Research Fellows Directory
Dr Yury Korolev
University of Cambridge
I am an applied mathematician having a deep interest in inverse problems and imaging.
In many situations, data directly describing the objects of interest may not be available due to the limitations of measurement techniques, high cost of direct measurements or specific requirements such as non-invasiveness in medical applications. In such situations, indirectly measured data are collected and models that describe the data acquisition process are used to infer about the quantities of interest from available data.
Mathematically, this typically leads to an inverse problem. Inverse problems often suffer from ill-posedness, which is why special regularisation techniques are needed to solve these problems numerically. Quantitative information acquired from the data is supplemented by qualitative information about the solution such as smoothness, sparseness or piecewise constancy.
Careful analysis of the uncertainties associated with the inversion is key to the design of stable numerical algorithms. There are two sources of uncertainty that need to be considered: uncertainty associated with the data and uncertainty associated with the models. Data uncertainty typically is due to noise and finite precision of measurement devices, whilst model uncertainty reflects the simplified nature of the available models that cannot capture the phenomena involved in data acquisition in their full complexity.
I am currently working on an approach to quantification of both types of uncertainty using the apparatus of partially ordered functional spaces. Partial order, as I endeavour to demonstrate, is attractive both from the methodological point of view, enabling a detailed description of data and model uncertainty, and from the point of view of computational efficiency, allowing to stay in the convex setting in situations when standard methods lead to non-convex optimisation problems.