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Erfolgreich abgeschlossene Projekte

• CH11

  Sensing with Nanopores

  Dr. Jürgen Fuhrmann / Dr. Clemens Guhlke

  Projektleiter: Dr. Jürgen Fuhrmann / Dr. Clemens Guhlke
  Projekt Mitglieder: -
  Laufzeit: 01.06.2017 - 31.12.2018
  Status: beendet
  Standort: Weierstraß-Institut
  

  Beschreibung

  Sensing with nanopores is a promising new technology to analyze macromolecules like DNA strands by low cost/high speed measurements. The sensing device is constructed based on a nanopore embedded into a membrane which separates two electrodes. The system is filled with an electrolyte containing macromolecules to be analyzed. An electric potential is applied to the electrodes and induces an ionic current through the pore. Sensing is based on the observation that this ionic current is influenced by the geometrical configurations of the pore and of the macromolecules positioned within the pore. Under controlled movement of the macromolecule through the pore a characteristic time dependent current signal is generated, which is correlated to the structure of the pore and the macromolecule. Therefore nanopores can be used to count and even to characterize macromolecules in an electrolytic solution. n order to achieve a better understanding of the of phenomena that control the passing time of the analytes (macromolecules) through the nanopore, and to derive a relation between characteristic properties of the macromolecule and the generated current, the project will focus on three groups of tasks: Development of an appropriate nanopore model in the context of non-equilibrium thermodynamics, which accounts for the geometrical properties of pore and analyte, the charged boundary layers, ion diffusion and fluid flow. Combination and analysis of novel numerical discretization schemes, like pressure robust methods for fluid flow and novel finite volume discretization approaches for the PNP system in order to provide physically meaningful numerical models of the double layer structure and its impact on the fluid flow. Use of asymptotic analysis to derive reduced models, which include the relevant features of the complete thermodynamic model in different regimes.
  
  https://www.wias-berlin.de/projects/ECMath-CH11/

• CH13

  Empirical Bayes methods for patient-specific prediction and control of pharmacological interventions

  Dr. Rainald Ehrig / Prof. Dr. Susanna Röblitz

  Projektleiter: Dr. Rainald Ehrig / Prof. Dr. Susanna Röblitz
  Projekt Mitglieder: Dr Ilja Klebanov
  Laufzeit: 01.06.2017 - 31.12.2018
  Status: beendet
  Standort: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Beschreibung

  One of the main goals of mathematical modeling related to medical applications is to obtain patient-specific parametrizations and model predictions. In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Empirical Bayes methods can provide a solution to this controversy. Instead of applying Bayes’ rule to each measurement separately, these methods usually boil down to combining all measurements in order to construct an informative prior as a first step and then using this prior for the Bayesian inference of the individual parametrizations in a second step. We want to demonstrate the applicability and benefit of this approach on a high-dimensional model system for predicting patient-specific treatment success rates related to in vitro fertilization in reproductive medicine.
  
  http://www.zib.de/projects/empirical-bayes-methods-patient-specific-prediction-and-control-pharmacological-interventions

• CH16

  Reliable joint simulations for orthopaedic decision making in hip surgery

  Prof. Dr. Ralf Kornhuber / Dr.-Ing. Stefan Zachow

  Projektleiter: Prof. Dr. Ralf Kornhuber / Dr.-Ing. Stefan Zachow
  Projekt Mitglieder: Dr. Jonathan Youett
  Laufzeit: 01.06.2017 - 31.12.2018
  Status: beendet
  Standort: Freie Universität Berlin
  

  Beschreibung

  This project aims at estimating the unknown parameters of a physics-based joint model together with a systematic sensitivity analysis, to ensure the reliability of a computer-assisted surgery planning tool developed in previous Matheon projects. The estimation shall be carried out using a Bayesian approach in combination with reduced basis methods to achieve feasible computing times. This calibration of the model will be accompanied by a clinical validation based on real patient data, in cooperation with the Orthopaedic Research Center of the university hospital Stavanger.
  
  http://www.mi.fu-berlin.de/en/math/groups/ag-numerik/projects/A-CH1/index.html

• CH19

  Estimating Dynamics of Macromolecular Systems by Low Rank Approximation Techn

  Priv.-Doz. Dr. Konstantin Fackeldey / Prof. Dr. Frank Noé / Prof. Dr. Reinhold Schneider / Dr. Hao Wu

  Projektleiter: Priv.-Doz. Dr. Konstantin Fackeldey / Prof. Dr. Frank Noé / Prof. Dr. Reinhold Schneider / Dr. Hao Wu
  Projekt Mitglieder: -
  Laufzeit: 01.06.2017 - 31.12.2018
  Status: beendet
  Standort: Freie Universität Berlin / Technische Universität Berlin
  

  Beschreibung

  The dynamics of a molecular system can be described by the propagation of probabilities. The project aims at estimating coarse grained models of probability densities for molecular dynamics (MD) by nonlinear projections from a high dimensional space onto a low dimensional space. Molecular processes such as protein kinetics from all-atom simulations and the like suffer from the high dimensionality of the underlying space. To overcome this, projections from the high dimensional space onto a low dimensional space have been introduced, such that the system can be described on a coarser scale by using less degrees of freedom. In the present project we apply low rank tensor approximations, to tackle the curse of dimensions. We will use Observable Operator models (OOM) to estimate the dynamics using data from short time simulation.
  
