Minisymposia Abstracts
Speaker:  Frank Schöpfer Helmut Schmidt Universitat 
Title:  Iterative regularization methods for parameter identification problems in Banach spaces 
Abstract:  In this talk we are concerned with nonlinear parameter identification problems in
Banach space settings. We discuss convergence and regularizing properties of two iterative
solution methods. One of them is a Landweber type iteration, the other one the iteratively regularized Gauss–Newton method. A discrepancy principle as stopping rule renders both iterative
schemes regularization methods. The applicability of the theoretical findings is illustrated by
two parameter identification problems for elliptic PDEs.

Speaker:  Christian Clason University of Graz 
Title:  L^{1} data fitting for parameter identification problems for PDEs 
Abstract: 
If the measured data is corrupted by impulsive noise (e.g., from malfunctioning pixels in camera sensors, faulty memory locations in hardware, or transmission in noisy channels),
L^{1} data fitting is more robust than conventional L^{2} fitting. However, this leads to a nonsmooth
problem, even if the parametertomeasurement mapping is differentiable. In this talk, a numerical approach for the solution of nonlinear L^{1} data fitting problems in the context of parameter
identification for PDEs is presented and analyzed, which is based on a semismooth Newton
method and shows locally superlinear convergence. The effectiveness is illustrated through
numerical examples.

Speaker:  Antonio Leitao Federal University of Santa Catarina 
Title:  Levelset approaches of L^{2}type for recovering shape and contrast in illposed problems 
Abstract:  We investigate levelset type approaches for solving illposed inverse problems, under the assumption that the solution is a piecewise constant function. Our goal is to identify the level sets as well as the level values of the unknown parameter function. Two distinct levelset frameworks are proposed for solving the inverse problem. In both of them the levelset function is assumed to be in L^{2}. Corresponding Tikhonov regularization approaches are derived and analyzed. Existence of minimizers for the Tikhonov functionals is proven. Moreover, convergence and stability results of the variational approaches are established, characterizing the Tikhonov approaches as regularization methods. 
Speaker:  Taufiquar Khan Clemson University 
Title:  Maximizing distinguishability and sparsity regularization for the inverse problem in diffuse optical tomography 
Abstract:  In this talk, an extension of the distinguishability criteria of Isaacson and Knowles
to optimal source in DOT will be discussed. The influence of the inner products of the function
spaces on the resulting optimal source will be demonstrated using simulations. We will further
discuss a minmax formulation of sparsity regularization with optimal source. We will present
some numerical results for sparse reconstruction of the optical parameters.

Speaker:  Cara Brooks RoseHulman Institute of Technology 
Title:  A generalized approach to the method of local regularization 
Abstract:  The convergence theory associated with local regularization for solving inverse illposed
problems includes finitely smoothing linear Volterra problems, nonlinear Hammerstein
and autoconvolution problems, as well as linear nonVolterra integral equations. We formulate
a new regularization method which generalizes the method of local regularization and provides
a framework for the design of nonglobal regularization methods and methods which make
use only of the data most relevant to the desired solution. We demonstrate application of the
generalized method to solving some linear inverse problems.

Speaker:  Stefan Anzengruber Radon Institute for Computational and Applied Mathematics 
Title:  Convergence rates for Morozov's discrepancy principle 
Abstract: 
In this talk we combine the techniques of variational inequalities, which generalize classical source
and nonlinearity conditions (see, e.g., [2]), with Morozov's discrepancy principle a widelyused
aposteriori parameter choice rule for Tikhonov regularization of inverse problems. By this means,
we are able to derive convergence rates with respect to the Bregman distance (compare [3,5]) for
nonlinear operators between Banach spaces and these findings together with a structural variational
inequality are used to obtain convergence rates of up to linear order in the norm topology (compare
[4]).
Moreover, we will see that for the special choice of sparsity promoting weighted ℓ_{p}norms,
with 1 ≤ p ≤ 2, as penalty terms such a structural variational inequality is satisfied in Hilbert
spaces if the searchedfor solution is known to be sparse, and that therefore a linear rate of convergence
is obtained in the limiting case p = 1. (Joint work with Ronny Ramlau.)
References

Speaker:  Christoph Schwarzbach University of British Columbia, Canada 
Title:  Convergence rates for Morozov's discrepancy principle 
Abstract:  TBA

Speaker:  Annamaria Barbagallo Universita degli Studi di Catania 
Title:  Competitive financial equilibrium problems with policy interventions: Variational formulation and inverse problems 
Abstract: 
The aim of the talk is to generalize the general model of competitive financial equilibrium
introduced in [1] incorporating policy interventions in the form of taxes and price controls.
In [1], the authors obtained a very interesting balance law and a liability formula, which
could be of topical interest considering the world economic disequilibrium. Now, we would
like to improve this model introducing the aspect of policy interventions. To this aim, let
us denote the price ceiling associated with instrument j by r_{j} and the price floor associated
with instrument j by r_{j}. The meaning of this constraint is that to each investor a minimal
price r_{j} for the assets held in the instrument j is guaranteed, whereas each investor is requested
to pay for the liability in any case a minimal price r_{j}. Analogously each investor
cannot obtain for an asset a price greater than r_{j} and as a liability the price cannot exceed
the maximum price r_{j}. After giving the evolutionary financial equilibrium condition,
an equivalent variational inequality formulation is proved, from which an existence result
follows. Moreover, we apply the infinite dimensional duality theory called empty interior
theory, obtaining the existence of Lagrange variables and their explicit expression. Then,
we investigate on the inverse theory by means of the Lagrange multipliers. (Joint work with Patrizia Daniele and Sofia Giuffré.)
References

Please address administrative questions to aipc@math.tamu.edu. Scientific questions should be addressed to the chair of the Scientific Program Committee: rundell AT math.tamu.edu