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Research Interestsimage processing
scientific computing
numerical analysis
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Mathematics Münster

C: Models and Approximations

C1: Evolution and asymptotics
C2: Multi-scale phenomena and macroscopic structures
C3: Interacting particle systems and phase transitions
C4: Geometry-based modelling, approximation, and reduction
Current PublicationsSassen, Josua; Hildebrandt, Klaus; Rumpf, Martin; Wirth, Benedikt Parametrizing product shape manifolds by composite networks. , 2023 online
Dirks C, Wirth B An adaptive finite element approach for lifted branched transport problems. Optimization and Control for Partial Differential Equations: Uncertainty quantification, open and closed-loop control, and shape optimizationRadon Series on Computational and Applied Mathematics Vol. tba, 2022 online
Effland A, Heeren B, Rumpf M, Wirth B Consistent Curvature Approximation on Riemannian Shape Spaces. IMA Journal of Numerical Analysis Vol. 42 (1), 2022 online
Potthoff Jonas, Wirth Benedikt Optimal fine-scale structures in compliance minimization for a uniaxial load in three space dimensions. ESAIM: Control, Optimisation and Calculus of Variations Vol. 28, 2022 online
Striewski Paul, Wirth Benedikt Elastic 3D-2D Image Registration. Journal of Mathematical Imaging and Vision Vol. 64, 2022 online
Lohmann J, Schmitzer B, Wirth B Formulation of branched transport as geometry optimization. Journal de Mathématiques Pures et Appliquées Vol. 163, 2022 online
Lohmann, Julius; Schmitzer, Bernhard; Wirth, Benedikt Duality in branched transport and urban planning. Applied Mathematics and Optimization Vol. 86, 2022 online
Braunsmann J, Rajkovic M, Rumpf M, Wirth B Learning low bending and low distortion manifold embeddings. 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), 2021 online
Dirks C, Striewski P, Wirth B, Aalto A, Olguin-Olguin A A mathematical model for cell polarization in zebrafish primordial germ cells. IMA Mathematical Medicine and Biology Vol. 38 (2), 2021 online
Current ProjectsCRC 1450 A05 - Targeting immune cell dynamics by longitudinal whole-body imaging and mathematical modelling We develop strategies for tracking and quantifying (immune) cell populations or even single cells in long-term (days) whole-body PET studies in mice and humans. This will be achieved through novel acquisition protocols, measured and simulated phantom data, use of prior information from MRI and microscopy, mathematical modelling, and mathematical analysis of image reconstruction with novel regularization paradigms based on optimal transport. Particular applications include imaging and tracking of macrophages and neutrophils following myocardial ischemia-reperfusion or in arthritis and sepsis. online
CRC 1450 A06 - Improving intravital microscopy of inflammatory cell response by active motion compensation using controlled adaptive optics We will advance multiphoton fluorescence microscopy by developing a novel optical module comprised of a high-speed deformable mirror that will actively compensate tissue motion during intravital imaging, for instance due to heart beat (8 Hz), breathing (3 Hz, in mm-range) or peristaltic movement of the gut in mice. To control this module in real-time, we will develop mathematical methods that track and predict tissue deformation. This will allow imaging of inflammatory processes at cellular resolution without mechanical tissue fixation. online
SPP 2256: Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials - Subproject: Multiscale structure in compliance minimization It is a classic engineering problem to identify which geometry of an elastic material best supports a load while consuming a minimum material amount. Typically, the optimal geometry would involve an infinitely fine microstructure with infinitely many, infinitely fine holes. To avoid dealing with the microstructure, the optimization problem is typically relaxed (meaning that microstructured regions are simply replaced by a non-microstructured material with same macroscopic elastic properties, adapting the optimization problem correspondingly) or strongly regularized (meaning that one adds something like production costs to the optimization problem, which will prevent structures with too many or too fine holes). In contrast, in this project we are interested in the case of very weak regularization, in which fine structures and coarsening phenomena over multiple scales will occur (that is, in some regions the structure will be very fine and in others quite coarse). Our aim is to better understand these structures via variational methods. Thus, rather than modelling and understanding the behaviour or response of an existing material, we here consider a material design problem that results in a complex structure. online
Mathematische Rekonstruktion und Modellierung der CAR T-Zell Verteilung in vivo in einem Tumormodell Krebstherapien, wie Bestrahlung oder Chemotherapie, liefern häufig nur unzureichende Behandlungserfolge, so dass der Bedarf an neuartigen Therapiestrategien groß ist. Immuntherapien verwenden das körpereigene Immunsystem, um die Krebszellen zu erkennen und zu bekämpfen. Dem Patienten werden hierzu Abwehrzellen (T-Zellen) entnommen und diese werden genetisch verändert, sodass sie in der Lage sind, Krebszellen zu erkennen. Die so modifizierten "CAR T-Zellen" werden angereichert und dem Patienten zurückgegeben (transfundiert).