  http://www.mi.fu-berlin.de/en/math/groups/comp-mol-bio/projects/ecmath19/index.html

• CH1

  Reduced basis methods in orthopedic hip surgery planning

  Prof. Dr. Ralf Kornhuber / Dr.-Ing. Stefan Zachow

  Projektleiter: Prof. Dr. Ralf Kornhuber / Dr.-Ing. Stefan Zachow
  Projekt Mitglieder: Dr. Jonathan Youett
  Laufzeit: -
  Status: beendet
  Standort: Freie Universität Berlin
  

  Beschreibung

  This project aims at the development, analysis and implementation of algorithms for computer-assisted planning in hip surgery and hip joint replacement by fast virtual test. Fast forward simulations of patient-specific motion of hip joints and implants in 3D shall be enabled by exploiting suitable a priori information. To this end, we will derive, analyze, and implement reduced basis methods for heterogeneous joint models (reduced approximation).
  
  http://www.mi.fu-berlin.de/en/math/groups/ag-numerik/projects-completed/A-CH1/index.html

• CH2

  Sparse compressed sensing based classifiers for -omics mass-data

  Prof. Dr. Tim Conrad / Prof. Dr. Gitta Kutyniok / Prof. Dr. Christof Schütte

  Projektleiter: Prof. Dr. Tim Conrad / Prof. Dr. Gitta Kutyniok / Prof. Dr. Christof Schütte
  Projekt Mitglieder: Nada Cvetkovic / Martin Genzel
  Laufzeit: -
  Status: beendet
  Standort: Freie Universität Berlin / Technische Universität Berlin
  

  Beschreibung

  Tumor diseases rank among the most frequent causes of death in Western countries coinciding with an incomplete understanding of the underlying pathogenic mechanisms and a lack of individual treatment options. Hence, early diagnosis of the disease and early relapse monitoring are currently the best available options to improve patient survival. In this project, we aim for the identification of disease specific sets of biological signals that reliably indicate a disease outbreak (or status) in an individual. Such biological signals (e.g. proteomics or genomics data) are typically very large (millions of dimensions), which significantly increases the complexity of algorithms for analyzing the parameter space or makes them even infeasible. However, these types of data usually exhibit a very particular structure, and at the same time, the set of disease specific features is very small compared to the ambient dimension. Such a high-dimensional setting naturally calls for the application of the concept of sparse classifiers, which has been extensively studied in the fields of compressed sensing and statistical learning during the last decade. Our research focuses on both algorithmic improvements of available methods as well as theoretical results such as recovery guarantees for general data models.
  
  http://medicalbioinformatics.de/research/projects/ecmath-ch2

• CH3

  Multiview geometry for ophthalmic surgery simulation

  Prof. Dr. Michael Joswig

  Projektleiter: Prof. Dr. Michael Joswig
  Projekt Mitglieder: André Wagner
  Laufzeit: -
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  A fundamental problem in machine vision asks to generate geometric information about a scene in 3-space from several camera images. This is relevant, e.g., in the context of augmented reality frameworks for eye surgery simulation. It is the goal of this project to apply techniques from geometric combinatorics and algebraic geometry for analyzing the picture space to allow for a profound computational preprocessing.
  
  http://page.math.tu-berlin.de/~wagner/CH3.htm

• CH4

  Optimal control of chemical reaction systems and application to drug resistance mitigating therapy

  Dr. Max von Kleist / PD Dr. Marcus Weber

  Projektleiter: Dr. Max von Kleist / PD Dr. Marcus Weber
  Projekt Mitglieder: Dr. Wei Zhang
  Laufzeit: -
  Status: beendet
  Standort: Freie Universität Berlin
  

  Beschreibung

  Development and spread of drug resistant microorganisms is a major health issue which, accompanied by an attrition in drug development, is expected to worsen in the near future. The source of drug resistance development is the inadequate use of antimicrobials: Inadequate therapies insufficiently suppress susceptible strains, which may give rise to a drug resistant type. At the same time, inadequate therapy exerts enough selective pressure to provide the newly emerged resistant strain with a selective advantage that allows it to become fixed in the population. In recent years, we have elaborated the idea, that an optimal switching between existing antimicrobial drugs may mitigate drug resistance development in the individual. Drug resistance development is an intrinsically stochastic process. This process can be accurately described by the chemical master equation (CME). A major mathematical drawback is the fact that the CME cannot be solved directly due to its numerical complexity. Therefore, computation of an optimal control/therapy based on a direct numerical solution of the CME is usually not feasible. The aim of the proposed project is to mathematically characterize and develop optimal control policies derived from approximations of the CME, and to use the developed methods to suggest drug mitigating therapies to clinical partners in the field of HIV-1 and antibiotic resistance.
  
  http://systems-pharmacology.de/?page_id=621

• CH5

  Model classification under uncertainties for cellular signaling networks

  Prof. Dr. Alexander Bockmayr / Prof. Dr. Susanna Röblitz / Prof. Dr. Heike Siebert

  Projektleiter: Prof. Dr. Alexander Bockmayr / Prof. Dr. Susanna Röblitz / Prof. Dr. Heike Siebert
  Projekt Mitglieder: Stefanie Kasielke / Adam Streck
  Laufzeit: -
  Status: beendet
  Standort: Freie Universität Berlin / Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Beschreibung

