Martin Burger

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• Dr.
• Leading Scientist at Deutsches Elektronen-Synchrotron

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a...

We propose a learning framework based on stochastic Bregman iterations to train sparse neural networks with an inverse scale space approach. We derive a baseline algorithm called LinBreg, an accelerated version using momentum, and AdaBreg, which is a Bregmanized generalization of the Adam algorithm. In contrast to established methods for sparse tra...

In this paper we provide a novel approach to the analysis of kinetic models for label switching, which are used for particle systems that can randomly switch between gradient flows in different energy landscapes. Besides problems in biology and physics, we also demonstrate that stochastic gradient descent, the most popular technique in machine lear...

This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and further discuss the derivation of quantitative estimates respectively needed ingredients such as Bregman distances...

Variational regularization has remained one of the most successful approaches for reconstruction in imaging inverse problems for several decades. With the emergence and astonishing success of deep learning in recent years, a considerable amount of research has gone into data-driven modeling of the regularizer in the variational setting. Our work ex...

Optimal transport tasks naturally arise in gas networks, which include a variety of constraints such as physical plausibility of the transport and the avoidance of extreme pressure fluctuations. To define feasible optimal transport plans, we utilize a $p$ -Wasserstein metric and similar dynamic formulations minimizing the kinetic energy necessary f...

This paper presents a method for finding a sparse representation of Barron functions. Specifically, given an L2 function f, the inverse scale space flow is used to find a sparse measure μ minimising the L2 loss between the Barron function associated to the measure μ and the function f. The convergence properties of this method are analysed in an id...

The aim of this paper is to provide a mathematical analysis of transformer architectures using a self-attention mechanism with layer normalization. In particular, observed patterns in such architectures resembling either clusters or uniform distributions pose a number of challenging mathematical questions. We focus on a special case that admits a g...

We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimizat...

Multiplicative Gamma noise remove is a critical research area in the application of synthetic aperture radar (SAR) imaging, where neural networks serve as a potent tool. However, real-world data often diverges from theoretical models, exhibiting various disturbances, which makes the neural network less effective. Adversarial attacks work by finding...

Hypergraph learning with $p$-Laplacian regularization has attracted a lot of attention due to its flexibility in modeling higher-order relationships in data. This paper focuses on its fast numerical implementation, which is challenging due to the non-differentiability of the objective function and the non-uniqueness of the minimizer. We derive a hy...

We analyze various formulations of the $\infty$-Laplacian eigenvalue problem on graphs, comparing their properties and highlighting their respective advantages and limitations. First, we investigate the graph $\infty$-eigenpairs arising as limits of $p$-Laplacian eigenpairs, extending key results from the continuous setting to the discrete domain....

We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typica...

We present the analysis of the stationary equilibria and their stability in case of an opinion formation process in presence of binary opposite opinions evolving according to majority-like rules on social networks. The starting point is a kinetic Boltzmann-type model derived from microscopic interactions rules for the opinion exchange among individ...

A popular method to perform adversarial attacks on neuronal networks is the so-called fast gradient sign method and its iterative variant. In this paper, we interpret this method as an explicit Euler discretization of a differential inclusion, where we also show convergence of the discretization to the associated gradient flow. To do so, we conside...

In this paper, we analyze a recently proposed algorithm for the problem of sampling from probability distributions $\mu^\ast$ in $\mathbb{R}^d$ with a Lebesgue density and potential of the form $f(Kx)+g(x)$, where $K$ is a linear operator and $f$, $g$ are convex and non-smooth. The algorithm is a generalization of the primal-dual hybrid gradient (P...

As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. This paper explores the benefit of the hypergraph structure for the interpolation of point cloud data that contain no explicit structural information. We define the εn-ball hypergraph and the kn-nearest neighbor hypergraph on a point cloud and study...

Optimal transport tasks naturally arise in gas networks, which include a variety of constraints such as physical plausibility of the transport and the avoidance of extreme pressure fluctuations. To define feasible optimal transport plans, we utilize a p-Wasserstein metric and similar dynamic formulations minimizing the kinetic energy necessary for...

This paper studies the p-biharmonic equation on graphs, which arises in point cloud processing and can be interpreted as a natural extension of the graph p-Laplacian from the perspective of hypergraph. The asymptotic behavior of the solution is investigated when the random geometric graph is considered and the number of data points goes to infinity...

