Käänteisongelmia

Käänteisongelmia, or inverse problems, are a class of mathematical problems where the goal is to determine the causes of observed effects. This is in contrast to forward problems, where the causes are known and the effects are predicted. In a forward problem, given a model and its parameters, we compute the observable data. In an inverse problem, we are given the observable data and a model, and we aim to infer the model's parameters.

Inverse problems are prevalent in many scientific and engineering disciplines. For instance, in medical imaging, such as CT scans or MRI, the observed data are projections or signals, and the inverse problem is to reconstruct the internal structure of the body. In geophysics, seismic waves recorded at the surface are used to infer the Earth's subsurface structure. In astronomy, telescope observations are used to determine properties of distant celestial objects.

A significant challenge in inverse problems is their ill-posedness. Often, inverse problems are not well-posed in the sense of Hadamard, meaning they may lack a unique solution, have no solution, or be highly sensitive to noise in the data. This means that small errors in the measurements can lead to large errors in the reconstructed parameters. Consequently, regularization techniques are frequently employed to stabilize the problem and obtain meaningful solutions. These methods introduce prior information or constraints to guide the solution towards a physically plausible outcome. The development of efficient and robust algorithms for solving inverse problems remains an active area of research.