The tangential cone conditions (TCCs) are sufficient conditions on a nonlinear forward operator for proving convergence of various iterative nonlinear regularization schemes such as Landweber iteration. Especially for parameter identification problems with boundary data, they have not been verified yet, even though numerical results for nonlinear iterative regularization method
usually show...
Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of...
We deal with the shape reconstruction of inclusions in elastic bodies and solve the inverse problem by means of a monotonicity-based regularization. In more detail, we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. In doing so, we introduce constraints on the...
We consider a holistic approach to find a closed formula for the generalized ray transform of a tensor field. This means that we take refraction, attenuation and time-dependence into account. We model the refraction by an appropriate Riemannian metric which leads to an integration along geodesics. The absorption appears as an attenuation coefficient in an exponential factor. The derived...
We study an inverse magnetization problem arising in geo- and planetary magnetism. This problem is non-unique and the null space can be characterized by the Hardy-Hodge decomposition. The additional assumption that the underlying magnetization is spatially localized in a subdomain of the sphere (which can be justified when interested, e.g., in regional magnetic anomalies) ameliorates the...
We review the Bayesian approach to inverse problems, and describe recent progress in our theoretical understanding of its performance in non-linear situations. Statistical and computational guarantees for such algorithms will be provided in high-dimensional, non-convex scenarios, and model examples from elliptic and transport (X-ray type) PDE problems will be discussed. The connection between...
We consider a linear inverse problem of the form $y=Ax+\epsilon \dot W$ where the action of the operator (matrix) $A$ on the unknown $x$ is corrupted by white noise (a standard Gaussian vector) $\dot W$ of level $\epsilon>0$. We study the candidate solutions $\hat x_m$ provided by the $m$-th conjugate gradient CGNE iterates. Refining Nemirovskii's trick, we are able to provide explicit error...
I will present a novel learning framework based on stochastic Bregman iterations. It allows to train sparse neural networks with an inverse scale space approach, starting from a very sparse network and gradually adding significant parameters. Apart from a baseline algorithm called LinBreg, I will also speak about an accelerated version using momentum, and AdaBreg, which is a Bregmanized...
With smart energy meters increasingly available to private households, new applications arise, such as identifying main power consuming devices and predicting human activity. One major obstacle is that smart energy meters typically provide aggregated data, where each source of energy consumption is summed. Further, obtaining training data can be intrusive. To counteract this, we propose an...
We analyze the use of the so-called general regularization scheme in the scenario of unsupervised domain adaptation under the covariate shift assumption. Learning algorithms arising from the above scheme are generalizations of importance weighted regularized least squares method, which up to now is among the most used approaches in the covariate shift setting. We explore a link between the...
In recent years deep/machine learning methods using convolutional networks have become increas- ingly popular also in inverse problems mainly due to their practical performance [1]. In many cases these methods outperform conventional regularization methods, such as total variation regulariza- tion, in particular when applied to more complicated data such as images containing texture. A major...
In this talk, we propose and analyze a generalized conditional gradient method for infinite dimensional variational inverse problems written as the sum of a smooth, convex loss function and a, possibly non-smooth, convex regularizer.
Our method relies on the mutual update of a sequence of extremal points of the unit ball of the regularizer and a sparse iterate given as a suitable linear...
In this talk I will discuss uniqueness, stability and reconstruction for infinite-dimensional nonlinear inverse problems with finite measurements, under the a priori assumption that the unknown lies in, or is well-approximated by, a finite-dimensional subspace or submanifold. The methods are based on the interplay of applied harmonic analysis, in particular sampling theory and compressed...
We present two families of regularization method for solving nonlinear ill-posed problems between Hilbert spaces by applying the family of Runge–Kutta methods to an initial value problem, in particular, to the asymptotical regularization method.
In Hohage [1], a systematic study of convergence rates for regularization methods under logarithmic source condition including the case of operator...
The singular-value decomposition (SVD) is an important tool for the analysis and solution of linear ill-posed problems in Hilbert spaces. However, it is often difficult to derive the SVD of a given operator explicitly, which limits its practical usefulness. An alternative in these situations are frame decompositions (FDs), which are a generalization of the SVD based on suitably connected...
We show convergence rates results for Banach space regularization in the case of oversmoothing, i.e. if the penalty term fails to be finite at the unknown solution. We present a flexible approach based on K-interpolation theory which provides more general and complete results than classical variational regularization theory based on various types of source conditions for true solutions...
We consider the dynamic Positron Emission Tomography (PET) reconstruction method proposed by Schmitzer et al. $[1]$ that particularly aims to reconstruct the temporal evolution of single or small numbers of cells by leveraging optimal transport. Using a MAP estimate the cells' evolution is reconstructed by minimizing a functional $\mathcal{E}_n$ - composed of a Kulback-Leibler-type data...
