We consider the efficient numerical minimization of Tikhonov functionals with nonlinear operators and non-smooth and non-convex penalty terms, which appear, e.g., in variational regularization. For this, we consider a new class of SCD semismooth* Newton methods, which are based on a novel concept of graphical derivatives, and exhibit locally superlinear convergence. We present a detailed description of these methods, and provide explicit algorithms in the case of sparsity ($\ell_p$, $0\leq p < \infty$) and TV penalty terms. The numerical performance of these methods is then illustrated on a number of tomographic imaging problems. This is joint work with Helmut Gfrerer and Ronny Ramlau.

In the standard photoacoustic tomography (PAT) measurement setup, the data used consist of time-dependent signals measured on an observation surface. In contrast, the measurement data of the recently invented full-field detection technique provides the solution of the wave equation in the spatial domain at a single point in time. While reconstruction using classical PAT data has been extensively studied, not much is known about the full-field PAT problem. In this work, we study full-field photoacoustic tomography with spatially variable sound velocity and spatially variable attenuation. In particular, we reconstruct the initial pressures $p\vert_{t=0}$ and $p_t\vert_{t=0}$ from 2D projections of the full 3D acoustic pressure distribution at a given time. (Joint work with Ngoc Do, Missouri State University, Markus Haltmeier, University of Innsbruck and Linh Nguyen, University of Idaho.

Focused ultrasound is a widely used non-invasive diagnostic and therapeutic tool in modern medicine. A crucial assumption in of its all applications is a constant sound speed in the observed medium. Non-constant sound speeds lead to actual times of flight of the ultrasound waves through the medium differing from calculated times of flight, which are accounted for in focusing algorithms. This leads to an aberrated focus, blurring ultrasound images. As real-time ultrasound imaging is computationally expensive, a fast aberration correction method is needed. In this talk, we present adapted ultrasound focusing algorithms based on geometrical acoustics that make a step into this direction. In a known layered medium setting, it is possible to calculate the correct times of flight. The resulting adapted focusing algorithms correct for the aberrations caused by the different sound speeds in the medium layers. Numerical simulations to determine the precision of our methods are conducted. And finally, the improvements obtained by our Methods in reconstructing Ultrasound Images are demonstrated.

We discuss the time-domain acoustic wave propagation in the presence of a subwavelength resonator modeling a Microbubble. A uniform point-approximation expansion of the wave field -- valid in both space and time -- is derived. The leading-order behavior consists of two components: 1. The primary wave, representing the field generated in the absence of the bubble; 2. The resonant wave, arising from the interaction between the bubble and the background medium. We show that the lifetime of the resonant wave is inversely proportional to the imaginary part of the relevant subwavelength resonance (here, the Minnaert resonance), while its oscillation period is determined by the real part. As an application, we exploit the resonant wave's characteristics -- particularly its lifetime and period -- to simultaneously recover the mass density and bulk modulus in heterogeneous background media. This approach has significant implications for imaging techniques employing contrast agents.

This talk addresses the problem of $\Gamma$-convergence of a family of Tikhonov functionals and assertions of the convergence of their respective infima. Such questions arise, if model uncertainties, inaccurate forward operators, finite dimensional approximations of the forward solutions and / or data, etc. make the evaluation of the original functional impossible and, thus, its minimizer not computable. But for applications it is of utmost importance that the minimizer of the replacement functional approximates the original minimizer. Under certain additional conditions this is satisfied if the approximated functionals converge to the original functional in the sense of $\Gamma$-convergence. We deduce simple criteria in different topologies which guarantee $\Gamma$-convergence as well as convergence of minimizing sequences. [1ex] The research presented in the talk is joint work with Alexey Belenkin and Michael Hartz (Saarland University Saarbr\"ucken).

We present a method for quantitatively estimating the permittivity distribution of an object from remotely measured electromagnetic fields. Our approach compares the fields measured before and after locally injecting plasmonic nanoparticles into the medium. The Hamiltonian generated by these nanoparticles supports subwavelength resonances, located in the lower complex plan but close to the real axis, known as plasmonic resonances. These resonances encode the unknown permittivity values at the nanoparticle locations. We propose an imaging functional that detects these plasmonic resonances, enabling the reconstruction of the permittivity distribution.

