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: L1 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), L1 data fitting is more robust than conventional L2 fitting. However, this leads to a non-smooth problem, even if the parameter-to-measurement mapping is differentiable. In this talk, a numerical approach for the solution of nonlinear L1 data fitting problems in the context of parameter identification for PDEs is presented and analyzed, which is based on a semi-smooth Newton method and shows locally superlinear convergence. The effectiveness is illustrated through numerical examples.

Speaker: Antonio Leitao
Federal University of Santa Catarina
Title: Level-set approaches of L2-type for recovering shape and contrast in ill-posed problems
Abstract: We investigate level-set type approaches for solving ill-posed 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 level-set frameworks are proposed for solving the inverse problem. In both of them the level-set function is assumed to be in L2. 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 min-max formulation of sparsity regularization with optimal source. We will present some numerical results for sparse reconstruction of the optical parameters.

Speaker: Cara Brooks
Rose-Hulman 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 non-Volterra integral equations. We formulate a new regularization method which generalizes the method of local regularization and provides a framework for the design of non-global 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 widely-used a-posteriori 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 ℓp-norms, with 1 ≤ p ≤ 2, as penalty terms such a structural variational inequality is satisfied in Hilbert spaces if the searched-for 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
  1. S.W. Anzengruber and R. Ramlau, Morozov's discrepancy principle for Tikhonov type functionals with nonlinear operators, Inverse Problems 26(2) (2010), 025001.
  2. R.I. Bot and B. Hofmann, An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems, J. Integral Equations Appl. 22(3) (2010).
  3. M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Problems 20 (2004), 1411–1421.
  4. M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with `q penalty term, Inverse Problems (2008), 24(5) 1–13.
  5. E. Resmerita and O. Scherzer, Error estimates for non-quadratic regularization and the relation to enhancement, Inverse Problems 22 (2006), 801–814.


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 rj and the price floor associated with instrument j by rj. The meaning of this constraint is that to each investor a minimal price rj 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 rj. Analogously each investor cannot obtain for an asset a price greater than rj and as a liability the price cannot exceed the maximum price rj. 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
  1. A. Barbagallo, P. Daniele and A. Maugeri, Variational formulation for a general dynamic financial equilibrium problem. Balance law and liability formula, Nonlinear Analysis, doi: 10.1016 j.na.2010.1



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