Inverse Problems with Learned Forward Operators
This is an incremental review for researchers in inverse problems, focusing on learned forward operators to reduce computational costs.
The paper tackles the challenge of solving inverse problems with computationally expensive forward operators by reviewing two paradigms for learning cheaper variants without compromising reconstruction quality, presenting their theory and numerical comparisons.
Solving inverse problems requires the knowledge of the forward operator, but accurate models can be computationally expensive and hence cheaper variants that do not compromise the reconstruction quality are desired. This chapter reviews reconstruction methods in inverse problems with learned forward operators that follow two different paradigms. The first one is completely agnostic to the forward operator and learns its restriction to the subspace spanned by the training data. The framework of regularisation by projection is then used to find a reconstruction. The second one uses a simplified model of the physics of the measurement process and only relies on the training data to learn a model correction. We present the theory of these two approaches and compare them numerically. A common theme emerges: both methods require, or at least benefit from, training data not only for the forward operator, but also for its adjoint.