# Abstract

Inverse problems appear in a variety of industrial applications. The mission statement of this project is to develop theory based solvers for inverse problems and to apply these results to real world problems. Whereas some of the industrial problems, presented here, serve only as test problems for theory based algorithms, other industrial problems are of their own interest, in term of analysis, solution strategies, ... In the following we explain some of the industrial applications, that we tackle by several methods, in some more detail.

# Inverse Problems in Structural Mechanics

Contact: martin.burger@jku.at, benjamin.hackl@ricam.oeaw.ac.at

#### Inclusion & Crack Detection:

To predict failures of materials it is desirable to know production failures or already appearing material failures like inclusions or cracks without destroying the material itself. There are several methods in non-destructive testing to achieve this goal. One method is to apply certain forces at the boundary of the object (device) and measure its deformation. From these measurements one can (uniquely) identify the inclusions or cracks in the material. Like usual for such inverse problems, they are ill-posed. Besides the ill-posed character of the problem we face an additional problem, namely that the number of inclusions or cracks is usually not a-priori known. To deal with the possible topology changes we use the level-set method with the additional information of topological derivatives.

- Hend Ben Ameur, Martin Burger, Benjamin Hackl,
*Level set methods for geometric inverse problems in linear elasticity,*Inv. Prob., 20, 2004, 673-696. Preprint: .pdf, .ps.gz - Martin Burger, Benjamin Hackl, Wolfgang Ring,
*Incorporating topological derivatives into level sets methods,*J. Comp. Phys., 194, 2004, 334-362. Preprint: .pdf - Hend Ben Ameur, Martin Burger, Benjamin Hackl,
*On some geometric inverse problems in linear elasticity,*UCLA CAM-03-55, 2003. Preprint: .pdf

#### Parameter Identification in Plasticity:

For aeronautics-, automotive- industry it is important to
predict the failure of their devices. This is usually done by
large numerical simulations. To get good predictions the
knowledge of the material parameters is crucial. Some of
these parameters can be measured directly by some tests like
compression-, shear- and tensile test but most parameters
have to be determined indirectly. Especially for metals
parameters like the hardening moduli, the damage variable or
even the whole yield surface need to be determined
indirectly. The identification of these parameter is usually
ill-posed. Furthermore the models that describe the failure
of metals are highly non-linear and usually not
differentiable. Hence a standard application of iterative
regularization method seems not possible. This industrial
application provides a rich source of theoretical as well as
practical problems in inverse problems, especially the
development of derivative free iterative regularization
methods, that exploit the rich but non-differentiable
structure of the original direct problem.

# Identification of Nonlinearities

Contact: philipp.kuegler@jku.at, herbert.egger@oeaw.ac.at

The goal of parameter identification is to indirectly retrieve parameters that appear in partial differential equations from (noisy) data related to the pde-solution, e.g., the solution itself at some specific time, some boundary traces or more complicated functionals. Parameter identification problems belong to the class of inverse problems and are typically ill-posed such that they need to be regularized depending on the quality of the data. Even if the parameters to be identified are assumed to be constant and the underlying PDE system is linear, the related inverse problem is usually nonlinear. This nonlinearity is strengthened if the parameters are considered as functions of space and/or time and/or the underlying PDE system is nonlinear due to known quantities.

In our project, we focus on the nonlinear inverse problem of identifying parameter functions that depend on the PDE solution, the norm of its gradient, i.e., the underlying PDE system is nonlinear due to unknown quantities. The applicability of classical regularization methods to the "identification of nonlinearities" is studied, furthermore especially taylored algorithms, that enable to abandon differentiability requirements, are designed, analyzed and implemented. When it comes to the identification of nonlinearities in dynamical systems, we utilize our results for the design of methods that allow for real-time estimation of the parameters.