  Mathematical modelling in biological and medical applications is almost always faced with the problem of incomplete and noisy data. Rather than adding unsupported assumptions to obtain a unique model, a different approach generates a pool of models in agreement with all available observations. Analysis and classification of such models allow linking the constraints imposed by the data to essential model characteristics and showcase different implementations of key mechanisms. Within the project, we aim at combining the advantages of logical and continuous modeling to arrive at a comprehensive system analysis under data uncertainty. Model classification will integrate qualitative aspects such as characteristics of the network topology with more quantitative information extracted from clustering of joint parameter distributions derived from Bayesian approaches. The theory development is accompanied by and tested in application to oncogenic signaling networks.
  
  http://www.mi.fu-berlin.de/en/math/groups/dibimath/projects/A-CH5/index.html

• CH6

  Uncertainty quantification for Bayesian inverse problems with applications to systems biology

  Prof. Dr. Susanna Röblitz / Prof. Dr. Christof Schütte

  Projektleiter: Prof. Dr. Susanna Röblitz / Prof. Dr. Christof Schütte
  Projekt Mitglieder: Dr Ilja Klebanov
  Laufzeit: -
  Status: beendet
  Standort: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Beschreibung

  In biotechnology, systems biology, or reaction engineering one is faced with large systems of ordinary differential equations (ODE) that are used to describe the kinetics of the reaction network of interest. These ODE models contain a large number of mostly unknown kinetic parameters that one needs to infer from usually sparse and noisy experimental data. Typically, inverse problems like classical parameter identification are associated with ill-posed behaviour. However, Bayesian approaches can be used to recover joint parameter distributions and allow for the quantification of uncertainty and risk in a way demanded by the applications. In this project, we want to overcome the computational limitations of classical Markov-chain Monte-Carlo methods by developing new algorithmic approaches to Bayesian inverse problems using, e.g., sparse approximation results or empirical Bayes methods. The methods will directly be applied to large-scale networks in systems biology.
  
  http://www.zib.de/projects/UQ-systems-biology

• CH7

  Network-of-Network based -omics data integration

  Prof. Dr. Tim Conrad / Prof. Dr. Christof Schütte

  Projektleiter: Prof. Dr. Tim Conrad / Prof. Dr. Christof Schütte
  Projekt Mitglieder: -
  Laufzeit: -
  Status: beendet
  Standort: Freie Universität Berlin
  

  Beschreibung

  Project Background

  Pancreatic cancer is the fifth leading cause of cancer death in Germany (see DKFZ Report, 2010). It is estimated that in 2030 it will be the second leading cause of cancer death incurring a cost of about 15,8 Billion US-Dollar worldwide to the public health systems.

  Cancer is a systems disease

  "Cancer is no more a disease of cells than a traffic jam is a disease of cars. A lifetime of study of the internal-combustion engine would not help anyone to understand our traffic problems.'" (Smithers1962). It is accepted that gene mutations are part of the process of cancer, but mutations alone are not enough. Cancer involves an interaction between neoplastic cells and surrounding tissue on many different levels, e.g. interaction of RNA molecules, proteins, and metabolites. But most available models are limited to only one or very few levels of interactions and describe a rather static view.

  From single to multi source: data integration on a systems level

  Current high-throughput -omics technologies have dramatically eased the production of part lists for a variety of organisms. What is still missing are the dynamic interactions among an organism's molecular parts, and the interactions between different biological levels, such as transcriptomics and proteomics. This is pivotal to better understanding of an organism's biology, and - in our case - to understand pancreas cancer.
  
  Therefore, the aim of this project is two-fold: (1) use data acquired in our earlier projects to create a holistic integration of the aforementioned sources and levels for modeling pancreas cancer, which we call Network-of-Networks or short: NoN (in our context networks of different -omics levels, such as genomics, transcriptomics, proteomics and metabolomics. (2) A NoN is a very large and complex object and its structure differs significantly from other biological networks. Thus, new methods for complexity reduction and analyzing NoNs will be developed in this project.

  The goal

  In this project we aim to develop a new method that can be used to solve this task: the identification of minimal, yet robust fingerprints from very high-dimensional, noisy -omics data. Our method will be based on ideas from the areas of compressed sensing and machine learning.
  
  http://medicalbioinformatics.de/research/projects/ecmath-ch7

• CH8

  X-ray based anatomy reconstruction with low radiation exposure

  Hon.-Prof. Hans-Christian Hege / Dr. Martin Weiser / Dr.-Ing. Stefan Zachow

  Projektleiter: Hon.-Prof. Hans-Christian Hege / Dr. Martin Weiser / Dr.-Ing. Stefan Zachow
  Projekt Mitglieder: Dennis Jentsch
  Laufzeit: -
  Status: beendet
  Standort: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Beschreibung

  Medical imaging is essential in diagnostics and surgery planning. For representation of bony structures different imaging modalities are used; the leading methods are X-ray projection (projectional radiography) and CT. Disadvantage of these imaging techniques is the ionization caused by X-rays, particularly in CT, where the dose is 250-500 times higher than in classic X-ray projection. From the clinical perspective therefore one would like to replace CT acquisitions by a few possible X-ray projections. The project deals with the ill-posed inverse problem of 3D reconstruction of bony structures from 2D radiographs. Virtual radiographs are generated from virtual bone structure models; these are compared with clinical patient images and incrementally changed until a sufficiently accurate bone model is found whose virtual projections fit to the measured data. By using a statistical shape model as prior knowledge it is possible to formulate a well-posed optimization problem in a Bayesian setting. Using gradient methods and multilevel/multiresolution methods for both the reconstruction parameters and image data, good computational performance is achieved. Uncertainty quantification techniques can be applied to describe the spatially varying accuracy of the reconstructed model. Finding best X-ray projections (recording directions) minimizing both uncertainty and radiation exposure leads to a design of experiments problem. Two flavors of this design optimization are considered: An all-at-once approach finding the best image acquisition setup before any X-ray projections are performed, and a sequential approach determining the best next projection direction based on the accumulated knowledge gained from the previously taken images.
  
