Efficient solvers for large-scale inverse problems

Inverse problems are ubiquitous in many areas of Science and Engineering. However, they are usually quite challenging to solve, especially if they are large-scale and severely ill-posed. The goal of this project is to solve inverse problems by devising innovative regularisation methods, which leverage advanced tools in numerical linear algebra, optimisation, and scientific computing.

This project mainly deals with imaging applications. For instance, regularization methods must be used when deblurring astronomical images, or when performing tomographic reconstructions in Industry and Medicine. Sometimes, the acquired data of an inverse problem can be heavily corrupted or incomplete: in these cases, the theory of compressive sensing specifies the necessary conditions to reconstruct a good solution, provided that sparsity constraints are imposed. Another potential application to be considered within this project is data assimilation for numerical weather forecasting.

Krylov subspace methods have proved to be very efficient to regularise linear large-scale and severely ill-posed problems. They are mainly employed as iterative solvers when some terms evaluated in the 2-norm are involved. A few Krylov subspace algorithms have recently been derived, which incorporate simple constraints on the solution, and/or terms evaluated in the 1-norm (in order to enforce sparsity). This project should contribute to the development of alternative or additional solvers based on Krylov subspace methods, which can incorporate more advanced constraints, and a variety of regularisation terms. Both linear and nonlinear problems should be considered. A theoretical analysis of the new methods should be performed, and their efficiency should be tested on the problems mentioned above.

This project will be completed under the supervision of an interdisciplinary team, with academics in the Department of Mathematics, the Faculty of Science, or the Faculty of Engineering.

Anticipated start date: 2 October 2017.

Applications may close earlier than the advertised deadline if a suitable candidate is found; therefore, early application is recommended.