Opportunities

The goal of this project is to implement a fast Matlab solver for integro-differential equations involving the Fractional Laplace operator. These equations are able to model anomalous diffusion processes and have been recently used in many applications such as material science, finance, bioengineering, continuum mechanics, graph theory, and machine learning.

In this case, classical numerical schemes such as the finite element method may be very inefficient. Indeed, they require the storage of large-sized densely populated matrices that waste memory and computational resources. In order to speed up the numerical computations, the idea is to employ some recent advanced techniques based on computational harmonic analysis and compressed sensing. These methods are able to exploit the sparsity of the solution with respect to a suitable set of basis functions and to compress the discretization using “short and fat” matrices, which are less demanding in terms of memory and more computationally tractable.

Proficiency with Matlab and familiarity with basic concepts of functional and numerical analysis are required. Previous knowledge of the finite element method is useful, but not mandatory.

Compressive imaging deals with the reconstruction of images from small numbers of measurements.  Its applications include medical imaging (MRI, CT), lensless imaging, electron and fluorescence microscopy, and infrared imaging.

A fundamental question in compressive imaging is the design of good (or optimal) sampling strategies.  For example, in Fourier-based imaging, this corresponds to the selection of frequencies in k-space.  This project will explore the design and analysis of optimal sampling strategies based on the mathematics of compressed sensing and nonlinear approximation.  In particular, the aim is to understand the conditions which ensure optimal nonlinear approximation of images via compressive sensing for popular sparse regularization techniques such as wavelets and their various generalizations.

This project will combine techniques from approximation theory, wavelets and compressed sensing.  Experience with Matlab is useful. 

Data from MRI scans is typically used to produce images that are interpreted for diagnosis.  However, there is substantial amounts of information in the data that is not extracted during the image formation process.  Discarded information such as temperature change, blood flow velocity, perfusion, diffusion, structure and physiology can provide practitioners with much broader clinical insight into the patient.

The purpose of this project is to develop numerical methods for extracting this information.  A major difficulty in extracting this information is that it requires the solution of a nonlinear and ill-posed inverse problem, which means that standard numerical methods typically produce poor reconstructions.  Nevertheless, progress has recently been made by exploiting the sparsity of the underlying quantities so as to regularize the underlying problems.  Aiming to enhance reconstruction fidelity, this project will explore the further development of these techniques.  Three particular areas of focus will be (i) the extension from single-coil to multiple-coil (i.e. parallel) MRI, (ii) the development of new gridding strategies for dealing with nonuniformly-acquired data, and (iii) the implementation of novel joint sparsity-promoting regularizers.

Linear algebra, numerical analysis and Matlab are essential.  Basic optimization and Fourier transforms are useful.