Simulations are a typical object of study in applied mathematics. Some examples are:
- Predicting the weather of tomorrow measuring the weather of today
- Given a sound source (a singer) and the shape of the auditorium, compute what is heard by the listeners
- Assume an X-Ray is attenuated by a bone. How much of the ray intensity is left after the bone?
These problems, which can also be solved by experiments, are direct problems. Often, their modeling involves Differential Equations. Typically, the solution to the problem exists, is unique, and is stable in the sense that small measurement errors will lead to small errors in the result. According to Hadamard, problems with these properties are well-posed.
However, many problems do not fall into this category. Some examples are:
- Given what a listener hears in an auditorium, can you identify the sound source? (Inverse Scattering Problem, in: Kac 1966, Can one hear the shape of a drum?)
- Given the amount of ray intensity that is lost when an X-ray travels through the body, can you identify the bone shape inside the body? (computerized tomography)
- Assume you measure resistance on the surface of the body. Can you identify the conductivity inside the body? (impedance tomography)
- Assume you measure an EKG, which more or less measures the electric current that the heart sends out. Can you identify the shape of the heart? (inverse electrocardiography)
- Measuring the weather of today, can you identify the weather of yesterday? (time-reversal)
These problems can not be solved with experiments, but only mathematically. Their name is inverse problems. Typically, they lose all the nice properties above: there is no unique solution, and if it exists, it does not depend on the data in a stable way. Problems with these properties are ill-posed.
We will start off the lecture by examining inverse problems in an analytical way. In the second part, we will use this to look at various problems and compute solution strategies.
While we always have an application in mind, of course, no physical knowledge is needed. However, it will be part of the homeworks to implement and analyze the algorithms we develop, so some programming experience is useful. The examples in the lecture will be given in Python, but all programming languages are acceptable.
Inverse Problems are a field that borrow stuff from many fields. While everything is covered in the lecture, it helps to have a basic understanding of Functional Analysis, PDE and/or stochastics.
I ask that all participants join the Learnweb course even before the lecture starts, so that I can pass on some information, if needed. The course already contains some videos of 2020 with an informal introduction to two inverse problems. The password is RadonTrafo.