Martin Burger

Martin Burger
Friedrich-Alexander-University of Erlangen-Nürnberg | FAU · Department of Mathematics

Dr.

About

359
Publications
55,249
Reads
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9,710
Citations
Additional affiliations
October 2018 - present
Friedrich-Alexander-University of Erlangen-Nürnberg
Position
  • Professor
October 2006 - September 2018
University of Münster
Position
  • Professor
Education
December 1998 - June 2000
Johannes Kepler University Linz
Field of study
  • Mathematics

Publications

Publications (359)
Article
Full-text available
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...
Preprint
Full-text available
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 in...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Article
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...
Preprint
Full-text available
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...
Article
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...
Preprint
Full-text available
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...
Chapter
Full-text available
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...
Article
Full-text available
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...
Conference Paper
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Conference Paper
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Article
Full-text available
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...
Preprint
Full-text available
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...
Article
Full-text available
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...
Article
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...
Article
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Article
Full-text available
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...
Article
Full-text available
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...
Preprint
Full-text available
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...
Article
Full-text available
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...
Article
Full-text available
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...
Article
Full-text available
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...
Preprint
Full-text available
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...
Article
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
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...
Article
Full-text available
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...
Preprint
Full-text available
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...
Preprint
Full-text available
The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller-Segel model. The model we study has the same entropy as the minimal Keller-Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the m...
Article
Full-text available
The orientation of microtubule networks is exploited by motors to deliver cargoes to specific intracellular destinations, and is thus essential for cell polarity and function. Reconstituted in vitro systems have largely contributed to understanding the molecular framework regulating the behavior of microtubule filaments. In cells however, microtubu...
Article
Full-text available
The aim of this paper is to further develop mathematical models for bleb formation in cells, including cell membrane interactions with linker proteins. This leads to nonlinear reaction–diffusion equations on a surface coupled to fluid dynamics in the bulk. We provide a detailed mathematical analysis and investigate some singular limits of the model...
Article
Controlling large particle systems in collective dynamics by a few agents is a subject of high practical importance, e.g., in evacuation dynamics. In this paper we study an instantaneous control approach to steer an interacting particle system into a certain spatial region by repulsive forces from a few external agents, which might be interpreted a...
Preprint
Full-text available
In this work we analyse the functional ${\cal J}(u)=\|\nabla u\|_\infty$ defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis is performed directly on the functional without the need to approximate with smooth $p$-norms. We prove that its ground states coincide with multiples of the distance function to the bo...
Article
Full-text available
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...
Preprint
Full-text available
We introduce a first order Total Variation type regulariser that decomposes a function into a part with a given Lipschitz constant (which is also allowed to vary spatially) and a jump part. The kernel of this regulariser contains all functions whose Lipschitz constant does not exceed a given value, hence by locally adjusting this value one can dete...
Article
Full-text available
The aim of this paper is to investigate the use of a Landweber-type method involving the Shannon entropy for the regularization of linear ill-posed problems. We derive a closed form solution for the iterates and analyze their convergence behaviour both in a case of reconstructing general nonnegative unknowns as well as for the sake of recovering pr...
Preprint
Full-text available
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 ($p \geq 2$). We give upper bounds for the finite extinction time and establish convergence rates of the flow. Moreover, we study next order asymptotics and prove that...
Preprint
Full-text available
The aim of this paper is to investigate the use of a Landweber-type method involving the Shannon entropy for the regularization of linear ill-posed problems. We derive a closed form solution for the iterates and analyze their convergence behaviour both in a case of reconstructing general nonnegative unknowns as well as for the sake of recovering pr...
Preprint
Full-text available
In this paper we analyse a class of nonlinear cross-diffusion systems for two species with local repulsive interactions that exhibit a formal gradient flow structure with respect to the Wasserstein metric. We show that systems where the population pressure is given by a function of the total population are critical with respect to cross-diffusion p...
Preprint
Full-text available
In this paper we analyse a class of nonlinear cross-diffusion systems for two species with local repulsive interactions that exhibit a formal gradient flow structure with respect to the Wasserstein metric. We show that systems where the population pressure is given by a function of the total population are critical with respect to cross-diffusion p...
Chapter
Full-text available
In this work we investigate the computation of nonlinear eigenfunctions via the extinction profiles of gradient flows. We analyze a scheme that recursively subtracts such eigenfunctions from given data and show that this procedure yields a decomposition of the data into eigenfunctions in some cases as the 1-dimensional total variation, for instance...
Chapter
Full-text available
We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections to total variation and infimal convolution type regularizers \({{\,\mathrm{{{\,\mathrm{TVL}\,}}^p}\,}}\) and,...
Article
Full-text available
We introduce a novel regulariser based on the natural vector field operations gradient, divergence, curl and shear. For suitable choices of the weighting parameters contained in our model it generalises well-known first- and second-order TV-type regularisation methods including TV, ICTV and TGV2 and enables interpolation between them. To better und...
Preprint
Full-text available
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 et al., see \cite{Puel2002}, and results in a optimal control problem. We analyse this problems and provide stability estimates for t...
Poster
Full-text available