- P. Kügler,
*A derivative-free Landweber iteration for parameter identification in certain elliptic PDEs,*Inverse Problems, 19, 2003, 1407-1426. Preprint: .pdf - P. Kügler,
*Identification of a temperature dependent heat conductivity from single boundary measurements,*SIAM J. Numer. Anal., 41, 2003, 1543-1563. Preprint: .pdf - B. Kaltenbacher, M. Kaltenbacher, S. Reitzinger,
*Identification of Nonlinear B-H Curves Based on Magnetic Field Computations and Multigrid Methods for Ill-Posed Problems,*Europ. J. of Appl. Math., 14, 2002, 15-38. Preprint: .ps - P. Kügler, H.W. Engl,
*Identification of a temperature dependent heat conductivity by Tikhonov regularization,*J. of Inverse and Ill-posed Problems, 10, 2002, 67-90. Preprint: .ps - H. Egger, H.W. Engl, M.V. Klibanov,
*Global Uniqueness and Hölder Stability for Recovering a Nonlinear Source Term in a Parabolic Equation,*Inverse Problems, 21, 2005, 271-290. Preprint: .pdf

# Inverse Problems in Semiconductors

Contact: martin.burger@jku.at

#### Inverse Dopant Profiling in Semiconductor Devices:

The identification and optimal design of doping profiles in semiconductors (e.g. p-n junctions) are from growing interest. As a geometrical problem one is concerned with topological changes, that can be handled with the level-set method, but this time any topological change results into a completely different device, which is not desirable. Nonetheless the level set method provides still more computational efficiency and more general shapes than classical parameterization approach. Hence the challenge is to develop efficient and convergent level-set methods that do not yield unwanted topology changes.

- M. Burger, H. W. Engl, A. Leitao, P. Markowich,
*On inverse problems for semiconductor equations,*Matematica Contemporanea (2004), to appear - M. Burger, P. Markowich,
*Model reduction for semiconductor inverse dopant profiling,*in: I.Troch, F.Breitenecker, eds.. Proceedings of the 4th IMACS Symposium on Mathematical Modelling (Vienna, 2003, electronic, ISBN 3-901608-24-9). - M. Burger, R. Pinnau,
*Fast optimal design of semiconductor devices,*SIAM J. Appl. Math., 64, 2003, 108-126. Preprint: .pdf - M.Burger, H.W.Engl, P.Markowich,
*Inverse doping problems for semiconductor devices,*Recent Progress in Computational and Applied PDEs (Kluwer Academic/Plenum Publishers, 2002), 27-38. - M. Burger, H. W. Engl, P. Markowich, P. Pietra,
*Identification of doping profiles in semiconductor devices,*Inverse Problems, 17, 2001, 1765-1795. Preprint: .ps.gz

#### Photonic crystals and Waveguides:

The optimal design of photonic bandgap structures and waveguides is an increasing issue in technological applications. As usual in optimal design problems, topology changes are one challenge, that might be tackled by the use of level-set methods. Besides the large scale, which is due to the solution of an eigenvalue problem of the Helmholtz or the Maxwell equations, another challenge of these problems is also the non-differentiability of the objective functional at multiple eigenvalues which make the development of new algorithms necessary.

- M. Burger, S. Osher, E. Yablonovitch,
*Inverse problem techniques for the design of photonic crystals,*IEICE Transactions E 87, 2004. - H. Engl, T. Felici,
*On shape optimization of optical waveguides,*Inverse Problems, 17, 2001, 1141-1162.