Für T-Zell-Therapien besteht in zweierlei Hinsicht Forschungsbedarf:
  • Spezifizität: Die CAR T-Zellen werden auf bestimmte Erkennungsmerkmale (sogenannte Antigene) der Tumorzellen abgerichtet. Allerdings treten diese Antigene teilweise auch bei gesunden Zellen auf, sodass die CAR T-Zellen auch gesunde Zellen angreifen, was zu unerwünschten Nebenwirkungen führt. Um dies zu verhindern, müssen spezifischere Antigene gefunden bzw. Methoden erforscht werden, eine spezifischere Aktivierung der CAR T-Zellen zu erreichen. Eine Idee besteht hier z.B. in der Kombination mehrerer Antigene.
  • Solide Tumoren: Während CAR T-Zelltherapien bei Leukämien (Blutkrebs) schon vielversprechende Erfolge zeigen, ist dies bei soliden Tumoren noch nicht der Fall. Der Grund wird in der Mikroumgebung solider Tumoren vermutet, wo verschiedene Barrieren ein effektives Eindringen der Immunzellen verhindern.
Bis heute ist die Verteilung und die Aktivität der transfundierten Zellen im Körper und im Tumor nur unzureichend bekannt.

Das Ziel dieses Projektes ist es, CAR T-Zellen im Körper mittels nicht-invasiver Bildgebungsverfahren wie PET/SPECT zu beobachten. Hierzu nutzen wir ein Tumormodell in der Maus. CAR T-Zellen werden mit nuklearmedizinischen Tracern markiert und ihre Verteilung und Aktivität wird in der Maus beobachtet. online
EXC 2044 - C1: Evolution and asymptotics In this unit, we will use generalisations of optimal transport metrics to develop gradient flow descriptions of (cross)-diffusion-reaction systems, rigorously analyse their pattern forming properties, and develop corresponding efficient numerical schemes. Related transport-type- and hyperbolic systems will be compared with respect to their pattern-forming behaviour, especially when mass is conserved. Bifurcations and the effects of noise perturbations will be explored.

Moreover, we aim to understand defect structures, their stability and their interactions. Examples are the evolution of fractures in brittle materials and of vortices in fluids. Our analysis will explore the underlying geometry of defect dynamics such as gradient descents or Hamiltonian structures. Also, we will further develop continuum mechanics and asymptotic descriptions for multiple bodies which deform, divide, move, and dynamically attach to each other in order to better describe the bio-mechanics of growing and dividing soft tissues.

Finally, we are interested in the asymptotic analysis of various random structures as the size or the dimension of the structure goes to infinity. More specifically, we shall consider random polytopes and random trees.For random polytopes we would like to compute the expected number of faces in all dimensions, the expected (intrinsic) volume, and absorption probabilities, as well as higher moments and limit distributions for these quantities. online
EXC 2044 - C2: Multi-scale phenomena and macroscopic structures In multi-scale problems, geometry and dynamics on the micro-scale influence structures on coarser scales. In this research unit we will investigate and analyse such structural interdependence based on topological, geometrical or dynamical properties of the underlying processes.

We are interested in transport-dominated processes, such as in the problem of how efficient a fluid can be stirred to enhance mixing, and in the related analytical questions. A major concern will be the role of molecular diffusion and its interplay with the stirring process. High Péclet number flow in porous media with reaction at the surface of the porous material will be studied. Here, the flowinduces pore-scale fluctuations that lead to macroscopic enhanced diffusion and reaction kinetics. We also aim at understanding advection-dominated homogenisation problems in random regimes.