Understanding the melting process of a constructed Electric Arc Furnace is crucial in today´s steel industry. To obtain that understanding, we utilize a controlled ODE system capable of simulating said process. Unknown parameters of the underlying furnace, that serve as input data to the ODE system, are needed to accurately simulate the melting process. Due to the complexity of the system, Inverse Parameter Estimation is not feasible. We therefore train a Neural Operator on simulation data which will then serve as a surrogate of the Forward Operator and thus make Parameter Estimation feasible.

The Jordan-Moore-Gibson-Thompson JMGT equation, a third order in time quasilinear PDE, is an advanced model in nonlinear acoustics that -- as opposed to classical second order in time models such as the Kuznetsov and the Westervelt equation takes into account finite speed of propagation. In this talk we aim to highlight the fact that this allows to prove either local or linearized uniqueness of space dependent coefficients as relevant for imaging, such as nonlinearity parameter, sound speed, and attenuation parameter, from measurement of a single time trace of the acoustic pressure on the boundary. Simultaneous identification of pairs of these coefficients only requires two such measurements, provided the excitation is chosen appropriately. This is a setting relevant to several ultrasound based tomography methods. Our approach relies on the Inverse Function Theorem, which requires to prove that the forward operator is a differentiable isomorphism in appropriately chosen topologies. In case of a harmonic excitation, we can use a multiharmonic expansion to work in frequency domain and illustrate the multiplication of information by nonlinearity.

In computerized tomography (CT) the measurement process can be modeled using the Radon transform, which maps the unknown material density to the corresponding absorption loss. In Nano-CT the scale is so small that vibrations of the measuring apparatus lead to unwanted rigid movement of the scanned object. Ignoring this and just building the linear Radon operator with the assumed motion leads to artifacts in the reconstruction. The true motion is not known, i.e. we can not construct the correct linear Radon operator and have to determine the motion parameters. Furthermore we are interested X-ray phase contrast. This can also be modeled using the Radon transform, but the phase information can not be measured and needs to be computed first. This phase retrieval leads to artifacts of its own, partially due to fluctuations in the illumination. In conclusion, the input data is misaligned and has background artifacts. Both need to be addressed for sufficient image quality. The currently used re-projection alignment algorithm uses re-projected filtered back projections and image registration to reconstruct shift and object. We improve on the algorithm by using a thresholded version of normalized cross correlation for the image registration and imposing additional constraints, specifically a non-negativity constraint on the object, smoothness on movement and taking the uncertainty in low frequencies due background data artifacts into account. We illustrate the algorithm on measurements of nano-porous glass. The data was recorded at the Göttinger Instrument for Nano-Imaging with X-rays (GINIX) operated by the Salditt group (University of Göttingen) located at the P10 beamline at the PETRA III storage ring at DESY in Hamburg.

In this talk, we consider a two-step iteratively regularised Landweber algorithm that incorporates two different priors. The model-driven step performs a gradient descent update of the current iterate, starting from an initial guess. The data-driven step integrates prior information by regularising the solution toward a regularisation point. We prove the convergence and stability of the method and demonstrate its effectiveness through numerical examples. This is joint work with Otmar Scherzer.

Asymptotic expansions of the solution to an elliptic PDE in the presence of inclusions of small size have found successful applications in inverse problems, as a means to detect inhomogeneities from boundary measurements in a robust way. In this talk, instead of perturbations in the bulk, we consider perturbations of the boundary conditions. For instance a homogeneous Neumann condition may be replaced by a homogeneous Dirichlet condition on a `small' set $\omega_e$, or vice-versa. We characterize the first term in the asymptotic expansion of the solution, in terms of the relevant measure of smallness of $\omega_\varepsilon$, and give explicit examples when $\omega_\varepsilon$ is a small surfacic ball in ${\mathbb R}^d$, $d=2,3$. We also investigate the case when the perturbation is of a Robin type. This is joint work with Charles Dapogny, Roman Moskalenko and Michael Vogelius.

We consider the Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level, so it is purely data-driven. According to a famous veto, convergence in the worst-case scenario cannot be expected in general. However, by imposing certain conditions on the noisy data, we establish a new convergence result which, in addition, requires neither the Gâteaux differentiability of the forward operator nor the reflexivity of the image space. Therefore, we also expand the applied range of the Landweber iteration to cover non-smooth ill-posed inverse problems and to handle the situation that the data is contaminated by various types of noise. Numerical simulations are also reported. We also discuss future directions in heuristic rules.