When computing causes for observed or desired effects, one is dealing with an inverse problem.
Such problems include the identification of parameters in differential equations, medical imaging problems,
non-destructive testing and deconvolution. Mathematically, inverse problems usually lead to ill-posed problems,
which might exhibit non-unique solutions (if any at all) and whose solution is unstable with respect to data
noise. Special techniques, so called regulatization methods have to be applied to stabilize inverse problems.
These regularization methods quite frequently lead to optimization problems, where the objective function needs
the solution of a partial differential equation; the optimization procedure then needs tha calculation of gradients
with repect to the unknown parameter functions.
Linz is one ot the world-wide leading centers for mathematical research on inverse problems.
See the Projects page for some sample projects on inverse problems.