Inverse problems are ubiquitous in many applications, and essentially arise every time that one wants to recover the cause of an observed effect. This happens, for instance, when one can only acquire indirect measurements of a quantity of interest: notable examples are the deblurring of images (in this setting, the quantity of interest is a sharp image of an object and the measurements are a blurred version thereof) and tomography (in this setting, the quantity of interest is the inside of an object and the measurements are projections thereof).
Once a mathematical model of the physical process linking the cause and the observed effect (for instance, a mathematical model of the measurement process) is established, one can proceed by "inverting" this model, i.e., by solving the inverse problem. This can rarely be achieved without recurring to a computer, after the original continuous mathematical model has been discretised. If the model is linear (as in many image deblurring and tomography applications), the task of solving the discrete inverse problem ultimately amounts to the solution of a linear system of equations. Despite this apparent simplicity, discrete inverse problems are typically very challenging to solve numerically, for at least two reasons: first, the size of the discretised problem is often huge and, second, the solution is very sensitive to the unavoidable perturbations (or noise) that affect the data. In other words, straightforwardly using a numerical solver for linear systems will not work in this setting, for at least two reasons: first, this may be unaffordable in terms of computing time and resources and, second, the solution is totally dominated by perturbations and therefore meaningless.
To fix these issues, first one should employ some form of regularisation, i.e., one should replace the original discretised problem with one that is less sensitive to perturbations; then one should devise suitable algorithms that can solve the regularised problem in an efficient way. This project mainly focuses on the latter important task, i.e., the derivation and the mathematical justification of new and fast numerical methods for the solution of regularised linear inverse problems.
Every successful regularisation method should incorporate a right amount of prior information about the unknown. This often leads to the introduction of nonlinearities, so that involved optimisation methods are used instead of simple linear solvers (indeed, many optimisation methods often require the solution of a sequence of intermediate linear systems arising within the nonlinear scheme). One of the main goals of this project is to remove nonlinearities by introducing adaptive and flexible terms, so that only one linear system needs to be solved (instead of a sequence thereof), thereby guaranteeing massive computational savings with respect to the available strategies, without penalising the quality of the solution. The new methods will be supported by strong mathematical principles and will be made available to the wider scientific community by producing free software that can be potentially used for a variety of computational tasks, even beyond regularisation methods.
The findings of this project can generate great impact in a variety of applications, with specific focus on tomography. In the medical context, many tomographic technologies are useful to diagnose health problems without a need for more invasive techniques. In the industrial context, a variety of tomographic approaches are employed to monitor the quality of production systems, allowing for improved safety of products and reduced economical costs. The new methods can play a dramatic role in making all these technologies computationally more affordable, assuring tangible impact on the society and the economy. Collaborations with academic groups in engineering will be pursued to make this happen.