The aim of this paper is to revisit the definition of differential operators on hypergraphs, which are a natural extension of graphs in systems based on interactions beyond pairs. In particular, we focus on the definition of Laplacian and p-Laplace operators for oriented and unoriented hypergraphs, their basic properties, variational structure, and...

The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (naïve) solution does not depend on the measured data continuously, regularization is needed to reestablish a continuous dependence. In this wor...

We present a comprehensive analysis of total variation (TV) on non-Euclidean domains and its eigenfunctions. We specifically address parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work sheds new light on the celebrated Beltrami and Anisotropic TV flows, and explains experimental findings from recent years on...

We investigate an interacting particle model to simulate a foraging colony of ants, where each ant is represented as an active Brownian particle. The interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of the pheromone field. Unlike conventional models, our study introduces a parameter that...

This workshop brought together researchers working on mathematical problems related to tomography, with a particular emphasis on novel applications and associated mathematical challenges. Examples of respective issues represented in the workshop were tomographic imaging with Compton cameras or coupled-physics imaging, resolution and aliasing, vecto...

The aim of this paper is to provide a theoretically founded investigation of state-of-the-art learning approaches for inverse problems. We give an extended definition of regularization methods and their convergence in terms of the underlying data distributions, which paves the way for future theoretical studies. Based on a simple spectral learning...

This paper presents a method for finding a sparse representation of Barron functions. Specifically, given an L 2 function f , the inverse scale space flow is used to find a sparse measure µ minimising the L 2 loss between the Barron function associated to the measure µ and the function f. The convergence properties of this method are analysed in an...

We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typica...

We prove the existence of solutions to a non-linear, non-local, degenerate equation which was previously derived as the formal hydrodynamic limit of an active Brownian particle system, where the particles are endowed with a position and an orientation. This equation incorporates diffusion in both the spatial and angular coordinates, as well as a no...

This paper focuses on the application of Compton cameras in nuclear decommissioning, aiming to develop efficient and affordable radiological characterization methods. Compton cameras offer advantages over traditional Anger cameras, including increased sensitivity and a larger field of view. We showcase the use of monolithic scintillation detectors...

We investigate the convergence of solutions of a recently proposed diffuse interface/phase field model for cell blebbing by means of matched asymptotic expansions. It is a biological phenomenon that increasingly attracts attention by both experimental and theoretical communities. Key to understanding the process of cell blebbing mechanically are pr...

We investigate the use of Fourier Neural Operators (FNOs) for image classification in comparison to standard Convolutional Neural Networks (CNNs). Neural operators are a discretization-invariant generalization of neural networks to approximate operators between infinite dimensional function spaces. FNOs---which are neural operators with a specific...

The aim of this paper is to revisit the definition of differential operators on hypergraphs, which are a natural extension of graphs in systems based on interactions beyond pairs. In particular we focus on the definition of Laplacian and p-Laplace operators, their basic spectral properties, variational structure, and their scale spaces.We shall see...

In this paper we investigate the use of Fourier Neural Operators (FNOs) for image classification in comparison to standard Convolutional Neural Networks (CNNs). Neural operators are a discretization-invariant generalization of neural networks to approximate operators between infinite dimensional function spaces. FNOs—which are neural operators with...

We study a dynamic optimal transport problem on a network. Despite the cost for transport along the edges, an additional cost, scaled with a parameter $\kappa$, has to be paid for interchanging mass between edges and vertices. We show existence of minimisers using duality and discuss the relationship of the model to other metrics such as Fisher-Rao...

In this paper we investigate the use of Fourier Neural Operators (FNOs) for image classification in comparison to standard Convolutional Neural Networks (CNNs). Neural operators are a discretization-invariant generalization of neural networks to approximate operators between infinite dimensional function spaces. FNOs - which are neural operators wi...

We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimizat...

We investigate the convergence of solutions of a recently proposed diffuse interface/phase field model for cell blebbing by means of matched asymptotic expansions. It is a biological phenomenon that increasingly attracts attention by both experimental and theoretical communities. Key to understanding the process of cell blebbing mechanically are pr...

The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (naive) solution does not depend on the measured data continuously, regularization is needed to re-establish a continuous dependence. In this wo...

We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion system. The purpose of this article is twofold: first, it aims at a systematic study of the question of existence of weak solutions and their long-time asymptotic behaviour. Second, it pr...

We propose a learning framework based on stochastic Bregman iterations, also known as mirror descent, to train sparse neural networks with an inverse scale space approach. In contrast to established methods for sparse training the proposed family of algorithms constitutes a regrowth strategy for neural networks that is solely optimization-based wit...

