The research activities of the IPMI group are organized around four main themes:

1. Regularization, Learning, and Data-Driven Inversion
2. Wave Imaging and Inverse Scattering
3. Tomography and Integral-Geometric Reconstruction
4. Coupled-Physics, Resonance-Based Imaging and Emerging Methods

Our research combines inverse problems, wave propagation, tomography, regularization, and
resonance-based modeling to develop rigorous mathematical methods for imaging, sensing,
and related applications.

Several members of the group (Hyojae Lim, Bochra Mejri, Sergei Pereverzev, Otmar Scherzer) work on regularization methods for nonlinear inverse problems, which has been one of the core research themes at RICAM since its foundation. Recent developments in machine learning have strengthened this direction further. In particular, the group has advanced the mathematical understanding of the interplay between regularization theory and data-driven methods, including the formulation of major open challenges for the field and the development of rigorous learning-theoretic frameworks for inverse problems.

[1] C. Kirisits, B. Mejri, S. Pereverzev, O. Scherzer and C. Shi, Regularization of Non- linear Inverse Problems–From Functional Analysis to Data-Driven Approaches,
https://arxiv.org/abs/2506.17465.

Regularization for Transfer Learning and Domain Adaptation

A major focus of the group concerns transfer learning and domain adaptation, where one seeks to train models under one probability law and deploy them under another. This challenge is intrinsically ill-posed and naturally requires regularization. The group has contributed both foundational theory and practical strategies for hyper-parameter choice in unsupervised domain adaptation, with applications ranging from learning theory to biomedical problems.

[1] S. V. Pereverzyev, An Introduction to Artificial Intelligence Based on Reproducing Kernel Hilbert Spaces, Birkhäuser Cham, Basel, 2022.

[2] M.-C. Dinu, M. Holzleitner, M. Beck, D. H. Nguyen, A. Huber, H. Eghbal-zadeh, B. Moser, S. V. Pereverzyev, S. Hochreiter, and W. Zellinger, “Addressing parameter choice issues in unsupervised domain adaptation by aggregation,” International Conference on Learning Representations (ICLR), 2023.

[3] E. R. Gizewski, L. Mayer, B. Moser, D. H. Nguyen, S. Pereverzyev Jr, S. V. Pereverzyev, N. Shepeleva, and W. Zellinger, “On a regularization of unsupervised domain adaptation in RKHS,” Applied and Computational Harmonic Analysis, 57 (2022), pp. 201–227.

Regularization of Inverse Problems for Fractional Differential Equations

The calibration of models based on fractional differential equations requires dedicated regularization techniques, due in particular to the nonlocal nature of fractional derivatives and the presence of memory effects. The group has developed and analyzed such methods, with rigorous results on the identification of fractional orders and memory parameters in semilinear subdiffusion models.

[1] M. Krasnoschok, S. Pereverzyev, S. V. Siryk, N. Vasylyeva, “Determination of the Fractional Order in Semilinear Subdiffusion Equations,” Fract. Calc. Appl. Anal. 23 (2020), 694–722.

[2] S. Pereverzyev, S. V. Siryk, N. Vasylyeva, “Identification of the Memory Order in Multi-Term Semilinear Subdiffusion,” Numerical Functional Analysis and Optimization 46(3) (2025), 275–309.

Regularization in Electrochemical Impedance Spectroscopy

Electrochemical impedance spectroscopy is an important diagnostic tool across energy sciences, chemistry, and biology. The extraction of distribution-of-relaxation-times information from measured data leads to ill-posed inverse problems. Within this direction, the group developed adaptive multi-parameter regularization techniques and contributed to recent advances in the mathematical analysis of the DRT methodology.

[1] M. Žic, S. Pereverzyev Jr., V. Subotić, S. Pereverzyev, “Adaptive multi-parameter regularization approach to construct the distribution function of relaxation times,” Int. J. Geomath. 11, 2 (2020).

