We consider time-dependent inverse problems, where the quantity of interest, the data, or the forward operator, mapping between (subsets of) two suitable function spaces, may depend on time.
Aside from the high computational effort that is typically introduced by the additional temporal dimension, we often have to deal with either inconsistent data or heavily undersampled problems. However, time is not simply an additional variable. It’s physical nature is fundamentally different from spatial variables: While we can usually move freely in space, we can only advance in
time. This is reflected in the concept of causality: A temporal quantity f(t) at a time t only depends on previous time steps and can not be influenced by events happening in the future, i.e., at time instances t′ > t.
We consider two types of time-dependent inverse problems:
1) Time-discrete inverse problems with operators depending on a single time point t, mapping
between suitable Hilbert or Banach spaces, and
2) time-continuous problems, formulated in function space settings with vector-valued Banach spaces such as Lebesgue-Bochner spaces.
For these problems we define the notions of temporal and uniform ill-posedness, and present regularization approaches that particularly take into account time-dependence.