Inverse problems arise when we reconstruct a sharper image from a blurred one, when we create an iamge of interior structures from X-ray data, when we reconstruct the underground mass density from measurements of the gravity above the ground, and when we map the interior conductivity from measurements of the exterior electromagnetic field. When we solve an inverse problem, we compute the source that gives rise to the observed data using a mathematical model for the relation between the source and the data.
Inverse problems arise in many technical and scientific applications, such as medical imaging, geoscience, electromagnetic scattering, and non-destructive testing. An important example is image deblurring which arises, e.g., in astronomy or in biometric applications that involve fingerprint or iris recognition. Another well-known is example is geophysical prospecting for oil and gas reservoirs.
Inverse problems have intriguing “built-in” difficulties: the solution is always extremely sensitive to errors in the data (the computational problem is ill conditioned), and there may not be a unique solution. It is necessary to use prior information to compute stable unique solutions through the use of regularization algorithms. Our current research focuses on geostatistical methods, large-scale iterative algorithms, total variation tomography, and paradigms for incorporation of all relevant prior information in the reconstruction.
For more details see the home pages of Professor Per Christian Hansen and Associated Professor Kim Knudsen. See also these ongoing projects:
The software developed in these projects is available from the department's software page.