We aim at classifying micro-scale geometry or topology with respect to the macroscopic behaviour of processes considered therein. Examples are meta material modelling and the analysis of processes in biological material. Motivated by network formation and fracture mechanics in random media, we will analyse the effective behaviour of curve and free-discontinuity energies with stochastic inhomogeneity. Furthermore, we are interested in adaptive algorithms that can balance the various design parameters arising in multi-scale methods. The analysis of such algorithms will be the key towards an optimal distribution of computational resources for multi-scale problems.

Finally, we will study multi-scale energy landscapes and analyse asymptotic behaviour of hierarchical patterns occurring in variational models for transportation networks and related optimal transport problems. In particular, we will treat questions of self-similarity, cost distribution, and locality of the fine-scale pattern. We will establish new multilevel stochastic approximation algorithms with the aim of numerical optimisation in high dimensions. online
EXC 2044 - C3: Interacting particle systems and phase transitions The question of whether a system undergoes phase transitions and what the critical parameters are is intrinsically related to the structure and geometry of the underlying space. We will study such phase transitions for variational models, for processes in random environments, for interacting particle systems, and for complex networks. Of special interest are the combined effects of fine-scalerandomly distributed heterogeneities and small gradient perturbations.

We aim to connect different existing variational formulations for transportation networks, image segmentation, and fracture mechanics and explore the resulting implications on modelling, analysis, and numerical simulation of such processes. We will study various aspects of complex networks, i.e. sequences of random graphs (Gn)n∈N, asking for limit theorems as n tends to infinity. A main task will be to broaden the class of networks that can be investigated, in particular, models which include geometry and evolve in time. We will study Ising models on random networks or with random interactions, i.e. spin glasses. Fluctuations of order parameters and free energies will be analysed, especially at the critical values where the system undergoes a phase transition. We will also investigate whether a new class of interacting quantum fields connected with random matrices and non-commutative geometry satisfies the Osterwalder-Schrader axioms. Further, we will study condensation phenomena, where complex network models combine the preferential attachment paradigm with the concept of fitness. In the condensation regime, a certain fraction of the total mass dynamically accumulates at one point, the condensate. The aim is a qualitative and quantitative analysis of the condensation. We willalso explore connections to structured population models. Further, we will study interacting particle systems on graphs that describe social interaction or information exchange. Examples are the averaging process or the Deffuant model.

We will also analyse asymmetric exclusion processes (ASEP) on arbitrary network structures. An interesting aspect will be how these processes are influenced by different distribution mechanisms of the particles at networks nodes. If the graph is given by a lattice, we aim to derive hydrodynamic limits for the ASEP with jumps of different ranges for multiple species, and for stochastic interactingmany-particle models of reinforced random walks. Formally, local cross-diffusion syste ms are obtained as limits of the classical multi-species ASEP and of the many-particle random walk. We will compare the newly resulting limiting equations and are interested in fluctuations, pattern formation, and the long-time behaviour of these models on the microscopic and the macroscopic scale. Further, we will analyse properties of the continuous directed polymer in a random environment. online
EXC 2044 - C4: Geometry-based modelling, approximation, and reduction In mathematical modelling and its application to the sciences, the notion of geometry enters in multiple related but different flavours: the geometry of the underlying space (in which e.g. data may be given), the geometry of patterns (as observed in experiments or solutions of corresponding mathematical models), or the geometry of domains (on which PDEs and their approximations act). We will develop analytical and numerical tools to understand, utilise and control geometry, also touching upon dynamically changing geometries and structural connections between different mathematical concepts, such as PDE solution manifolds, analysis of pattern formation, and geometry. online
Phone+49 251 83-35144
FAX+49 251 83-32729
Secretary   Sekretariat Wernke
Frau Silvia Wernke
Telefon +49 251 83-35052
Fax +49 251 83-32729
Zimmer 120.001
AddressProf. Dr. Benedikt Wirth
Angewandte Mathematik Münster: Institut für Analysis und Numerik
Fachbereich Mathematik und Informatik der Universität Münster
Orléans-Ring 10
48149 Münster
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