In this work we present a comprehensive analysis of total variation (TV) on non Euclidean domains and its eigenfunctions. We specifically address parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work sheds new light on the celebrated Beltrami and Anisotropic TV flows, and explains experimental findings from re...

In this paper we investigate a generalisation of a Boltzmann mean field game (BMFG) for knowledge growth, originally introduced by the economists Lucas and Moll [23]. In BMFG the evolution of the agent density with respect to their knowledge level is described by a Boltzmann equation. Agents increase their knowledge through binary interactions with...

We propose a learning framework based on stochastic Bregman iterations, also known as mirror descent, to train sparse neural networks with an inverse scale space approach. We derive a baseline algorithm called LinBreg, an accelerated version using momentum, and AdaBreg, which is a Bregmanized generalization of the Adam algorithm. In contrast to est...

The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between...

Goal oriented autonomous operation of space rovers has been known to increase scientific output of a mission. In this work we present an algorithm, called the RoI Prioritised Sampling (RPS), that prioritises Region-of-Interests (RoIs) in an exploration scenario in order to utilise the limited resources of the imaging instrument on the rover effecti...

This paper studies the convergence of solutions of a nonlocal interaction equation to the solution of the quadratic porous medium equation in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use the gradient flow structure of the equations to derive bounds...

This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how these can be approximated using Γ-convergence. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to nor...

This work is devoted to study a class of singular perturbed elliptic systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behavior of limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain...

This chapter focuses on the mathematical modelling of active particles (or agents) in crowded environments. We discuss several microscopic models found in the literature and the derivation of the respective macroscopic partial differential equations for the particle density. The macroscopic models share common features, such as cross-diffusion or d...

This paper discusses basic results and recent developments on variational regular-ization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and further discuss the derivation of quantitative estimates respectively needed ingredients such as Bregman distances...

We propose a general strategy for solving nonlinear integro-differential evolution problems with periodic boundary conditions, where no direct maximum/minimum principle is available. This is motivated by the study of recent macroscopic models for active Brownian particles with repulsive interactions, consisting of advection-diffusion processes in t...

The aim of this paper is to discuss the mathematical modelling of Brownian active particle systems, a recently popular paradigmatic system for self-propelled particles. We present four microscopic models with different types of repulsive interactions between particles and their associated macroscopic models, which are formally obtained using differ...

We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion system. The purpose of this article is twofold: first, it aims at a systematic study of the question of existence of weak solutions and their long-time asymptotic behaviour. Second, it pr...

The vulnerability of deep neural networks to small and even imperceptible perturbations has become a central topic in deep learning research. Although several sophisticated defense mechanisms have been introduced, most were later shown to be ineffective. However, a reliable evaluation of model robustness is mandatory for deployment in safety-critic...

Goal oriented autonomous operation of space rovers has been known to increase scientific output of a mission. In this work we present an algorithm, called the RoI Prioritised Sampling (RPS), that prioritises Region-of-Interests (RoIs) in an exploration scenario in order to utilise the limited resources of the imaging instrument on the rover effecti...

The aim of this paper is to develop suitable models for the phenomenon of cell blebbing, which allow for computational predictions of mechanical effects including the crucial interaction of the cell membrane and the actin cortex. For this sake we resort to a two phase-field model that uses diffuse descriptions of both the membrane and the cortex, w...

This chapter focuses on the mathematical modelling of active particles (or agents) in crowded environments. We discuss several microscopic models found in literature and the derivation of the respective macroscopic partial differential equations for the particle density. The macroscopic models share common features, such as cross diffusion or degen...

This work applies Bayesian experimental design to selecting optimal projection geometries in (discretized) parallel beam x-ray tomography assuming the prior and the additive noise are Gaussian. The introduced greedy exhaustive optimization algorithm proceeds sequentially, with the posterior distribution corresponding to the previous projections ser...

We propose a novel strategy for Neural Architecture Search (NAS) based on Bregman iterations. Starting from a sparse neural network our gradient-based one-shot algorithm gradually adds relevant parameters in an inverse scale space manner. This allows the network to choose the best architecture in the search space which makes it well-designed for a...

The aim of this paper is to study the derivation of appropriate meso- and macroscopic models for interactions as appearing in social processes. There are two main characteristics the models take into account, namely a network structure of interactions, which we treat by an appropriate mesoscopic description, and a different role of interacting agen...

