Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Why are inverse problems ill-posed?

Talk by Prof. Dr. Mikko Salo

**Date:** 10.03.21 **Time:** 16.15 - 17.45 **Room:** Online ZHACM

Many inverse and imaging problems, such as image deblurring or electrical/optical tomography, are known to be highly sensitive to noise. In these problems small errors in the measurements may lead to large errors in reconstructions. Such problems are called ill-posed or unstable, as opposed to being well-posed (a notion introduced by J. Hadamard in 1902). The inherent reason for instability is easy to understand in linear inverse problems like image deblurring. For more complicated nonlinear imaging problems the instability issue is more delicate. We will discuss a general framework for understanding ill-posedness in inverse problems based on smoothing/compression properties of the forward map together with estimates for entropy and capacity numbers in relevant function spaces. The methods apply to various inverse problems involving general geometries and low regularity coefficients. We will use Electrical Impedance Tomography as a guiding example in the presentation. This talk is based on joint work with Herbert Koch (Bonn) and Angkana Rüland (Heidelberg).