In this talk we study the discretization of a well-posed nonlinear problem.
It may happen that discretized solutions do not converge. However, this effect disappears for a suitable chosen optimal control problem.
We consider an inverse source problem in the stationary radiating transport through a two dimensional absorbing and scattering medium. The attenuation and scattering properties of the medium are assumed known and the unknown vector field source is isotropic. For scattering kernels of finite Fourier content in the angular variable, we show how to recover the isotropic vector field sources from...
Constrained optimization problems represent a challenge when the objective function is non-differentiable, multimodal and the feasible region lacks regularity. In our talk, we will introduce a swarm-based optimization algorithm which is capable of handling generic non-convex constraints by means of a penalization technique. The method extends the class of consensus-based optimization (CBO)...
Reliable and fast medical imaging techniques are indispensable for diagnostics in clinical everyday life. A promising example of those is given by magnetic particle imaging (MPI) invented by Gleich and Weizenecker [1]. MPI is a tracer-based imaging method allowing for the reconstruction of the spatial distribution of magnetic nanoparticles via exploiting their non-linear magnetization response...
Diseases like cancer or arteriosclerosis often cause changes of tissue stiffness on the micrometer scale. Elastography is a common technique for medical diagnostics developed to detect these changes. We consider a complex problem of estimating both the internal displacement field and the material parameters of an object which is being subjected to a deformation. In particular, we present our...
We consider a class of nonlinear inverse problems, encompassing e.g. Polarimetric Neutron Tomography (PNT), where one seeks to recover a magnetic field by probing it with Neutron beams and measuring the resulting spin change. In recent years there has been great progress on fundamental theoretical questions regarding injectivity and stability properties for PNT and we survey some of the latest...
We deal with an inverse elastic scattering problem for the shape determination of a rigid scatterer in the time-harmonic regime. We prove a local stability estimate of log log type for the identification of a scatterer by a single far-field measurement.
The needed a priori condition on the closeness of the scatterers is estimated by the universal constant appearing in the Friedrichs...
We consider the statistical nonlinear inverse problem of recovering the absorption term f > 0 in the heat equation, with given boundary and initial value functions, from N discrete noisy point evaluations of the solution u_f. We study the statistical performance of Bayesian nonparametric procedures based on Gaussian process priors, that are often used in practice. We show that, as the number...
We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The high-dimensionality of the data space implies extremely high computational cost already for the evaluation of the forward operator, which makes a numerical solution of the inverse problem, e.g., by iterative...
Stochastic gradient descent (SGD) is a promising method for solving large-scale inverse problems, due to its excellent scalability with respect to data size. The current mathematical theory in the lens of regularization theory predicts that SGD with a polynomially decaying stepsize schedule may suffer from an undesirable saturation phenomenon, i.e., the convergence rate does not further...
The ensemble Kalman filter (EnKF) is a widely used metheodology for data assimilation problems and has been recently generalized to inverse problems, known as ensemble Kalman inversion (EKI). We view the method as a derivative free optimization method for a least-squares misfit functional and we present various variants of the scheme such as regularized EKI methods. This opens up the...
Human brain activity is based on electrochemical processes, which can only be measured invasively. For this reason, induced quantities such as magnetic flux density (via MEG) or electric potential differences (via EEG) are measured non-invasively in medicine and research. The reconstruction of the neuronal current from the measurements is a severely ill-posed problem though the visualization...
In the last 10 years, the Inverse Problem Matching Pursuits (IPMPs) were proposed as alternative solvers for linear inverse problems on the sphere and the ball, e.g. from the geosciences. They were constantly further developed and tested on diverse applications, e.g. on the downward continuation of the gravitational potential. This task remains a priority in geodesy due to significant...
The Calderon problem, known also as the inverse conductivity problem, regards the determination of the conductivity inside a domain by the knowledge of the boundary data. For the isotropic case, the stability issue is almost solved. However, for the anisotropic case things get more complicated, since Tartar observation that any diffeomorphism of the domain which keeps the boundary points fixed...
Electrical Impedance Tomography gives rise to the severely ill-posed Calder?ón problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The electrical conductivity of an object is of interest in many fields, notably medical imaging, where applications may vary from stroke detection...
The Ensemble Kalman inversion (EKI) is a powerful tool for the solution of Bayesian inverse problems of type $y=Au^\dagger+\varepsilon$, with $u^\dagger$ being an unknown parameter and $y$ a given datum subject to measurement noise $\varepsilon$. It evolves an ensemble of particles, sampled from a prior measure, towards an approximate solution of the inverse problem. In this talk I will...
