Inverse Problems

Inverse problems appear where unknown quantities are measured through an indirect and error-prone process. For instance, one can think of a tumor which is located with the help of radiologic images, the capacity of a subterranean oil reservoir which is estimated from hydraulic pressure measurements, or the depth of coastal waters which is inferred from observing surface waves.

The underlying physical process typically has a certain direction:

The tumor influences the attenuation of electromagnetic radiation in the tissue
The capacity of the oil reservoir influences the outcome of the pressure measurement
The depth of the water implies a specific wave pattern at the surface

This process — the so-called forward model — is often described in terms of a partial differential equation or a linear map like the Radon transform. However, one is usually not interested in the outcome of the forward model but rather in the quantity which caused this outcome, i.e., the position of the tumor, the capacity of the reservoir, or the depth of the water. Inferring the cause from the effect explains the term “Inverse problem”.

In general, however, the forward model cannot be inverted naively since the measurements process suffers from loss of information and stochastic corruptions. Such inverse problems are called ill-posed.

Our research area “Inverse problems” studies methods to circumvent this ill-posedness compute good approximations of the solution. Here we follow two different approaches: in the Bayesian framework, one models the unknowns such that they explain the measured data with highest probability. The variational approach, however, seeks for minimizers of an energy functional which enforces specific properties that an approximate solution of the inverse problem should have.