  http://www.zib.de/projects/x-ray-based-anatomy-reconstruction-low-radiation-exposure

• CH9

  Adaptive algorithms for optimization of hip implant positioning

  Dr. Martin Weiser / Dr.-Ing. Stefan Zachow

  Projektleiter: Dr. Martin Weiser / Dr.-Ing. Stefan Zachow
  Projekt Mitglieder: Marian Moldenhauer
  Laufzeit: -
  Status: beendet
  Standort: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Beschreibung

  This project aims at a software environment supporting computer-assisted planning for total hip joint replacement by suggesting implant positions optimized for longevity of bone implants. The aim is to pre-operatively assess stress distribution in bone and to determine an optimal implant position with respect to natural function and stress distribution to prevent loosening, early migration, stress shielding, undesired bone remodeling, and fracture. Increasing the longevity of implants will help to enhance quality of life and reduce the cost of health care in aging societies. Focus of the research is the development of efficient optimization algorithms by adaptive quadrature of the high-dimensional space of daily motions and appropriate choice of tolerances for the underlying dynamic contact solver.
  
  http://www.zib.de/projects/adaptive-algorithms-optimization-hip-implant-positioning

• CH10

  Analysis and numerics of the chemical master equation

  Prof. Dr. Harry Yserentant

  Projektleiter: Prof. Dr. Harry Yserentant
  Projekt Mitglieder: -
  Laufzeit: 01.06.2014 - 31.05.2017
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes the stochastic effects into account that cannot be neglected in the case of small population numbers.
  
  There is an ongoing effort to tackle the chemical master equation numerically. The major challenge is its high dimensionality: for a system of d interacting species the chemical master equation is a differential equation with state space N_0^d, N_0 the set of nonnegative integers.
  
  The main goal of project A-CH10 is build a sound mathematical basis for the numerical approximation of the chemical master equation and to put numerical methods for this equation on a firm mathematical ground.
  
  http://www.tu-berlin.de/?id=168383

• CH-AP2

  Genealogies and inference for populations with highly skewed offspring distributions under further evolutionary forces

  Prof. Dr. Jochen Blath

  Projektleiter: Prof. Dr. Jochen Blath
  Projekt Mitglieder: -
  Laufzeit: 01.10.2012 - 30.06.2018
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  Multiple merger coalescent modeling and analysis has up to now been mainly focused on neutral, haploid, single-locus set-ups. The central aim of this project is to develop the stochastic models, theoretical results and inference methods required to effectively describe and analyse the observed patterns of genetic variation in sequence data in real populations with skewed offspring distributions under the influence of further evolutionary forces, especially recombination, selection and population structure; in other words, the systematic development of the basics of a `mathematical population genetics for highly variableoffspring distributions'. Given recent progress in DNA sequencing technology, and insight in the limitations of inference methods based single locus set-ups, particular emphasis will be put on realistic diploid multi-locus models and the corresponding statistical machinery for data analysis.
  
  http://www.dfg-spp1590.de/abstracts.php#5

• CH-AP12

  pcCells - physicochemical principles of cellular information processing

  Prof. Dr. Frank Noé

  Projektleiter: Prof. Dr. Frank Noé
  Projekt Mitglieder: -
  Laufzeit: 01.01.2013 - 31.12.2017
  Status: beendet
  Standort: Freie Universität Berlin
  

  Beschreibung

  Biological cells are able to perform complex signal transduction tasks quickly, energy-efficiently and yet in a robust and noise-tolerant manner. These signal transduction tasks rely on intracellular information processing mechanisms in which chemical signals are sent, transmitted and received and the state of the overall machinery is stored in chemical or conformational switches. The physicochemical principles of information processing in cells is still not well understood, owing to fundamental restrictions in resolution in experiments and in sampling of molecular dynamics simulations. Here, we will develop new simulation methods based on adaptive molecular dynamics and Markov models. These methods, together with new statistical mechanical theories and single-molecule experimental analyses will be employed to investigate the molecular basis of intracellular signal processing mechanisms. Central to our proposal is the hypothesis that intracellular signal processing relies on spatiotemporal order of molecules arising from dynamical sorting. This hypothesis will be tested using examples of protein-ligand and protein-protein sorting in neuronal signalling. The proposed project is highly multidisciplinary, involving physical chemistry, computer science, mathematics and biology.
  
  http://compmolbio.biocomputing-berlin.de/index.php/projects/80-erc

• CH-AP16

  Projection theory of transfer operators

  Prof. Dr. Christof Schütte / PD Dr. Marcus Weber

  Projektleiter: Prof. Dr. Christof Schütte / PD Dr. Marcus Weber
  Projekt Mitglieder: -
  Laufzeit: 01.01.2014 - 31.12.2017
  Status: beendet
  Standort: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Beschreibung