[2] A. Maradesa, B. Py, J. Huang, Y. Lu, P. Iurilli, A. Mrozinski, H. M. Law, Y. Wang, Z. Wang, J. Li, S. Xu, Q. Meyer, J. Liu, C. Brivio, A. Gavrilyuk, K. Kobayashi, A. Bertei, N. J. Williams, C. Zhao, M. Danzer, M. Zic, P. Wu, V. Yrjänä, S. Pereverzyev, Y. Chen, A. Weber, S. V. Kalinin, J. P. Schmidt, Y. Tsur, B. A. Boukamp, Q. Zhang, M. Gaberšček, R. O’Hayre, F. Ciucci, “Advancing electrochemical impedance analysis through innovations in the distribution of relaxation times method,” Joule 8(7) (2024), 1958–1981.

Mathematical understanding of wave phenomena and imaging with waves has been a central activity of the group over the past years. In particular, the group has led the SFB “Tomography Across the Scales” (2018–2026) and is involved in the Christian Doppler Laboratory MaMSi on medical ultrasound tomography, in collaboration with the University of Vienna, GE Healthcare, and the Medical University of Vienna. This line of research combines rigorous wave analysis, inverse problems, and imaging design across multiple length scales.

Elastic Diffraction Tomography

The group has developed a novel two-step inversion process for reconstructing elastic parameters in inverse scattering problems for elasticity. The method is based on a careful separation of wave modes and opens a promising route for stable parameter reconstruction in elastic diffraction tomography.

[1] B. Mejri, O. Scherzer, “An inversion scheme for elastic diffraction tomography based on mode separation,” SIAM Journal on Applied Mathematics, 2024.

Mathematical Imaging with Random Contrast Agents

Over the last few years, we have modeled and analyzed ultrasound, photoacoustic, and elastographic imaging modalities using resonant contrast agents in deterministic settings. A natural next step is to incorporate randomness in the distribution of the contrast agents and quantify its impact on stability, resolution, and imaging performance. This direction is currently being developed within the group, including ongoing PhD work on photoacoustic imaging with random contrast agents.

[1] S. Senapati and M. Sini, “Minnaert Frequency and Simultaneous Reconstruction of the Density, Bulk and Source in the Time-Domain Wave Equation,” Archive for Rational Mechanics and Analysis, 2025.

[2] A. Ghandriche and M. Sini, “Simultaneous reconstruction of optical and acoustical properties in photoacoustic imaging using plasmonics,” SIAM Journal on Applied Mathematics, 2023.

[3] A. Ghandriche, S. Senapati and M. Sini, “Inverse Problems in Imaging using Resonant Contrast Agents,” in Inverse Problems for Mechanical Systems: Methods, Simulations and Experiments, CISM Courses and Lectures.

Mathematical Imaging in Vision

The group also develops mathematical methods for computer vision, with particular emphasis on the interpretation of vertex classifications in 3D scenes from 2D images. This problem is closely related to geometric imaging, occlusion analysis, and image understanding. A recent approach based on topological asymptotics provides a mathematically rigorous framework for such questions and opens further perspectives at the interface of imaging and vision.

[1] P. Gangl, B. Mejri, O. Scherzer, “Vertex characterization via second-order topological
derivatives,” Journal of Mathematical Imaging and Vision 67(53) (2025).

Dead Water Problem and Hydrodynamic Pressure

The group is also interested in wave propagation in fluid systems, including the dead-water phenomenon arising in stratified maritime environments. Motivated by emerging Arctic and Greenland navigation routes, this topic concerns the generation of internal waves by vessels moving across layered fluids with different densities. A first step in this direction has been the analysis of hydrodynamic pressure in the presence of an underlying current, with extensions to two-layer configurations expected to require a combination of analysis and numerical methods.

[1] A. Constantin, N. Gindrier, O. Scherzer, “Pressure beneath a periodic travelling water-wave in constant-vorticity flow over a flat bed,” http://arxiv.org/abs/2602.21077.

The group studies several tomographic inverse problems using A-analytic function theory in the spirit of Bukhgeim. This includes the inversion of attenuated moment ray transforms, range characterizations of tensor X-ray transforms, local tomography from partial data, and inverse source problems in radiative transport. This research has been carried out in part within the FWF project “Weighted X-ray Transform and Applications” and has also supported doctoral training within the group.

[1] H. Fujiwara, D. Omogbhe, K. Sadiq, A. Tamasan, “Inversion of the attenuated momenta ray transform of planar symmetric tensors,” Inverse Problems 40(7), 2024.

