turkmath.org

Ill-posed inverse problems are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. The talk illustrates the potential difficulties with reversing the process; namely, given a measured state of the model, try to determine what exact equation could have produced this outcome. We will primarily consider operator equations Au = y, with the operator A : E  Y  mapping between function spaces E and Y . We assume that the operator equation is ill-posed in the sense that small perturbations of y can lead to arbitrarily large perturbations on u (or even lead to non-existence of a solution). To obtain a stable estimate of the solution of such problems, it is often necessary to implement a regularization strategy. This requires the solution approach to satisfy some regularizing properties. Regularization techniques are the subject of this talk and we will attempt to highlight two approaches that are wide-spread in the literature:  (i) Variational methods are based on minimizing a weighted sum of a discrepancy term and a suitable regularization term (Tikhonov regularization) or on minimizing one of these terms under a constraint on the other (Ivanov or Morozov regularization, respectively); (ii) Iterative methods construct a sequence of iterates approximating the solution u†. Keywords: Regularization, Ill-posed Problems, Variational Methods, Iterative Methods