To understand this term, let us consider a model for obstetric sonography. An ultrasonic scanner produces a sound wave by using a transducer, the wave propagates in the body and echoes back to the transducer that records the echo. The propagation is mathematically modelled by the wave equation which gives a connection between the wave speed (a property of the medium) and the sound wave (oscillation in the medium). If we are given the wave speed as function of the position in the body and the vibrations of the transducer as a function of time, then we can solve the wave equation for the received echoes. In other words, the wave equation gives a model for ultasound measurements. Solving this equation with a wave speed corresponding to the mixture of different tissues in the human body requires computationally demanding simulations and is indeed a problem, the direct problem.
In addition to sonography, reflection seismology gives an example of an imaging method where the wave equation can be used as a model of measurements. Seismic reflection method seeks to create an image of the Earths crust from recording of echoes stimulated for example by explosions. In contrast to medical sonography, seismic reflection method reconstructs an image of the wave speed.
The inverse problem for the wave equation is closely related to several inverse problems of geometric nature. By a geometric inverse problem we mean a problem to recover a Riemannian manifold from a geometric data set, that is, a data set derived from the Riemannian structure only. A widely studied example is the boundary rigidity problem for which the data set is the distances between each pair of boundary points.
For example, consider Earth or some other solid body with a variable speed of wave propagation. The travel time of a wave between two points defines a natural non-Euclidean distance between the points. This is called the travel time metric and it corresponds to the distance function of a Riemannian manifold.
A classical inverse problem is to determine the wave speed inside an object when we know the travel times between the boundary points. This problem is an idealization of the geophysical problem where the structure of the Earth is to be found from the travel times of first arrivals of the earthquakes through the Earth. Moreover, measuring the wavefronts of elastic waves scattered from discontinuities inside the Earth, one obtains information on the broken geodesics in the Riemannian metric. Thus seismic measurements give information on the internal structure of the Earth. Almost all of our knowledge of the deep Earth is obtained using such data.
We have applied geometric methods to study the inverse problems related to the wave, heat, and conductivity equations, as well as Maxwell's and Dirac systems. Often inverse problems for different physical models can be solved with the same geometric method.
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Due to the ill-posed nature of inverse problems, it is important to understand how any uncertainty in the data or in the mathematical model is propagated to the solution. Problems taking such probabilistic viewpoints to inverse problems are called statistical inverse problems and have natural ties to the popular field of uncertainty quantification.
One of our main research topics is the Bayesian approach to inverse problems, where the problem is recast in the form of a statistical quest of information. Randomness is seen as a lack of information, whereas probability distributions correspond to 'degree of belief'. In such a formulation the input and the observed quantities are treated as random variables. The goal becomes to find the posterior distribution, i.e. the probability distribution of the input variables conditioned on the observation.
In practice, the direct problem is used to construct a statistical model (so-called likelihood distribution) for the possible outputs of a given input. In addition, the prior information about the input gives rise to a priori probability model. The posterior distribution is obtained from Bayes' rule as a product of the prior and likelihood distributions, and hence it is often referred to as Bayesian inversion.
Our group is also active in other aspects of statistical inverse problems. For example, our ongoing research includes topics such as correlation-based imaging and scattering from random media.
Applications for statistical inverse problems are far-reaching and include fields such as atmospheric sciences, geophysics, machine learning, medical imaging and econometrics.