Course about Inverse Problems

The course will treat the classical theory for linear inverse problems. Inverse problems occur in many applications in physics, engineering, biology and medical imaging. Loosely speaking, solving the forward problem consists of computing the outcome of a known model given the model parameters. The inverse problem consists of computation of the unknown parameter of interest given the physical model and noisy measurements of the outcome. Typical examples are parameter identification problems, image reconstruction in computer tomography, deconvolution problems in microskopy, denoising of images etc.  The underlying physical model can often be described by a (partial) differential equation or integral operator such that die observations are measurements of solution of differential equation or evaluation of the integral. Die unkown input parameter are then inital conitions, parameter functions or modell parameter.

The challenge in solving inverse problems is due to the fact that inverse problems are usually ill-posed problems: different inputs can cause similar observatations such that in particular in the case of noisy observation a reconscrution of the input is difficult. That's why regularization methods are applied.

In particular the following topics are discussed:

  • Examples of ill-posed inverse problems
  • Reconstruction in Computer tomography
  • Ill-posed operator equations
  • Regularization of linear inverse problems
  • Iterative reconstruction methods
  • Tikhonov-regularization
  • Outlook: dynamic inverse problems in medical imaging applications

 

Literature:

  • Engl, Hanke, Neubauer, Regularization of inverse problems
  • Rieder, Keine Probleme mit inversen Problemen
  • Hansen, Discrete inverse problems
  • Louis, Inverse und schlecht gestellte Probleme
  • Mueller, Siltanen, Linear and nonlinear inverse problems with practical applications
  • T. G. Feeman, The mathematics of medical imaging, Springer, 2010
  • F. Natterer, The Mathematics of Computerized Tomography, Classics in Applied Mathematics 32,SIAM, 2001

 

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