Inverse Problems in Computer Science
Within Computer Science, Inverse Problems is a strong aspect of the Vision and Imaging Sciences group (VIS). Imaging Science is a fast-growing interdisciplinary area, and so is mathematical imaging inside computer science. From its very beginning, it has been an area incorporating various different aspects of mathematics e.g. differential geometry, stochastic processes, functional analysis, optimisation theory and numerical methods to name a few. Moreover mathematical imaging has always been driven by applications, e.g., in computer vision or medicine.
The application of IP to imaging science is well established. However this field is vastly diverse as there is a substantial difference between problems depending on whether they are linear or non-linear, unique or non-unique, mildly or severely ill-posed, small or large scale. As medical and industrial imaging applications continue to address new physical measurements regimes (microwave, electromagnetic, optical, low energy X-Ray, and hybrid schemes), to pursue faster acquisition through data subsampling, and to address higher dimensional information including time and multispectral axes, there is increasing need to bring together appropriate state-of-the-art techniques in analysis, computation and statistics. Further areas in computer science where Inverse Problems arise are e.g. Bioinformatics and Machine Learning.
Inverse Problems in Mathematics
Inverse Problems are a complex and highly dynamic branch of mathematics with steadily arising challenges which require new mathematical methods and ideas. This is due to the non-linear, global and ill-posed nature of Inverse Problems. Therefore, their successful mathematical analysis require simultaneous use of techniques and ideas from analysis (spectral theory, harmonic and microlocal analysis, PDE-control, homogenization theory), geometry (metric and comparison geometry, geometric convergence, non-smooth geometry), sophisticated numerical models for the forward problem and numerical methods for the solution of the forward and inverse problem (accuracy, stability, regularisation) etc.
Theoretical problems include uniqueness/stability and development of reconstruction procedures and relations between discrete and continuous models. From the point of view of scientific computing, the development of fast numerical algorithms on traditional and novel (e.g. multicore/GPU) computing architectures poses significant challenges. Another topic is the relation between Inverse Problems and the theory of metamaterials, especially questions of electromagnetics, quantum and acoustic cloaking.
Inverse Problems in Statistics
Modern statistics naturally intersects with inverse problems at the interface of the estimation of overspecified or ill-conditioned problems via regularization, as well as errors in measurement problems. Regularization can be viewed as a Bayesian formulation of estimation, or in a frequentist framework as using the regularization to specify admissible estimators. It has become necessary in high dimensional problems to apply such techniques to enable estimation at all, and important problems include defining and ensuring consistency of any proposed estimation strategy. At UCL a number of problems are motivated by specific applications, such as deconvolution (e.g. mass spectra in analytical chemistry or spectroscopy). The inference of continuous time processes is another important problem, as the nature of the discrete observations naturally restrict the types of ordinary, partial and stochastic differential equations that can be inferred from the sampled observations. Finding suitable summaries of the observations in the high dimensional context is also pivotal to efficient inference in such scenarios. Constraints must be put in place both to enable computationally feasible estimation, considering the mathematical model of the relevant problems, but also to ensure good properties of the ensuing estimators.
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