This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how convergence of (discretized) approximations can be verified. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all...

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by works for the total variation, where interesting results on the eigenvalue problem and the relation to the total v...

The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between...

We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian appr...

The susceptibility of deep neural networks to untrustworthy predictions, including out-of-distribution (OOD) data and adversarial examples, still prevent their widespread use in safety-critical applications. Most existing methods either require a re-training of a given model to achieve robust identification of adversarial attacks or are limited to...

We consider a statistical inverse learning problem, where the task is to estimate a function $f$ based on noisy point evaluations of $Af$, where $A$ is a linear operator. The function $Af$ is evaluated at i.i.d. random design points $u_n$, $n=1,...,N$ generated by an unknown general probability distribution. We consider Tikhonov regularization with...

The vulnerability of deep neural networks to small and even imperceptible perturbations has become a central topic in deep learning research. Although several sophisticated defense mechanisms have been introduced, most were later shown to be ineffective. However, a reliable evaluation of model robustness is mandatory for deployment in safety-critic...

In this paper we study a dynamical optimal transport problem on a network that allows for transport of mass between different edges if a penalty $\kappa$ is paid. We show existence of minimisers using duality and discuss the relationships of the distance-functional to other metrics such as the Fisher-Rao and the classical Wasserstein metric and ana...

In this paper, we derive and analyse mean-field models for the dynamics of groups of individuals undergoing a random walk. The random motion of individuals is only influenced by the perceived densities of the different groups present as well as the available space. All individuals have the tendency to stay within their own group and avoid the other...

The biophysical and biochemical properties of live tissues are important in the context of development and disease. Methods for evaluating these properties typically involve destroying the tissue or require specialized technology and complicated analyses. Here, we present a novel, noninvasive methodology for determining the spatial distribution of...

We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and convex duality for general data fidelity terms and regularisation functionals. Both for a priori and a posterior...

This paper is devoted to the investigation of inverse problems related to stationary drift-diffusion equations modeling semiconductor devices. In this context we analyze several identification problems corresponding to different types of measurements, where the parameter to be reconstructed is an inhomogeneity in the PDE model (doping profile). For...

The aim of this paper is to investigate superresolution in deconvolution driven by sparsity priors. The observed signal is a convolution of an original signal with a continuous kernel. With the prior knowledge that the original signal can be considered as a sparse combination of Dirac delta peaks, we seek to estimate the positions and amplitudes of...

The propagation of gradient flow structures from microscopic to macroscopic models is a topic of high current interest. In this paper, we discuss this propagation in a model for the diffusion of particles interacting via hard-core exclusion or short-range repulsive potentials. We formulate the microscopic model as a high-dimensional gradient flow i...

We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian appr...

This work is concerned with the gradient flow of absolutely p-homogeneous convex functionals on a Hilbert space, which we show to exhibit finite (p<2) or infinite extinction time (p≥2). We give upper bounds for the finite extinction time and establish sharp convergence rates of the flow. Moreover, we study next order asymptotics and prove that asym...

The aim of this paper is to investigate superresolution in deconvolution driven by sparsity priors. The observed signal is a convolution of an original signal with a continuous kernel.With the prior knowledge that the original signal can be considered as a sparse combination of Dirac delta peaks, we seek to estimate the positions and amplitudes of...

In this paper, we extend the results of [8] by proving exponential asymptotic H^1 -convergence of solutions to a one-dimensional singular heat equation with L^2 -source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a t...

The aim of this paper is to study the derivation of appropriate meso-and macroscopic models for interactions as appearing in social processes. There are two main characteristics the models take into account, namely a network structure of interactions, which we treat by an appropriate mesoscopic description, and a different role of interacting agent...

This work applies Bayesian experimental design to selecting optimal projection geometries in (discretized) parallel beam X-ray tomography assuming the prior and the additive noise are Gaussian. The introduced greedy exhaustive optimization algorithm proceeds sequentially, with the posterior distribution corresponding to the previous projections ser...

Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and there exist very few results in this direction. In this work, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in Ehrlacher and Bakhta (ESAIM Math Model N...

We consider the problem of estimating the density of buyers and vendors in a nonlinear parabolic price formation model using measurements of the price and the transaction rate. Our approach is based on a work by Puel (Puel J-P 2002 C. R. Acad. Sci., Paris 335 (2) 161–166), and results in an optimal control problem. We analyze this problems and prov...