  The main object is analysing sufficent ways to compute a galerkin approximation of the transfer operator. This includes to study the theoretical properties of a galerkin approximation of specific systems and to development algorithms which guarantee those properties for the numerical approximation. Furthermore, one is interested in making this computation as cheap as possible.
  
  http://www.zib.de/projects/projection-theory-transfer-operators

• CH-AP19

  Meth4SysPharm: Modeling methods for systems pharmacology and application to HIV-1

  Dr. Max von Kleist

  Projektleiter: Dr. Max von Kleist
  Projekt Mitglieder: -
  Laufzeit: 01.05.2015 - 30.04.2019
  Status: beendet
  Standort: Freie Universität Berlin
  

  Beschreibung

  'Systems pharmacology' denotes the application of systems biology approaches to research questions arising in pharmacology. The aim is to understand the interaction of drugs with com- plex biological networks and to use this knowledge to develop- and improve medical therapy. Within the proposed project we will address unsolved mathematical challenges arising from this novel, interdisciplinary approach. We will integrate knowledge and data from three subtopics in order to study the mechanisms of drug resistance development in HIV-1, as a model system. In close cooperation with experts from the respective fields, the systems' response to drug interference, in terms of 'evolutionary dynamics' will be assessed alongside with the temporal resolution of drug interference ('drug action and pharmacokinetics') and their implications for the 'optimal use of therapy'. The proposed research program is expected to provide methodologi- cal advance that allows projecting these interrelations into measurable clinical outcomes, while addressing a relevant medical problem at the same time.
  
  http://page.mi.fu-berlin.de/vkleist/CurrentProjects.htm

• CH-AP20

  Integrative mathematical modeling of physiological- and molecular factors of osteoarthritis of the knee

  Dr. Max von Kleist

  Projektleiter: Dr. Max von Kleist
  Projekt Mitglieder: -
  Laufzeit: 01.05.2015 - 30.04.2019
  Status: beendet
  Standort: Freie Universität Berlin
  

  Beschreibung

  Osteoarthritis of the knee (OAK) is a complex multi-factorial condition that is characterised by a lack of hyaline cartilage self repair 1, inflammation & pain 2. As for many other multi-factorial conditions, computational models may be useful tools of direct clinical relevance that allow studying the interaction of putative factors. Within this subproject, we want to develop a comprehensive computational model of cartilage homeostasis that will help us to understand and to evaluate the onset and progression of OAK. While the underlying mechanisms of OAK remain unknown, several factors have been previously associated with osteoarthritis and studied in isolation, such as cell density-dependent extracellular matrix (EM) generation 3,4, EM metabolism 5, the effect of nutrient gradients 5,6 and the influence of (mechano-) growth factors 7.
  
  Our research group has a broad expertise in interdisciplinary research in biomedicine 8,11 with a particular focus on mechanistic mathematical modelling 8,9 including in vitro to in vivo extrapolation, as well as the analysis of complex clinical samples 10. Within this consortium, we aspire to combine biochemical data from PrevOP subprojects SP1-3 (inflammation, cartilage self-repair & pain) and OVERLOAD projects SP5, 7-8 (fluid transport, mechano-sensitive signalling, cartilage self-repair), to successively develop mathematical models of cartilage homeostasis. Particularly, integration of results from OVERLOAD SP5 may allow to couple image-derived clinical data to metabolic events in the cartilage. The developed comprehensive model of OAK will further our understanding of the disease and the interplay of the mentioned factors, provide insights into disease mechanisms and strategies for its prevention (like, e.g. physical training). The aim of the project is thus to provide a translational framework between in vitro bio-molecular studies, ex vivo analysis, animal models and human patients with OAK (clinical projects in PrevOP/OVERLOAD).
  
  http://overload-prevop.charite.de/verbund/m_v_kleist_cschuette/

• CH-AP24

  Free Boundary Problems and Level Set Methods

  Prof. Dr. Michael Hintermüller

  Projektleiter: Prof. Dr. Michael Hintermüller
  Projekt Mitglieder: -
  Laufzeit: 01.05.2011 - 31.03.2018
  Status: beendet
  Standort: Humboldt Universität Berlin
  

  Beschreibung

  Project part FREELEVEL will focus on two research streams: (i) shape and topological sensitivity-based solvers in tomography and (ii) the extension of spatially adapted regularization to more general image restoration problems, e.g., involving blind deconvolution, and non-convex regularization. Concerning tomography problems, a level-set-based algorithm relying on shape and extended topological sensitivities will be realized for FDOT and MIT, respectively. For MIT, first a reduced model leading to an elliptic PDE-system will be studied and, in a next step, the full Maxwell system will be taken into account. For numerical efficiency purposes, a shape-aware adaptive finite element method will be intertwined with the level-set solver (partly with FEMBEM). In the area of image restoration, we are motivated by optical diffusion tomography problems for detecting objects located behind turbid media and by convolution identification in dual-MR techniques (MRI). We formulate these problems in terms of blind deconvolution, preferably with non-convex regularization with respect to the image. The resulting problems will be studied and solved numerically. With respect to the latter - split Bregman - iteratively re-weighted total variation and semismooth Newton solvers will be investigated. Further, motivated by sparse magnetic resonance imaging, problems in compressed sensing with convex (OPTIM) and non-convex relaxation (MRI) of the 0-norm will be treated. The latter is interesting as there is evidence that non-convex regularization goes along with a possible reduction in acquired data. Further, FREELEVEL will focus on topics supporting other projects, such as piecewise polynomial Mumford-Shah based image segmentation using topological sensitivities (INVERSE).
  
