On the computation of the minimal-norm solution of linear and nonlinear problems

PES, FEDERICA

2022-02-25

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Abstract

The main topic of the thesis is the study of inverse problems and, in particular, it is the study of numerical methods for the computation of the minimal-norm solution of linear inverse problems in the continuous case and nonlinear ones in the discrete case. Inverse problems arise in many areas of science and engineering, from the need to interpret indirect and incomplete measurements. The most usual situation giving rise to an inverse problem is the need to interpret indirect physical measurements of an unknown object of interest, for instance, if one is interested in determining the internal structure of a physical system from the system’s measured behavior, or in determining the unknown input that gives rise to a measured output signal. Inverse problems are a recent topic in mathematics. Their study is motivated by the technological development of the last decades; for example, some of the more sophisticated medical diagnostic machines solve inverse problems, such as X-ray computed tomography, in which the inverse problem is to reconstruct the inner structure of an unknown physical body from the knowledge of X-ray images taken from different directions. An example of inverse problem, which will be treated in some numerical experiments of this thesis, concerns the study of the subsoil in a non-destructive way, through the propagation of electromagnetic waves, in order to know some properties of the subsoil. Another example concerns image processing, where the goal is to find the sharp photograph from a given blurry image. An inverse problem takes the form F(x) = b, where F is a linear or nonlinear operator, x represents the unknown solution, and b is the information available, that is, the measurements dataset. The goal is to reconstruct x starting from b. Inverse problems are closely related to the concept of ill-posed problems. To understand this concept we need to resort to the definition given by Hadamard at the beginning of the last century: such problems may not have a solution, or may have more than one, or that solution is not stable with respect to perturbation in the data. In applications, ill-posed problems are common whenever there is little available measured data compared to the number of unknowns. In this case, it is necessary to reformulate the original ill-posed problem into a well-posed problem. A typical approach is to resort to a least-squares problem, in which the mean squared error between F(x) and b is required to be minimal. In this thesis, we are concerned with problems that have no unique solution. Among the different solutions, we want to determine the minimal-norm solution. The subjects discussed in this thesis can be divided into two themes: nonlinear least-squares problems and systems of linear integral equations of the first kind.