Models and Approximations

© FB10 - D. Münsterkötter

Research Area C

Alsmeyer, Böhm, Dereich, Engwer, Friedrich (until 2021), Gusakova (since 2021), Hille, Holzegel (since 2020), Huesmann, Jentzen (since 2019), Kabluchko, Lohkamp, Löwe, Mukherjee, Ohlberger, Pirner (since 2022), Rave, Schedensack (until 2019), F. Schindler, Schlichting (since 2020), Seis, Simon (since 2021), Stevens, Weber (since 2022), Wilking, Wirth, Wulkenhaar, Zeppieri.

In research area C, we will focus on the development and foundation of mathematical models and their approximations that are relevant in the life sciences, physics, chemistry, and engineering. We will rigorously analyse the dynamics of structures and pattern formation in deterministic and stochastic systems. In particular, we aim at understanding the interplay of macroscopic structures with their driving microscopic mechanisms and their respective topological and geometric properties. We will develop analytical and numerical tools to understand, utilise, and control geometry-driven phenomena, also touching upon dynamics and perturbations of geometries. Structural connections between different mathematical concepts will be investigated, such as between solution manifolds of parameterised PDEs and non-linear interpolation, or between different metric, variational, and multi-scale convergence concepts for geometries. In particular, we aim to characterise distinctive geometric properties of mathematical models and their respective approximations.

  • 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.

  • 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 flow induces 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.

  • 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-scale randomly 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 will also 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 interacting many-particle models of reinforced random walks. Formally, local cross-diffusion systems 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.

  • 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.

    We will interpret data from different contexts (in particular measurements from the life sciences and physics, shapes from computer graphics applications, and solutions to parameterised PDEs) as elements of an underlying non-linear (infinite-dimensional) geometric space, e.g. a Riemannian manifold. This geometric structure will be exploited for the development of data processing tools. A focus will lie on variational and numerical methods for data fitting and regression via submanifolds, on singular perturbation methods, and on asymptotics and model reduction for parameterised PDEs by decomposing each solution into an element of a linear space and a Lie group element acting on it.

    The geometry of spatial patterns often determines the average, effective properties of such structures, e.g. in a material, its effective material properties. Motivated by particular patterns and their defects, as observed in biological organisms or materials, we will examine their macroscopic, homogenised properties and their stability with respect to pattern perturbations. The effective, homogenised structure will again be described in geometric terms: For instance, the homogenised free energy of carbon nanotubes may depend on their bending curvature, the transport efficiency of molecules in strongly layered biological membranes of cell organelles may depend on an effective distance metric, and defects in atomic crystals can be related to specific singularities of two-dimensional surfaces.

    Applications like shape optimisation or shape reconstruction problems are concerned with the identification of a geometry. Typically, there is an additional PDE constraint for which the sought geometry serves as the PDE domain. It is a challenge to efficiently approximate this geometry. We will develop concepts that quantify the efficiency of the geometry approximation in terms of the involved computational effort per desired accuracy, and we will investigate numerical schemes that can efficiently deal with complex (time-)varying PDE domains without the need for remeshing.

Further research projects of Research Area C members

• Interdisziplinäres Lehrprogramm zu maschinellem Lernen und künstlicher Intelligenz

The aim of the project is to establish and test a graduated university-wide teaching programme on machine learning (ML) and artificial intelligence (AI). AI is taught as an interdisciplinary cross-sectional topic that has diverse application possibilities in basic research as well as in economy and society, but consequently also raises social, ethical and ecological challenges.

The modular teaching program is designed to enable students to build up their AI knowledge, apply it independently and transfer it directly to various application areas. The courses take place in a broad interdisciplinary context, i.e., students from different departments take the courses together and work together on projects.