  Within the SFB, FREELEVEL acts as a center of expertise for shape and topological sensitivity-based level-set solvers for tomography problems. Moreover, FREELEVEL contributes expertise and software in image restoration to the SFB through various cooperations with practitioners (MRI) as well as the applied mathematics group within the SFB (OPTIM, INVERSE).
  
  http://math.uni-graz.at/mobis/freelevel.html

• CH-TU23

  Tractable recovery of multivariate functions from limited number of samples

  Dr. Jan Vybiral

  Projektleiter: Dr. Jan Vybiral
  Projekt Mitglieder: -
  Laufzeit: 01.06.2014 - 30.04.2015
  Status: beendet
  Standort: Technische Universität Berlin
  

• CH-AP1

  Interacting stochastic partial differential equations, combinatorial stochastic processes and duality in spatial population dynamics

  Prof. Dr. Jochen Blath

  Projektleiter: Prof. Dr. Jochen Blath
  Projekt Mitglieder: -
  Laufzeit: 01.03.2013 - 30.09.2015
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  The method of duality is a mathematical formalism that allows one to establish close connections between two stochastic Markov processes with respect to a class of `duality functions'. If a formal duality is established, it is often possible to study important properties of a `complicated' spatial stochastic system, such as longtime-behaviour or properties of its genealogy, by analysing the properties of a simpler, typically discreteor combinatorial, dual process. This method has been used with great success for many processes in the theory of interacting particle systems and interacting stochastic (P)DEs modeling the evolution of populations (e.g. the stepping stone or the Wright-Fisher model). In the last years, important progress has been achieved. However, there is still no systematic theory of duality (“finding dual processes is something of a black art", A. Etheridge [Eth06] p.519), and many systems of theoretical and practical importance await further analysis. This project has three main objectives. Firstly, we would like to transfer several concrete questions about certain SPDEs to questions about their dual processes (I). Secondly, we are interested in the long-term properties of the dual processes themselves (II). Finally, we aim towards a systematic analysis of the method of duality.
  
  http://www.dfg-spp1590.de/abstracts.php#27

• CH-AP3

  Multiple testing under unspecified dependency structure

  Dr. Thorsten Dickhaus

  Projektleiter: Dr. Thorsten Dickhaus
  Projekt Mitglieder: -
  Laufzeit: 01.04.2012 - 31.03.2015
  Status: beendet
  Standort: Humboldt Universität Berlin
  

  Beschreibung

  Multiple hypotheses testing has emerged as one of the most active research fields in statistics over the last 10-15 years, contributing at present approximately 8% of all articles in the four leading methodological statistics journals (data from Benjamini, 2010). This growing interest is especially driven by large-scale applications, such as in genomics, proteomics or cosmology. Many new multiple type I and type II error criteria like the meanwhile quite popular “false discovery rate” (FDR) have recently been propagated and published together with explicit algorithms for controlling them. A broad class of these methods employs marginal test statistics or p-values, respectively, for each individual hypothesis and a set of critical constants with which they have to be compared. Up to now, only under joint independence of all marginal p-values the behaviour of such methods is understood well. Moreover, under unspecified dependence the type I error level is often not kept accurately or not fully exhausted. This holds true especially for the FDR or related measures and offers room for improvements of those procedures with respect to type I error control and power. An adaptation to the dependency structure can therefore lead to a gain in validity (type I error rate is kept accurately) and efficiency (quantified by multiple power measures). In this project, a general theory of the usage of parametric copulae methods in this multiple testing shall be developed. This will be flanked by structural assumptions regarding the multivariate distribution of p-values reducing the complexity of the problem, for instance, the dimensionality of the copula parameter. Moreover, we will develop resampling techniques for empirical calibration of multiple testing thresholds in the case of unspecified dependency.
  
  https://www.mathematik.hu-berlin.de/de/for1735/projects_old/multipleTesting

• CH-AP4

  Statistical inference methods for behavioral genetics and neuroeconomics

  Dr. Thorsten Dickhaus

  Projektleiter: Dr. Thorsten Dickhaus
  Projekt Mitglieder: -
  Laufzeit: 01.07.2013 - 31.03.2015
  Status: beendet
  Standort: Humboldt Universität Berlin
  

  Beschreibung

  The proposed project contributes to fundamental research in behavioral genetics and neuroeconomics by developing refined statistical inference methods for data generated in these fields. In particular, techniques for multiple hypotheses testing will be refined, adapted and newly worked out. Multiple tests are needed in behavioral genetics in order to analyze associations between many genetic markers and behavioral phenotypes simultaneously. In neuroeconomics, high-dimensional and spatially clustered functional magnetic resonance imaging time series have to be analyzed with multiple testing techniques. We will apply the methods resulting from the research in this project to risk preference and genetics data that we have compiled in prior work. Furthermore, our methodological contributions will be applicable in many other fields, too: High-dimensional categorical data are also prevalent, for example, in genetic epidemiology and high-dimensional hierarchical data structures occur for instance in spatial statistics or in the context of the analysis of variance with many groups.
  
  http://gepris.dfg.de/gepris/projekt/239049500

• CH-AP5

  EPILYZE - DNA Methylierungs-Signaturen als innovative Biomarker für die quantitative und qualitative Analyse von Immunzellen, Subproject C

