Minisymposium "Dynamic Inverse Problems"
(Organizer: A. Leitão)

Held during the conference: Applied Inverse Problems (Pdf)
(UBC, Vancouver, June 25-29, 2007)

• Prof. Dr. Alfred K. Louis (Saarland University, Saarbrcken, Germany)

  Title: Dynamic Inverse Problems: Efficient Algorithms and Approximate Inverse
  Abstract:
  We consider dynamic problems, where the investigated object is allowed to change during the measurement. Hence we regularize both the spatial and the temporal behaviour of the solution. In a first step we show how two different regularization terms can be coupled in a Tikhonov regularization approach such that an efficient solution for underdetermined problems is possible. Especially we consider temporal smoothness of the object. We apply the results to tomography problems and to current density reconstructions. Finally we study this approach as special case of the approximate inverse regularization method. (Slides Pdf)

• Dr. Hanna Katriina Pikkarainen (RICAM, Linz, Austria)

  Title: State estimation approach to nonstationary inverse problems
  Abstract:
  We examine nonstationary inverse problems in which the time evolution of the unknown quantity is modelled by a stochastic partial differential equation. We consider the problem as a state estimation problem. The time discrete state evolution equation is exact since the solution is given by an analytic semigroup. For the practical reasons the space discretization of the time discrete state estimation system must be performed. However, space discretization causes an error and inverse problems are known to be very intolerant to both measurement errors and errors in models. We analyse the discretization error so that the statistics of the discretization error can be taken into account in the estimation. We are interested in the related filtering problem. A suitable Filtering method is presented. We also verify the method using numerical simulation. (Slides Pdf)

• Dr. Stefan Kindermann (Industrial Mathematics Institute, University of Linz, Austria)

  Title: Dynamic programming principle for static and dynamic inverse problems
  Abstract:
  Dynamic inverse problem are problems, where the solution, the data and/or the operator depend on time. We investigate a Tikhonov-type regularization method for linear dynamic problems, where the regularization term involves the time-derivative of the regularized solution. The minimizer of this problem is computed using Bellman's principle of dynamic programming. In this way we obtain a Hamilton-Jacobi equation for the value function. This equation can be solved in the time-discrete case by an Riccati equation and a backward and forward iteration. In this way we find an iterative way to compute a regularized solution. For this method it can be shown using spectral theory and filter function that it gives a convergent regularization. (Slides Pdf)

• Dr. Philipp Kügler (University of Linz, Austria and RICAM)

  Title: Online Parameter Identification in Time Dependent Differential Equations
  Abstract:
  Online parameter identification becomes necessary whenever model parameters are needed in order to support decisions that have to be taken during the operation of the real system. Based on ideas from adaptive control and regularization of inverse problems we suggest an online method that works both for ODEs and time-dependent PDEs. It allows for partial state observations and does neither require a linear parameterization nor data differentiation or filtering. Numerical examples in context of aircraft control are presented. (Slides Pdf)

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Last update: Friday, 24-August-2007