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Project members: Xiaoyi Jiang, Christian Engwer

• Dynamical systems and irregular gradient flows The central goal of this project is to study asymptotic properties for gradient flows (GFs) and related dynamical systems. In particular, we intend to establish a priori bounds and related regularity properties for solutions of GFs, we intend to study the behaviour of GFs near unstable critical regions, we intend to derive lower and upper bounds for attracting regions, and we intend to establish convergence speeds towards global attrators. Special attention will be given to GFs with irregularities (discontinuities) in the gradient and for such GFs we also intend to reveal sufficient conditions for existence, uniqueness, and flow properties in dependence of the given potential. We intend to accomplish the above goals by extending techniques and concepts from differential geometry to describe and study attracting and critical regions, by using tools from convex analysis such as subdifferentials and other generalized derivatives, as well as by employing concepts from real algebraic geometry to describe domains of attraction. In particular, we intend to generalize the center-stable manifold theorem from the theory of dynamical systems to the considered non-smooth setting. Beside finite dimensional GFs, we also study GFs in their associated infinite dimensional limits. The considered irregular GFs and related dynamical systems naturally arise, for example, in the context of molecular dynamics (to model the configuration of atoms along temporal evoluation) and machine learning (to model the training process of artificial neural networks).
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Project members: Christoph Böhm, Arnulf Jentzen

• Mathematical Research Data Initiative - TA2: Scientific Computing Driven by the needs and requirements of mathematical research as well as scientific disciplines using mathematics, the NFDI-consortium MaRDI (Mathematical Research Data Initiative) will develop and establish standards and services for mathematical research data. Mathematical research data ranges from databases of special functions and mathematical objects, aspects of scientific computing such as models and algorithms to statistical analysis of data with uncertainties. It is also widely used in other scientific disciplines due to the cross-disciplinary nature of mathematical methods. online
Project members: Mario Ohlberger, Stephan Rave

• Personalised diagnosis and treatment for refractory focal paediatric and adult epilepsy Epilepsy is among the most common neurological diseases, affecting between 0.5% and 1% of the general population. Therefore, new diagnosis and treatment methods have a high impact on society. Epilepsy is also among the most frequently diagnosed neurological paediatric disorders, with long-term implications for the quality of life of those affected. Only in two-thirds of cases, seizures can be adequately controlled with anticonvulsant drug treatment. For the remaining drug-refractory patients with focal epilepsy (up to about 2 Mill. in Europe), epilepsy surgery is currently the most effective treatment. However, only 15-20% of those patients are eligible for epilepsy surgery. That is either because the epileptogenic zone in the brain cannot be localized with sufficient accuracy with standard diagnostic means, or because the epileptogenic zone overlaps with eloquent cortical areas, so that it cannot be surgically removed without considerable neurological deficit. PerEpi aims to bring together a group of experts at the European level to improve this situation in two ways, both of which use concepts of non-invasive personalised medicine: The first one focuses on a new individualised multimodal approach to set a new milestone in localization accuracy of the epileptogenic zone in order to offer the most appropriate personalised therapy. The second one focuses on a new individually optimized transcranial electric brain stimulation technique as a new treatment option to reduce seizure frequency and severity. This is particularly attractive for those focal refractory patients where surgery is not an option because of an overlap with eloquent cortical areas. A dedicated ethics work package will ensure that the research in the consortium is designed and conducted following the highest ethical standards. In addition, this work package will study the translational pathways of the new approaches to foster clinical integration that is ethically and socially responsible. online
Project members: Christian Engwer

• CRC 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
Project members: Benedikt Wirth

• 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
Project members: Benedikt Wirth

• SPP 2265: Random Geometric Systems - Subproject: Optimal transport for stationary point processes Optimal transport by now has found manifold applications in various areas of mathematics, in particular it has turned into a powerful tool in the analysis of stochastic processes, particle dynamics, and the associated evolution equations, mostly however in a finite-dimensional setting. The goal of this project is to develop a counterpart to this theory in the framework of stationary point processes or more general random (infinite) measures and to employ these novel tools e.g. in the analysis of infinite particle dynamics or to attack questions for particular point process models of interest. online
Project members: Martin Huesmann

• CRC 1442: Geometry: Deformation and Rigidity - Geometric evolution equations Hamilton’s Ricci flow is a geometric evolution equation on the space of Riemannian metrics of a smooth manifold. In a first subproject we would like to show a differentiable stability result for noncollapsed converging sequences of Riemannian manifolds with nonnegative sectional curvature, generalising Perelman’s topological stability. In a second subproject, next to classifying homogeneous Ricci solitons on non-compact homogeneous spaces, we would like to prove the dynamical Alekseevskii conjecture. Finally, in a third subproject we would like to find new Ricci flow invariant curvature conditions, a starting point for introducing a Ricci flow with surgery in higher dimensions. online
Project members: Christoph Böhm, Burkhard Wilking