  Dr. Thorsten Dickhaus

  Projektleiter: Dr. Thorsten Dickhaus
  Projekt Mitglieder: -
  Laufzeit: 01.12.2012 - 31.03.2015
  Status: beendet
  Standort: Humboldt Universität Berlin
  

  http://foerderportal.bund.de/foekat/jsp/SucheAction.do?actionMode=view&fkz=031A191A#

• CH-AP6

  Numerische Analysis Hamiltonscher partieller Differentialgleichungen und hochdimensionaler Probleme

  Dr. Ludwig Gauckler

  Projektleiter: Dr. Ludwig Gauckler
  Projekt Mitglieder: -
  Laufzeit: 01.06.2014 - 31.05.2016
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  Numerical discretizations of Hamiltonian partial differential equations and differential equations in high dimensions shall be analysed in the project. On the one hand, qualitative properties of numerical methods for the discretization in time such as splitting and Runge-Kutta methods will be investigated. In particular, we will pursue the question if and on which time intervals a numerical method is able to reproduce the stability of waves, which is studied in detail in the mathematical analysis of the equations. On the other hand, the analysis of approximations in high spatial dimensions will be the second key activity in the project. Approximations on tensor manifolds shall be analysed with respect to their approximation properties, but also their long-time behaviour. Such approximations are used successfully in quantum dynamics in the case of the high dimensional linear Schrödinger equation. In addition, the convergence of numerical methods for the chemical master equation, an important equation in biology and chemistry, will be studied on the basis of recent regularity results.
  
  http://www.tu-berlin.de/?id=149224

• CH-AP11

  Wear Simulation of Knee Implants and Shape Optimization for Patient-group specific Wear Minimization

  Prof. Dr. Ralf Kornhuber / Dr. Martin Weiser

  Projektleiter: Prof. Dr. Ralf Kornhuber / Dr. Martin Weiser
  Projekt Mitglieder: -
  Laufzeit: 01.07.2013 - 30.12.2016
  Status: beendet
  Standort: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Beschreibung

  For the market admittance of joint implants, a standardized wear test has to be performed. During the design phase, similar tests are necessary as well. Those tests are very cost and time expensive. The project aims at the development of simulation and optimization methods for substituting some of the design phase tests by simulations. Additionally, the design process shall be accelerated by shape optimization, and the offered implants be tailored to the patient population by taking different patient groups into account.
  
  Focus of the work at ZIB is the long-time integration of wear trajectories. The implant geometry is modified due to wear, which in turn changes the wear rate. The evolution is determined by the wear of one load cycle, the simulation of which is computationally expensive. We develop adaptive methods for controlling tolerance, order, and time step for an efficient simulation of many load cycles.
  
  http://www.zib.de/projects/wear-simulation-knee-implants-and-shape-optimization-patient-group-specific-wear-minimization

• CH-AP13

  Adaptive Konformationsdynamik mit Anwendung auf Rhodopsinaktivierung

  Prof. Dr. Frank Noé

  Projektleiter: Prof. Dr. Frank Noé
  Projekt Mitglieder: -
  Laufzeit: 01.07.2012 - 30.06.2015
  Status: beendet
  Standort: Freie Universität Berlin
  

  Beschreibung

  Rare molecular events such as folding of proteins or nucleic acids, ligand binding, conformational changes or macromolecular aggregation are the basis of all life processes. Besides experimental techniques, molecular dynamics (MD) simulation is an established tool to analyze such processes. However, the usefulness of MD for investigating biological processes is limited by the sampling problem: Due to the high computational effort involved in simulating biomolecules at atomistic resolution, the accessible simulation times are much too short to find the biologically relevant conformations and make statistically reliable statements about transition rates. This problem also hinders the improvement of molecular models towards the reliable prediction of experimental observables. In the proposed work we will develop an adaptive conformation dynamics (ACD) which facilitates the simulation of slow biomolecular processes on small CPU clusters using atomistic models. This method will be applied in order to elucidate the detailed structural mechanism of the activation of the G-protein coupled receptor Rhodopsin.
  
  http://compmolbio.biocomputing-berlin.de/index.php/projects/93-dfg825-3-1

• CH-AP15

  Generalized tensor methods in quantum chemistry

  Prof. Dr. Reinhold Schneider

  Projektleiter: Prof. Dr. Reinhold Schneider
  Projekt Mitglieder: -
  Laufzeit: 01.06.2013 - 31.05.2016
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  The computation of the electronic structure is of utmost importance for the task of molecular engineering in modern chemistry and material science. In this context, the accurate computation of the electron correlation is a fundamental and extremely difficult problem. In contrast to the tremendous progress made in calculating weakly correlated systems by Density Functional Theory (DFT) for extended systems or Coupled Cluster Methods for highly accurate calculations, there are two major types of systems for which current quantum chemical methods have deficiencies: (1) Open-shell systems with a large number of unpaired electrons, as they occur in multiple transition metal complexes or in molecular magnets; (2) Extended or periodic systems without a band gap, where the limit of the applicability of the available size consistent methods is reached.The aim of this proposal is to develop a general tensor network state (TNS) based algorithm that can be applied efficiently to these open problems of quantum chemistry. Realization of such an algorithm relies on carrying out a variety of complex tasks. Several new formal methods and methodological concepts of tensor decompositions will have to be designed to comply with the specific, nonlocal nature of the Hamiltonian, and to this end, the applicants will join their rather complementary expertise regarding the powerful DMRG method and similar recent developments from physics, mathematics and information technology. To arrive at an efficient implementation of the quantum chemistry TNS algorithm, our contributions will be implemented and tested based on existing program structures of the QC-DMRG- Budapest [Legeza-2011] and the TTNS-Vienna [Murg-2010c] codes.
  
