The SFB program expired on September 30, 2008. For the link to the successor project click DK Computational Mathematics


-  F1308 Iterative

-  F1308 Geometric

-  F1308 New Concepts

-  F1308 Applications


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


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.

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


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.

Inverse Problems in Semiconductors


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.

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.

Direct and Inverse Problems in Polymer Crystallization


Multiscale Modelling and Simulation of Crystallization Processes

Some short text about the problem statement

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.

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

Inverse Problems in glass

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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.

Volatility Estimation in Finance

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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.

Inverse Scattering and Impedance Tomography


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.

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