Large part of numerical problems from the Real World and from Scientific Computing is reduced, at the end, to solving linear algebra problems, such as linear systems or eigenvalue problems. The large size of the associated matrices implies a high computational complexity (in both time and memory) of their solutions. General and very popular solution algorithms cannot often be applied for their large cost while methods specifically tailored on the peculiar features of the problem can be very effective.
Once translated into the linear algebra language, the specific features of the original problem appear as sparsity and structure properties (sometimes hidden) of the involved matrices.
This way, the analysis and exploitation of structural properties is a fundamental step in the design of effective solution methods. A paradigmatic example is given by the PageRank problem of Google.
Goals of the research group are:
- To conduct theoretical research on classes of structured matrices in order to develop mathematical tools which can be used for the design and analysis of new efficient algorithms for solving important computational problems.
- To develop, analyze, and implement new efficient solution algorithms for important computational problems arising in emerging real problem applications, by exploiting the analytic properties of the associated continuous mathematical model.
Problems and subjects of main interest are:
- Theoretical tools: iterative methods, regularization, preconditioning, Toeplitz-like structured matrices, spectral properties.
- Applications: inverse problems (imaging, scattering, remote sensing), integral equations models, applications involving PDEs.