## PhD research topic - Silvia Noschese

#### Research areas

Numerical Linear Algebra

#### Keywords

Discrete ill-posed problems

#### Research topics

Discrete ill-posed problems are linear least squares problems with ill-conditioned coefficient matrix whose singular values cluster at zero. Such problems arise in image deblurring as well as from the discretization of linear ill-posed problems, such as Fredholm integral equations of the first kind with a continuous kernel. In discrete ill-posed problems that arise in applications in science and engineering, the large sized data are typically contaminated by error, whose propagation causes the classical solution given by the Moore-Penrose pseudo-inverse to be meaningless.

A common approach to determine a useful approximate solution is to replace the least squares problem by a nearby problem that is less sensitive to the errors in the data and to round-off errors introduced during the solution process. Several strategies for the regularization have been proposed to reduce the error and thereby obtain an accurate approximation of the desired solution; Tikhonov regularization, truncated singular value decomposition (TSVD), truncated generalized singular value decomposition (TGSVD), and Krylov subspace methods are the most popular approaches.

The aim of the research is both the choice of suitable regularization matrices employed in the penalized least squares problem, and the determination of appropriate filter factors and regularization parameters in order to obtain approximate solutions of high quality. The subject of the research has been extensively studied over the past decade and effective strategies develop continuously.

Knowledge of MATLAB is imperative in the research project.

#### Bibliography

[1] D. Calvetti e L. Reichel, Tikhonov regularization of large linear problems, BIT, 43 (2003), 263–283.

[2] P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, Philadelphia, 2010.

[3] P. C. Hansen, Regularization tools version 4.0 for MATLAB 7.3, Numer. Algorithms 46 (2007), 189-194.