# Mission Statement

The core research topics of the Computational Science Center are Inverse Problems and Image Analysis. The common thread among these areas is a canonical problem of recovery of an object (function or image) from partial or indirect information. Particular research topics are:

- Geometrical Modeling for Image Analysis.
- Mathematical Modeling and numerical simulations in Coupled Physics Imaging.
- Variational regularization methods for Image Analysis and Inverse Problems.
- Mathematical models for visual attention.

# News

The special semester is postponed to 2022, but a smaller numbers of workshops will still take place. (→ more information).

We kindly invite you to the talk "On some inverse source problems governed by fractional diffusion equations" by Mohamed Ben Salah. Monday, Oct. 4th, 9:00am Vienna Time.

Contact to get an invite link.

This topic is concerned with inverse source problems related to fractional operators. Three problems have been addressed in this research study.

The first one is related to the fractional Laplacian operator. The aim is to reconstruct an unknown source term from internal noisy measured data. The leading term of the mathematical model equation is governed by the fractional spectral Laplacian. The inverse problem is formulated as a regularized optimiza- tion one minimizing a least square type functional. The existence, uniqueness and stability of the unknown source term have been established. In the numer- ical part, we develop a numerical reconstruction approach for identifying the unknown source term and solving the inverse problem.

The second inverse problem is governed by a time-fractional diffusion equa- tion. It consists in identifying an unknown source term support from boundary measurements of the potential field. In this study, we have proposed a fast and accurate approach combining the robustness of the Kohn-Vogelius formulation and the rapidity of the topological gradient method. In the theoretical part of this work, We have derived a topological asymptotic expansion, with respect to a small geometric perturbation of the source term, valid for a large class of source functions. In the numerical part, we have developed a reconstruction algorithm for solving the considered geometric inverse source problem and iden- tifying the location, size and shape of the unknown sub-domains.

The third inverse source problem is related to a space-time fractional diffu- sion equation. Our aim is to identify an unknown source term from partially observed data. The employed model involves the Caputo fractional derivative in time and the non-local fractional Laplacian operator in space. The well- posedness of the forward problem is discussed. The considered ill-posed inverse source problem is formulated as a minimization one. The existence, uniqueness and stability of the solution of the minimization problem are examined. An iter- ative process is developed for identifying the unknown source term. A numerical implementation of the proposed approach is performed. The convergence of the discretized fractional derivatives is analyzed. The efficiency and accuracy of the proposed identification algorithm are confirmed by some numerical experiments.

We kindly invite you to the talk "A realistic electrode model in electrical impedance tomography" by Noemi Naujoks. Monday, Sep. 27th, 10:00am Vienna Time.

Contact to get an invite link.

Electrical impedance tomography (EIT) is a non-invasive imaging technique based on the reconstruction of electrical conductivities in the human body. For this purpose, various current patterns are injected into the body via electrodes and the resulting voltages are measured. From a mathematical point of view, the reconstruction of conductivity using these measured data provides an ill-posed, non-linear inverse problem. The task is to include the use of electrodes into the mathematical model. The gap model describes the simplest type of electrode modeling by assuming that the current, injected through a specific electrode, has a uniform strength on the entire area of the electrode. However, in reality, the distribution of a current flow along an electrode is unknown. Furthermore, electrodes are usually built from a highly conductive material, which we consider a perfect conductor. Hence the potential along each electrode is assumed to be constant. The shunt model, which forms a more realistic approach considering these properties, is investigated in this work. To perform the reconstruction of conductivity using this model, an inexakt Newton-type method is applied. The aim is to implement the theoretical findings and investigate whether the realistic electrode modeling can achieve a positive effect in reconstruction. A comparison of the numerical results of the two models indicates that the shunt model is more accurate at reconstructing images than the gap model.

The Tenth International Conference "Inverse Problems: Modelling and Simulation" (IPMS 2022) will be held during May 22-28, 2022 at the Congress Center of the Paradise Bay Hotel, Mellieha, Malta. The IPMS conference series is one of the main scientific meetings of the field which has been organized every two years since 2002, under the auspices of The Eurasian Association on Inverse Problems (EAIP). The Conference IPMS 2022 is the tenth (Jubilee) in the series. The objective of this meeting is to be multidisciplinary and international, bringing together scientists working on various topics of inverse problems in diverse areas. An important goal of the IPMS conferences is to encourage the participation of young researchers by offering them the opportunity to deliver invited talks and by partially supporting them. We look forward to welcoming you to Malta.

(→ more information)We are offering a Post Doc Research Position within the Christian-Doppler Laboratory on Mathematical Modeling and Simulations of Next Generations of Ultrasound Devices at the Faculty of Mathematics, University of Vienna, funded by the Christian-Doppler Research Association. Full time, 40 hours per week, for an initial period of two year (with possible extensions up to a maximum of seven years), starting on January 1st, 2022.

The researcher will work on developing and implementing inverse reconstruction formulas for ultrasound imaging. In particular strong interactions with the experimental partners and medical doctors at the Medical University of Vienna are required. We anticipate that the software components are directly implemented on programmable medical devices.

(→ detailed information)We are offering a Doctorate Research Position within the Christian-Doppler Laboratory on Mathematical Modeling and Simulations of Next Generations of Ultrasound Devices at the Faculty of Mathematics, University of Vienna, funded by the Christian-Doppler Research Association. 30 working hours per week, for an initial period of two year (with possible extensions up to four year, or until the doctorate is finished), starting on January 1st, 2022.

The researcher will work on developing and implementing image processing tools for ultrasound imaging. In particular strong interactions with the experimental partners and medical doctors at the Medical University of Vienna are required, which will evaluate the software developments.

(→ detailed information)