle specific classes of inverse problems, computational tools, and experimental evidence to better understand the most recent advances on the subject including those involving probabilistic deep learning.