Modelle und Approximationen

© FB10 - D. Münsterkötter

Forschungsschwerpunkt C

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

Anwendungen aus den Natur- und Lebenswissenschaften bestimmen die Herausforderungen in diesem Forschungsschwerpunkt. Dabei zielen wir auf die Entwicklung und Analyse von grundlegenden dynamischen und geometrischen Modellierungs- und Approximationsansätzen zur Beschreibung deterministischer und stochastischer Systeme ab. Wir untersuchen beispielsweise das Zusammenspiel von makroskopischen Strukturen mit zugrundeliegenden mikroskopischen Prozessen und deren jeweiligen topologischen und geometrischen Eigenschaften. Ein weiterer Fokus ist die Untersuchung, Ausnutzung und Optimierung der zugrundeliegenden Geometrie in mathematischen Modellen. Wir untersuchen strukturelle Verbindungen zwischen unterschiedlichen mathematischen Konzepten, wie z.B. zwischen Lösungsmannigfaltigkeiten von partiellen Differentialgleichungen und nicht-linearer Interpolation oder zwischen verschiedenen metrischen, variationellen oder mehrskaligen Konvergenzkonzepten für Geometrien. Speziell zielen wir auf die Charakterisierung kennzeichnender geometrischer Eigenschaften von mathematischen Modellen und deren Approximationen.

 

 

 

 

 

 

 

Weitere Forschungsprojekte von Mitgliedern des Forschungsschwerpunkts C

 

Invariante Mannigfaltigkeiten für schnelle Diffusionen nahe Auslöschung online
Project members: Christian Seis

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, René Fritze (Milk), 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

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
Project members: Benedikt Wirth

SPP 2256: Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials - Subproject: Variational modelling of fracture in high-contrast microstructured materials: mathematical analysis and computational mechanics After the seminal work of Francfort and Marigo, free-discontinuity functionals of Mumford-Shah type have been established as simplified and yet relevant mathematical models to study fracture in brittle materials. For finite-contrast constituents, the homogenisation of brittle energies is by-now well-understood and provides a rigorous micro-to-macro upscaling for brittle fracture.Only recently, explicit high-contrast brittle microstructures have been provided, which show that, already for simple free-discontinuity energies of Mumford-Shah type, the high-contrast nature of the constituents can induce a complex effective behaviour going beyond that of the single constituents. In particular, macroscopic cohesive-zone models and damage models can be obtained by homogenising purely brittle microscopic energies with high-contrast coefficients. In this framework, the simple-to-complex transition originates from a microscopic bulk-surface energy-coupling which is possible due to the degeneracy of the functionals.Motivated by the need to understand the mathematical foundations of mechanical material-failure and to develop computationally tractable numerical techniques, the main goal of this project is to characterise all possible materials which can be obtained by homogenising simple high-contrast brittle materials. In mathematical terms, this amounts to determine the variational-limit closure of the set of high-contrast free-discontinuity functionals. This problem has a long history in the setting of elasticity, whereas is far less understood if fracture is allowed.For the variational analysis it will be crucial to determine novel homogenisation formulas which “quantify” the microscopic bulk-surface energy-coupling. Moreover, the effect of high-contrast constituents on macroscopic anisotropy will be investigated by providing explicit microstructures realising limit models with preferred crack-directions.The relevant mathematical tools will come from the Calculus of Variations and Geometric Measure Theory. Along the way, new ad hoc extension and approximation results for SBV-functions will be established. The latter will be of mathematical interest in their own right, and appear to be widely applicable in the analysis of scale-dependent free-discontinuity problems.The computational mechanics results will build upon the mathematical theory, and will complement it with relevant insights when the analysis becomes impracticable. High performance fast Fourier transform and adaptive tree-based computational methods will be developed to evaluate the novel cell formulas. The identified damage and cohesive-zone models will be transferred to simulations on component scale.The findings are expected to significantly enhance the understanding of the sources and mechanisms of material-failure and to provide computational tools for identifying anisotropic material-models useful for estimating the strength of industrial components. online
Project members: Caterina Zeppieri

SPP 2265: Random Geometric Systems - Subproject: Optimal matching and balancing transport The optimal matching problem is one of the classical problems in probability. By now, there is a good understanding of the macroscopic behaviour with some very detailed results, several challenging predictions, and open problems. The goal of this project is to develop a refined analysis of solutions to the optimal matching problem from a macroscopic scale down to a microscopic scale. Based on a recent quantitative linearization result of the Monge-Ampère equation developed in collaboration with Michael Goldman, we will investigate two main directions. On the one hand, we aim at rigorously connecting the solutions to the optimal matching problem to a Gaussian field which scales as the Gaussian free field. On the other hand, we seek to close the gap and show that rescaled solutions to the optimal matching problem converge in the thermo-dynamic limit to invariant allocations to point processes, or more generally to balancing transports between random measures. In the long run, combining these limit results with the Gaussian like behaviour of solutions to the matching problem, we seek to analyse the random tessellations induced by the limiting invariant allocations. online
Project members: Martin Huesmann

