PhD Defence: Double Regularised Total Least Squares Method

Double Regularised Total Least Squares Method

In the last decades many researches have drawn their attentions for ill-posed problems and designed methods to overcome the stability issue whenever the data is contaminated by some noisy. However, often also the operator is not known exactly. Therefore a more realistic approach is the case when both sides of the underlying operator equation have some incorrectness. In our work we investigate the latter case in the infinite dimensional setting. More precisely, we propose a Tikhonov-type functional with a generalised misfit term and an additional penalty term, which promotes sparsity. For the sake of the PhD defense we aim in this talk to introduce the proposed method as well as the main results for a general audience. Therefore our talk is divided into two parts: in the first one we give the motivation to our work compared to standard inverse problems. Additionally we raise some research questions that led to fully understand the target problem to be solved.

In the second half of this talk, now oriented to the inverse problem community, we present a new method: the dbl-RTLS (double regularised total least squares). In this section we also provide both theoretical and numerical key results obtained during our research. In more details, we study the analytical properties for the non-quadratic regularisation with two regularisation parameters, as well as a parameter choice rule depending on both noise levels. Convergence results for both regularised solution and operator is given, a novelty result in the literature. Moreover, we discuss computational aspects, develop an algorithm based on alternating minimisation strategy and we provide a numerical illustration.

Finally, we conclude this talk summarising the results so far presented.