[2] K. Sadiq, “Range characterization of the X-ray transform of symmetric complex valued tensor fields,” Inverse Problems 42(1), 015012, 2026.

[3] K. Sadiq, A. Tamasan, “On the range of the X-ray transform of symmetric tensors compactly supported in the plane,” Inverse Problems and Imaging 17(3) (2023), 660–685.

[4] H. Fujiwara, K. Sadiq, A. Tamasan, “Partial inversion of the 2D attenuated X-ray transform with data on an arc,” Inverse Problems and Imaging 16(1) (2022), 215–228.

[5] H. Fujiwara, K. Sadiq, A. Tamasan, “A Fourier approach to the inverse source problem in an absorbing and anisotropic scattering medium,” Inverse Problems 36(1), 015005, 2019.

Superresolution Microscopy

Recent progress in superresolution microscopy, pursued within the SFB “Tomography Across the Scales”, has enabled the identification of polarization properties of loaded dipole emitters. These measurements allow for more accurate localization of dipoles and a more refined identification of molecular and material parameters. This creates new opportunities at the interface of inverse problems, microscopy, and applied sciences.

Recent work of the group has characterized broad families of resonances generated by high-transmission structures, including high-frequency resonances of Fabry-Pérot type. Theseresults open the way to high-frequency homogenization theories and to imaging concepts with potentially very high resolution, combining resonant media, wave localization, and effective models.

[1] L. Li and M. Sini, “High contrast transmission and Fabry-Pérot-type resonances,”
https://arxiv.org/abs/2510.19096.

[2] L. Li and M. Sini, “High-Contrast Transmission Resonances for the Lamé System,”
https://arxiv.org/pdf/2601.10290.

Wave Control for Mathematical Imaging and Therapy

A recent direction of the group concerns the use of nanoparticles and resonant structures as actuators in controlled wave and heat processes. This combines wave propagation, sub-wavelength resonators, and feedback stabilization, with applications to imaging, therapy, thermo-plasmonics, and acoustic cavitation. The framework is flexible enough to be adapted to acoustics, electromagnetism, elasticity, and coupled models.

[1] A. Mukherjee, A. S. Rodrigues, and M. Sini, “Feedback Stabilization and Tracking for Heat Equations Using Thermo-Plasmonic Nanoparticles as Actuators,”
https://arxiv.org/abs/2602.14581.

[2] X. Cao, A. Mukherjee and M. Sini, “Effective medium theory for heat generation using plasmonics: a parabolic transmission problem driven by the Maxwell system,” Mathematische Annalen, 2025.

[3] A. Mukherjee and M. Sini, “Time-dependent acoustic waves generated by multiple resonant bubbles: application to acoustic cavitation,” Journal of Evolution Equations, 2024.

Electromagnetism with Extreme and Resonant Materials

The group develops mathematical models and asymptotic theories for electromagnetic fields generated by high-contrast and resonant nanostructures, with particular emphasis on dielectric nanoparticles, plasmonic inclusions, and hybrid dielectric-plasmonic dimers. This research addresses both the wave generation mechanisms and the collective behavior of clusters of resonant particles, with applications to subwavelength imaging, effective media, and the design of advanced electromagnetic materials. Recent contributions include the analysis of electromagnetic waves generated by high-index nanoparticle clusters, the study of hybrid dielectric-plasmonic dimers, and the extension to clusters of such hybrid resonators. Such hybrid structures open the way to generating (double) negative effective permittivity and permeability and bi-anisotropic constitutive laws and eventually hyperbolic media.

[1] X. Cao, A. Ghandriche, M. Sini, “The electromagnetic waves generated by a cluster of nanoparticles with high refractive indices,” Journal of the London Mathematical Society (2) 108 (2023), no. 4, 1531–1616.

[2] X. Cao, A. Ghandriche, M. Sini, “Electromagnetic waves generated by a hybrid dielectric-plasmonic dimer,” SIAM Journal on Applied Mathematics 85 (2025), no. 5, 1949–1975.

[3] X. Cao, A. Ghandriche, M. Sini, “Electromagnetic Scattering by a Cluster of Hybrid Dielectric-Plasmonic Dimers,” https://arxiv.org/abs/2601.08242.