SPP 2265: Random Geometric Systems - Subproject: Random polytopes The aim of the project is to investigate random polytopes. online
Project members: Zakhar Kabluchko

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

Transport Equations, mixing and fluid dynamics Advection-diffusion equations are of fundamental importance in many areas of science. They describe systems, in which a quantity is simultaneously diffused and advected by a velocity field. In many applications these velocity fields are highly irregular. In this project, several quantitative aspects shall be investigated. One is related to mixing properties in fluids caused by shear flows. The interplay between the transport by the shear flow and the regularizing diffusion leads after a certain time, to the emergence of a dominant length scales which persist during the subsequent evolution and determine mixing rates. A rigorous understanding of these phenomena is desired. In addition, stability estimates for advection-diffusion equations will be derived. These shall give a deep insight into how solutions depend on coefficients and data. The new results shall subsequently be applied to estimate the error generated by numerical finite volume schemes approximating the model equations. online
Project members: Christian Seis

HyperCut – Stabilized DG schemes for hyperbolic conservation laws on cut cell meshes The goal of this project is to develop new tools for solving time-dependent, first order hyperbolic conservation laws, in particular the compressible Euler equations, on complex shaped domains.In practical applications, mesh generation is a major issue. When dealing with complicated geometries, the construction of corresponding body-fitted meshes is a very involved and time-consuming process.In this proposal, we will consider a different approach: In the last two decades so called cut cell methods have gained a lot of interest, as they reduce the burden of the meshing process. The idea is to simply cut the geometry out of a Cartesian background mesh. Theresulting cut cells can have various shapes and are not bounded from below in size. Compared to body-fitted meshes, this approach is fully automatic and much cheaper. However, standard explicit schemes are typically not stable when the time step is chosen with respect to the background mesh and does not reflect the size of small cut cells. Thisis referred to as the small cell problem.In the setting of standard meshes, both Finite Volume (FV) and Discontinuous Galerkin (DG) methods have been used successfully for solving non-linear hyperbolic conservation laws. For FV schemes, there already exist several approaches for extending thesemethods to cut cell meshes and overcoming the small cell problem while keeping the explicit time stepping. For DG schemes, this is not the case.The goal of this proposal is to develop stable DG schemes for solving time-dependent hyperbolic conservation laws, in particular the compressible Euler equations, on cut cell meshes using explicit time stepping.We particularly aim at a method that(1) solves the small cell problem and permits explicit time stepping,(2) preserves mass conservation,(3) is high-order along the cut cell boundary, where many important quantities are evaluated,(4) satisfies theoretical properties such as monotonicity and TVDM stability for model problems,(5) works for non-linear hyperbolic conservation laws, in particular the compressible Euler equations,(6) is robust in the presence of shocks or discontinuities,(7) and sufficiently simple to be implemented in higher dimensions.We base the spatial discretization on a DG approach to enable high accuracy. We plan to develop new stabilization terms to overcome the small cell problem for this setup. The starting point for this proposal is our recent publication for stabilizing a DG discretizationfor linear advection using piecewise linear polynomials. We will extend these results in different directions, namely to non-linear problems, including the compressible Euler equations, and to higher order, in particular to piecewise quadratic polynomials.We will implement these methods using the software framework DUNE and publish our code as open-source. online
Project members: Christian Engwer

GRK 2149 - Starke und schwache Wechselwirkung - von Hadronen zu Dunkler Materie 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

Multidimensional Stochastic Models and Their Applications The developments of Probability, Statistics, and their applications lead to increasingly complex models, involving a large number of variables as well as hidden or explicit parameters including a variety of random factors as well. The objects of our research like random matrices and random polynomials, Gaussian fields, random polytopes, fractional Laplacians as generators of Levy processes, U-max statistics and kernel density estimators are topics which are currently intensively studied. Research in this area requires the solution of basic open questions in modern Probability and Statistics using techniques relying on the latest mathematical achievements. In particular we shall concentrate on the solution of open problems concerning U-max statistics, the relation of random polytopes and intrinsic volumes to Gaussian random matrices and Gaussian processes, complexity measures for quantization and coding errors and the distribution of fractional processes as well as problems of small deviations. online
Project members: Zakhar Kabluchko

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

Localized Reduced Basis Methods for PDE-constrained Parameter Optimization This projects is concerned with model reduction for parameter optimization of nonlinear elliptic partial differential equations (PDEs). The goal is to develop a new paradigm for PDE-constrained optimization based on adaptive online enrichment. The essential idea is to design a localized version of the reduced basis (RB) method which is called Localized Reduced Basis Method (LRBM). online
Project members: Mario Ohlberger, Felix Schindler, Tim Keil