# Direct and Inverse Problems in Polymer Crystallization

Contact: martin.burger@jku.at

#### Multiscale Modelling and Simulation of Crystallization Processes

Some short text about the problem statement

- M. Burger, V. Capasso, A. Micheletti,
*Mathematical modelling of the crystallization process of polymers,*in: E.A.Lipitakis, HERCMA 2001, Proceedings of the 5th Hellenic European Conference on Computer Mathematics and its Applications (LEA, Athens, 2002), 51-63. Reprint in HERMIS International Journal, 3, 2003, 135-164. - M. Burger,
*Growth of multiple crystals in polymer melts,*European J. Appl. Math., 2003, to appear. - M. Burger,
*Growth and impingement in polymer melts,*in: P. Colli et. al., eds., Free Boundary Problems (Birkhaeuser, Basel, 2003). - V. Capasso, M. Burger, A. Micheletti, C. Salani,
*Mathematical models for polymer crystallization processes,*in: V.Capasso et. al., eds., Mathematical modelling for polymer industry (Springer, 2002), 167-242. - M. Burger, V. Capasso, G. Eder,
*Modelling crystallization of polymers in temperature fields,*ZAMM,82, 2002, 51-63. - M. Burger, V. Capasso, C. Salani,
*Modelling multi-dimensional crystallization of polymers in interaction with heat transfer,*Nonlinear Analysis, Series B, 3, 2002, 139-160. - M. Burger, V. Capasso,
*Mathematical modelling and simulation of non-isothermal crystallization of polymers,*Math. Models and Methods in Appl. Sciences, 11, 2001, 1029-1054. - A. Micheletti, M. Burger,
*Stochastic and deterministic simulation of nonisothermal crystallization of polymers,*J. Math. Chem., 30, 2001, 169-193. - M. Burger,
*Mathematical Models and Methods in Polymer Processing,*(Diploma Thesis, JK University Linz, June 1998).

#### Identification of Nucleation Rates

An inverse problem of particular interest is the
identification of nucleation rates of polymeric materials,
which can be modeled as functions of temperature. With
traditional techniques, the value of the nucleation rate for
each temperature value has to be determined by one specific
quasi-isothermal experiment and an explicit counting process,
which is difficult to automize since the crystal structures
are difficult to recognize by mathematical imaging
techniques. Therefore, and opposed to normal growth rates,
only few data on nucleation rates where available. By using
iterative regularization methods, we were able to identify
the nucleation rate as a function of temperature in a single
non-isothermal experiment only, and from the mathematical
structure of the inverse problem (source conditions,
identifiability and considerations) we were even able to
obtain information on the design of such experiments.

- M. Burger, V. Capasso, H. W. Engl,
*Inverse problems related to crystallization of polymers,*Inverse Problems, 15, 1999, 155-173. - M. Burger,
*Iterative regularization of a parameter identification problem occurring in polymer crystallization,*SIAM J. Numer. Anal., 39, 2001, 1029-1055. Preprint: .ps - M. Burger,
*Direct and Inverse Problems in Polymer Crystallization Processes,*(PhD-Thesis, JK University Linz, May 2000). - M. Burger, V. Capasso, G. Eder, H. W. Engl,
*Modelling and parameter-identification in non-isothermal crystallization of polymers,*in: L. Arkeryd, J. Bergh, P. Brenner, and R. Pettersson (eds.), Progress in Industrial Mathematics at ECMI 98, Teubner, Stuttgart, Leipzig, 1999, 114-121.

#### Optimal Control of Crystallization Processes

An ultimate goal in the production of high-quality
polymeric material is the optimal control of the cooling
process to improve the mechanical properties of the final
solidified product. Since the mechanical
properties Using the models determined above, one
can formulate

- M. Burger, V. Capasso, A. Micheletti,
*Optimal control of polymer morphologies,*J. Eng. Math., 2004, to appear.

# Inverse Problems in glass

Contact person: kuegler@indmath.uni-linz.ac.at or fusek@indmath.uni-linz.ac.at

We study the identification of a parameter in a fourth-order elliptic partial differential equation that models the optimal design of car windshields to be manufactured by the sagging process. Considered as a second-order equation for the unknown parameter, the problem is of mixed type, i.e., changing between elliptic and hyperbolic. Numerical routines for directly solving this equation are not available. Instead, we apply iterative regularization methods where special care has to be taken since the course of the iteration depends markedly on the mixed type of the second-order equation. By means of symmetry techniques for PDEs, the performance of the algorithms shall be enhanced.

#### Identification of the heat transfer coefficient

The knowledge of the heat transfer coefficient as a
function of temperature is of decisive importance for a
precise numerical simulation of the heat transfer between the
glass and the molds. We consider a fast algorithm for
numerical identification of the heat transfer coefficient
based on a natural linearization of the corresponding inverse
problem. Numerical tests show that it is suitable for
problems with perturbed data. The proposed approach can also
be used for other parameter identification problems, where
one wants to recover an unknown nonlinear parameter ß(u) from noisy observations
of the state u.

