July 14, 2024

What are inverse problems?

Inverse problems usually consist in finding causes for observed effects. Examples are imaging methods in medicine, engineering, and the natural sciences such as computerized tomography, magnetic resonance imaging, electron microscopy, seismic imaging or optical nanoscopy (e.g. STED or SMS), among others. In many cases the determination of effects from given causes, the so-called forward problem, can be modelled by differential equations. This leads to inverse problems where some part of a differential equation such as a coefficient function, an initial or boundary condition or the shape of a domain has to be determined from observations of part of the solution to the differential equation.

A main difficulty in the solution of inverse problem is the phenomenon of ill-posedness: Very different cause may have similar effect. Therefore, it is difficult to reconstruct such causes from noisy observations of their effects. Hence, stable reconstruction methods must use a-priori knowledge on the solution (the unknown cause). This may for example be smoothness or sparsity information or non-negativity constraints. In regularization theory the aim is to show that despite of ill-posedness the reconstruction error tends to zero with the noise level and to analyze the rate of this convergence.

Besides generic regularization methods, which are based on formulating an ill-posed inverse problem as an operator equations F(x)=y with a forward operator F that does not have a continuous inverse, also problem-specific reconstruction methods are investigated. Such methods often determine only some particularly relevant part of the unknown cause x, e.g. a functional of x, the support or function, or a singularity. Examples are sampling and probe methods for inverse scattering problems, reconstruction methods for electrical impedance tomography based on geometrical optics solutions, time reversal and generalized Kirchhoff migration methods or the determination of polarization and moment tensors of small inclusions.

A basic question for any inverse problem concerns identifiability, i.e. if the unknown cause is uniquely determined by idealized noise free observations of its effect. Moreover, for ill-posed problems one tries to bound the instability under a-priori information on the solution. To study these questions quite different techniques are employed for different problem classes. Therefore, from a mathematical perspective the field of inverse problems is at the interface of theory and numerical analysis of PDEs, differential geometry, integral geometry, numerical analysis, statistics, uncertainty quantification, and machine learning.