In the last few years, following the work of Michaël Unser and collaborators, the inverse problem community has taken a keen interest in representing the solutions of variational problems.
That kind of representation results, which rely on convex analysis, make it possible to derive some theoretical properties of minimizers and to design efficient numerical methods.
In this talk, I will give...
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...
Elastography, as an imaging modality in general, aims at mapping the mechanical properties of a given material sample. For estimating the values of stiffness and strain quantitatively, we look at Elastography from the perspective of Inverse Problems. In particular, we start with theoretical ideas on how to perform Elastography and continue all the way to implementing Optical Coherence...
Standard regularization methods for inverse problems with non-injective forward operators typically introduce bias by favoring components orthogonal to the null space of the forward map. This often poses challenges in source recovery tasks, particularly in inverse source problems. To address this, we generalize previous work that mitigated the issue using the pseudo-inverse $A^\dagger$, by now...
In many application areas (mechanics, geophysics, image processing,...), a large number of real-life problems can be cast as inverse problems. This is the case in the field of energy, where key problems related to engineering, maintenance and management of power systems like, for example, the non-destructive evaluation of some components of nuclear power plants, are inverse problems. Solving...
Recent advancements in photon induced near-field electron microscopy (PINEM) enable the preparation, coherent manipulation and characterization of free-electron quantum states. The available measurement consists of electron energy spectrograms and the goal is to reconstruct a density matrix which represents the quantum state. This requires the solution of an ill-posed inverse problem, where a...
Tomographic inverse problems remain a cornerstone of medical investigations, allowing the visualization of patients' interior features. While the infinite-dimensional operators modeling the measurement process (e.g., the Radon transform) are well understood, in practice one can only observe finitely many measurements and employ finitely many computations in reconstruction. Thus, proper...
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...
Quantitative dynamic positron emission tomography (PET) attempts to reconstruct kinetic tissue parameters based on a time series of images showing the concentration of the PET tracer over time. The underlying problem is a nonlinear parameter identification problem which is based on an estimation of the arterial input function obtained by costly and time-consuming blood sample analysis. Our...
Atmospheric tomography, the problem of reconstructing atmospheric turbulence profiles from wavefront sensor measurements, is an integral part of
many adaptive optics systems. It is used to enhance the image quality of
ground-based telescopes, such as for the Multiconjugate Adaptive Optics Relay For ELT Observations (MORFEO) instrument on the Extremely Large
Telescope (ELT).
Singular-value...
Sparse data tomography is a challenging testing ground for several theoretical and numerical studies, for which both variational regularization and data-driven techniques have been investigated. In this talk, I will present hybrid reconstruction frameworks that combine model-based regularization with data-driven approaches by relying on the interplay between sparse regularization theory,...
Ptychography is a type of coherent diffraction imaging which uses a strongly coherent X-ray source from a synchrotron to reconstruct high resolution images. By shifting the illumination source, it exploits redundancy of multiple diffraction patterns to robustly solve the related phase retrieval problem. The shift parameters are often subject to uncertainty and introduce additional...
Physics-informed neural networks (PINNs) have emerged as a powerful tool in the scientific machine learning community, with applications to both forward and inverse problems. While they have shown considerable empirical success, significant challenges remain—particularly regarding training stability and the lack of rigorous theoretical guarantees, especially when compared to classical...
Advanced adaptive optics (AO) instruments use Fourier-type wavefront sensors (WFSs) to measure and correct wavefront distortions caused by the Earth's atmosphere. Conventionally, the wavefront reconstruction relies on matrix-vector-multiplications (MVMs). However, these linear estimators assume small wavefront aberrations and may fail to capture the nonlinear behavior of Fourier-type wavefront...
In this talk we study the minimization of convex, $L$-smooth functions defined on a separable real Hilbert space. We analyze regularized stochastic gradient descent (reg-SGD), a variant of stochastic gradient descent that uses a Tikhonov regularization with time-dependent, vanishing regularization parameter. We prove strong convergence of reg-SGD to the minimum-norm solution of the original...
The Calderón problem consists in recovering an unknown coefficient of a partial differential equation from boundary measurements of its solution. These measurements give rise to a highly nonlinear forward
operator. As a consequence, the development of reconstruction methods for this inverse problem is challenging, as they usually suffer from the problem of local convergence. To circumvent...
We introduce two reconstruction schemes that enable the recovery of a function in the entire Euclidean space $\mathbb{R}^n$ from local data $(u|_W, (-\Delta)^s u|_W)$, where $W$ is an arbitrarily small nonempty open set. These procedures rely crucially on the weak UCP for the fractional Laplacian.
We apply these schemes to two distinct inverse problems. Following the seminal work from...
In inverse problem research, both the mathematical foundations and the development of implementation technologies for industrial applications play essential roles. This presentation highlights two specific applications among several industrial cases, including inspection of wind turbine blades, quality control of smartphone cameras, maintenance of marine structures, and state estimation in...
This study focuses on estimating psychological parameters that influence decision-making, based on virtually collected consumer purchase data, and constructs a consumer purchasing behavior model grounded in prospect theory. A virtual shopping environment was developed in which consumers were presented with various combinations of the same product differing in expiration dates and prices,...