- H. W. Engl, P. Fusek, S. V. Pereverzev,
*Natural linearization for the identification of nonlinear heat transfer laws,*submitted to Inverse Problems. - N. Bila,
*Application of symmetry analysis to a PDE arising in the car windshield design,*SIAM J. Appl. Math. 65, 2004, 113-130. Preprint: .pdf - P. Kügler,
*A parameter identification problem of mixed type related to the manufacture of car windshields,*SIAM J. Appl. Math. 64, 2004, 858-877. Preprint: .pdf

# Volatility Estimation in Finance

Contact person: herbert.egger@ricam.oeaw.ac.at

During the last decades mathematics have had a huge impact
in finance, e.g., for the modelling of market behaviour, for
model prediction, for valuating trading strategies, for
pricing all sort of financial assets. Many mathematically
interesting problems arise in the field of derivative
securities, whose trading volumes multiply exceeds the one of
traditional assets in the meanwhile. Starting from the
seminal paper of Black and Scholes, a rigorous framework for
pricing and hedging of all sorts of financial derivatives has
been developed. In the absence of arbitrage, a fair price of
an option can in principle be determined by specifying a
stochastic process for the underlying asset. Usually, such
models depend on certain model parameters which have to be
estimated in order to be able to predict market behaviour of
the future: *volatility*, which essentially measures
the uncertainty of the evolution of the underlying asset,
turns out to be of special importance. While volatility, to
be more precise, markets expectation of future volatility,
cannot be observed directly, it is possible to identify the
volatility from quoted market prices of liquidly traded
option. This inverse problem turns out to be ill-posed and
thus regularization has to be used for stable identification.
Similar parameter identification problems also arise for
interest products.

- H. Egger,
*Identification of Volatility Smile in the Black Scholes Equation via Tikhonov Regularization,*Master's thesis, Johannes Kepler University Linz, Industrial Mathematics Institute, 2001. - H. Egger,
*Recovering volatility in the Black-Scholes model,*In I. Troch and F. Breitenecker, editors, Proceedings of the 4th MATHMOD Vienna, 2003, ARGESIM Report no. 24. - H. Egger, H. W. Engl,
*Tikhonov Regularization Applied to the Inverse Problem of Option Pricing: Convergence Analysis and Rates,*Inverse Problems, 21, 2005. to appear. Preprint: .pdf

# Inverse Scattering and Impedance Tomography

Contact: martin.burger@jku.at

Very challenging inverse problems are severely ill-posed problems like inverse scattering and impedance tomography. Inverse scattering is concerned with the identification of objects by there scattered farfield and has its applications for example in radar. Impedance tomography is concerned with the identification of (for example) the density of a material from "full" dirichlet to neumann measurements at the boundary. Applications are in medicine. Besides the severely ill-posed nature of both problems, possible topology changes and the large scale are further challenges.

- T. Hohage,
*On the numerical solution of a three-dimensional inverse medium scattering problem,*Inverse Probl. 17, 2001, 1743-1763. - T. Hohage,
*Iterative Methods in Inverse Obstacle Scattering: Regularization Theory of Linear and Nonlinear Exponentially Ill-Posed Problems,*PhD thesis, University of Linz, 1999. - T. Hohage,
*Iterative regularization methods in inverse scattering,*In K. A. Woodbury, editor, Inverse Problems in Engineering: Theory and Practice, New York, 1999. The American Society of Mechanical Engineers. - T. Hohage,
*Convergence rates of a regularized Newton method in sound-hard inverse scattering,*SIAM J.Numer.Anal., 36, 1998, 125-142. - T. Hohage, C. Schormann,
*A Newton-type method for a transmission problem in inverse scattering,*Inverse Problems, 14, 1998, 1207-1227.

SpezialForschungsBereich SFB F013 | Special Research Program of the FWF - Austrian Science Fund