• CRC 1442: Geometry: Deformation and Rigidity - D03: Integrability The project investigates a novel integrable system which arises from a quantum field theory on noncommutative geometry. It is characterised by a recursive system of equations with conjecturally rational solutions. The goal is to deduce their generating function and to relate the rational coefficients in the generating function to intersection numbers of tautological characteristic classes on some moduli space. online
Project members: Raimar Wulkenhaar

• CRC 1442: Geometry: Deformation and Rigidity - B01: Curvature and Symmetry The question of how far geometric properties of a manifold determine its global topology is a classical problem in global differential geometry. In a first subproject we study the topology of positively curved manifolds with torus symmetry. We think that the methods used in this subproject can also be used to attack the Salamon conjecture for positive quaternionic Kähler manifolds. In a third subproject we study fundamental groups of non-negatively curved manifolds. Two other subprojects are concerned with the classification of manifolds all of whose geodesics are closed and the existence of closed geodesics on Riemannian orbifolds. online
Project members: Michael Wiemeler, Burkhard Wilking

• ML-MORE: Machine learning and model order reduction to predict the efficiency of catalytic filters. Subproject 1: Model Order Reduction Reaktiver Stofftransport in porösen Medien in Verbindung mit katalytischen Reaktionen ist die Grundlage für viele industrielle Prozesse und Anlagen, wie z.B. Brennstoffzellen, Photovoltaikzellen, katalytische Filter für Abgase, etc. Die Modellierung und Simulation der Prozesse auf der Porenskala kann bei der Optimierung des Designs von katalytischen Komponenten und der Prozessführung helfen, ist jedoch derzeit dadurch eingeschränkt, dass solche Simulationen zu grossen Datenmengen führen, zeitaufwändig sind und von einer grossen Anzahl von Parametern abhängen. Außerdem werden auf diese Weise die im Laufe der Jahre gesammelten Versuchsdaten nicht wiederverwendet. Die Entwicklung von Lösungsansätzen für die Vorhersage der chemischen Konversionsrate mittels moderner datenbasierter Methoden des Maschinellen Lernens (ML) ist essenziell, um zu schnellen, zuverlässigen prädiktiven Modellen zu gelangen. Hierzu sind verschiedene Methodenklassen erforderlich. Neben den experimentellen Daten sind voll aufgelöste Simulationen auf der Porenskala notwendig. Diese sind jedoch zu teuer, um einen umfangreichen Satz an Trainingsdaten zu generieren. Daher ist die Modellordnungsreduktion (MOR) zur Beschleunigung entscheidend. Es werden reduzierte Modelle fur den betrachteten instationären reaktiven Transport entwickelt, um große Mengen an Trainingsdaten zu simulieren. Als ML-Methodik werden mehrschichtige Kern-basierte Lernverfahren entwickelt, um die heterogenen Daten zu kalibrieren und nichtlineare prädiktive Modelle zur Effizienzvorhersage zu entwickeln.Hierbei werden große Daten (bzgl. Dimensionalität und Sample-Zahl) zu behandeln sein, was Datenkompression und Parallelisierung des Trainings erfordern wird. Das Hauptziel des Projekts ist es, alle oben genannten Entwicklungen in einem prädiktiven ML-Tool zu integrieren, das die Industrie bei der Entwicklung neuer katalytischer Filter unterstützt und auf viele andere vergleichbare Prozesse übertragbar ist. online
Project members: Mario Ohlberger, Felix Schindler

• RTG 2149: Strong and Weak Interactions - from Hadrons to Dark Matter The Research Training Group (Graduiertenkolleg) 2149 "Strong and Weak Interactions - from Hadrons to Dark Matter" funded by the Deutsche Forschungsgemeinschaft focuses on the close collaboration of theoretical and experimental nuclear, particle and astroparticle physicists further supported by a mathematician and a computer scientist. This explicit cooperation is of essence for the PhD topics of our Research Training Group.Scientifically this Research Training Group addresses questions at the forefront of our present knowledge of particle physics. In strong interactions we investigate questions of high complexity, such as the parton distributions in nuclear matter, the transition of the hot quark-gluon plasma into hadrons, or features of meson decays and spectroscopy. In weak interactions we pursue questions, which are by definition more speculative and which go beyond the Standard Model of particle physics, particularly with regard to the nature of dark matter. We will confront theoretical predictions with direct searches for cold dark matter particles or for heavy neutrinos as well as with new particle searches at the LHC.The pillars of our qualification programme are individual supervision and mentoring by one senior experimentalist and one senior theorist, topical lectures in physics and related fields (e.g. advanced computation), peer-to-peer training through active participation in two research groups, dedicated training in soft skills, and the promotion of research experience in the international community. We envisage early career steps through a transfer of responsibilities and international visibility with stays at external partner institutions. An important goal of this Research Training Group is to train a new generation of scientists, who are not only successful specialists in their fields, but who have a broader training both in theoretical and experimental nuclear, particle and astroparticle physics. online
Project members: Raimar Wulkenhaar

• 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
Project members: Benedikt Wirth

• Mathematical Theory for Deep Learning It is the key goal of this project to provide a rigorous mathematical analysis for deep learning algorithms and thereby to establish mathematical theorems which explain the success and the limitations of deep learning algorithms. In particular, this projects aims (i) to provide a mathematical theory for high-dimensional approximation capacities for deep neural networks, (ii) to reveal suitable regular sequences of functions which can be approximated by deep neural networks but not by shallow neural networks without the curse of dimensionality, and (iii) to establish dimension independent convergence rates for stochastic gradient descent optimization algorithms when employed to train deep neural networks with error constants which grow at most polynomially in the dimension. online
Project members: Arnulf Jentzen, Benno Kuckuck

• Existence, uniqueness, and regularity properties of solutions of partial differential equations The goal of this project is to reveal existence, uniqueness, and regularity properties of solutions of partial differential equations (PDEs). In particular, we intend to study existence, uniqueness, and regularity properties of viscosity solutions of degenerate semilinear Kolmogorov PDEs of the parabolic type. We plan to investigate such PDEs by means of probabilistic representations of the Feynman-Kac type. We also intend to study the connections of such PDEs to optimal control problems. online
Project members: Arnulf Jentzen

• Regularity properties and approximations for stochastic ordinary and partial differential equations with non-globally Lipschitz continuous nonlinearities A number of stochastic ordinary and partial differential equations from the literature (such as, for example, the Heston and the 3/2-model from financial engineering, (overdamped) Langevin-type equations from molecular dynamics, stochastic spatially extended FitzHugh-Nagumo systems from neurobiology, stochastic Navier-Stokes equations, Cahn-Hilliard-Cook equations) contain non-globally Lipschitz continuous nonlinearities in their drift or diffusion coefficients. A central aim of this project is to investigate regularity properties with respect to the initial values of such stochastic differential equations in a systematic way. A further goal of this project is to analyze the regularity of solutions of the deterministic Kolmogorov partial dfferential equations associated to such stochastic differential equations. Another aim of this project is to analyze weak and strong convergence and convergence rates of numerical approximations for such stochastic differential equations. online
Project members: Arnulf Jentzen

• Overcoming the curse of dimensionality: stochastic algorithms for high-dimensional partial differential equations Partial differential equations (PDEs) are among the most universal tools used in modeling problems in nature and man-made complex systems. The PDEs appearing in applications are often high dimensional. Such PDEs can typically not be solved explicitly and developing efficient numerical algorithms for high dimensional PDEs is one of the most challenging tasks in applied mathematics. As is well-known, the difficulty lies in the so-called ''curse of dimensionality'' in the sense that the computational effort of standard approximation algorithms grows exponentially in the dimension of the considered PDE. It is the key objective of this research project to overcome this curse of dimensionality and to construct and analyze new approximation algorithms which solve high dimensional PDEs with a computational effffort that grows at most polynomially in both the dimension of the PDE and the reciprocal of the prescribed approximation precision. online
Project members: Arnulf Jentzen