  http://gepris.dfg.de/gepris/projekt/234056486?language=en

• CH-AP17

  The mathematical analysis of interacting stochastic oscillators

  Prof. Dr. Wilhelm Stannat

  Projektleiter: Prof. Dr. Wilhelm Stannat
  Projekt Mitglieder: -
  Laufzeit: 01.11.2011 - 30.04.2016
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  State-the-art, own contribution: Rigorous mathematical models for spatialy extended neurons and neural systems under the influence of noise will be developed and analysed using the mathematical theory of stochastic evolution equations, in particular stochastic partial differential equations (see [6]). We will take into account thermal noise modelling local exterior forces acting on a couple of adjacent neurons but also parametric noise modelling uncertainties in the parameters. The impact of noise on the whole system will then be analyzed rigorously, to quantify, e.g., the probability for the propagation failure of an action potential. There are only few applications of the mathematical theory of stochastic evolution equations to neural systems subject to noise (see [1,2,8,10]). In particular, the recent developments of the theory based on the semigroup approach for mild solutions and the analysis of the associated Kolmogorov operator (see [9]) has so far only been applied to stochastic FitzHugh Nagumo systems in [4,5].
  
  Cited references:

  • [1] Albeverio S, Cebulla C (2007) Synchronizability of Stochastic Network Ensembles in a Model of Interacting Dynamical Units. Physica A Stat. Mech. Appl. 386, 503-512.
  • [2] Albeverio S, Cebulla C (2008) Synchronizability of a Stochastic Version of FitzHugh-Nagumo Type Neural Oscillator Networks, Preprint, SFB 611, Bonn.
  • [3] Blömker D (2007) Amplitude Equations for Stochastic Partial Differential Equations, World Scientific, New Jersey.
  • [4] Bonaccorsi S, Marinelli C, Ziglio G (2008) Stochastic FitzHugh-Nagumo equations on networks with impulsive noise, EJP 13, 1362-1379.
  • [5] Bonaccorsi S, Mastrogiacomo E (2007) Analysis of the stochastic FitzHugh-Nagumo system, Technical Report UTM 719, Mathematics, Trento, arXiv:0801.2325.
  • [6] Da Prato G, Zabczyk, J (1992) Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge.
  • [7] Es-Sarhir A Stannat W (2008) Invariant measures for Semilinear SPDE's with local Lipschitz Drift Coefficients and applications, Journal of Evolution Equations 8, 129-154.
  • [8] Kallianpur G, Wolpert R (1984) Infinite dimensional stochastic differential equation models for spatially distributed neurons, Appl. Math. Optim. 12, 125-172.

  
  
  https://www.bccn-berlin.de/Research/Projects_II/Branch_A/A11/

• CH-AP18

  Numerical analysis and simulation of cooperative phenomena in interacting stochastic oscillators

  Prof. Dr. Wilhelm Stannat

  Projektleiter: Prof. Dr. Wilhelm Stannat
  Projekt Mitglieder: -
  Laufzeit: 01.05.2013 - 30.04.2016
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  State-the-art, own contribution: In contrast to the case of deterministic reaction diffusion systems there are only few publications on the numerical analysis of stochastic reaction diffusion systems arising in neuroscience, like e.g. stochastic FitzHugh Nagumo systems (see [2,6,7]). From the neuroscience perspective in particular the numerical analysis of stochastic reaction diffusion systems exhibiting various spatial patterns based e.g. on partial synchronization are of interest (see [8] and references therein). From the mathematical viewpoint a major difficulty comes from the fact that the coefficients of the systems typically only satisfy a one-sided Lipschitz condition that cannot be controlled easily if perturbed with stochastic forcing terms. Recent results in the numerical analysis of stochastic differential equations with non-Lipschitz coefficients show that their numerical approximation has to be carried out with additional care (see [3]) in order to validate simulation results. We will be therefore interested in development and rigorous mathematical analysis of the numerical approximation of stochastic reaction diffusion systems in the excitable regime. There is a considerable amount of work in the physics literature on the influence of noise in excitable reacton diffusion systems (see [5] for a survey). On the other hand there is only a limited quantitative understanding of the influence of the stochastic forcing terms on the various effects like wave speed or nucleation of wave patterns. Certainly, spatial correlation of the noise terms will play a crucial role, which will be studied also systematically within this project.
  
  Cited references:

  • [1] Dahlem M A, Graf R,Strong A J, Dreier J P, Dahlem Y A, Sieber M, Hanke W, Podoll K, Schöll E (2010) Two-dimensional wave patterns of spreading depolarization: Retracting, re-entrant, and stationary waves, Physica D 239, 889-903.
  • [2] DeVille R E L, Vanden-Eijnden E (2007) Wavetrain response of an excitable medium to local stochastic forcing, Nonlinearity 20, 51-74.
  • [3] Hutzenthaler M, Jentzen A, Kloeden P E (2011) Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467, 1563–1576.
  • [4] Laing C, Lord G J (Eds.), Stochastic Methods in Neuroscience (2010) Oxford University Press, Oxford.
  • [5] Lindner B, Garcia-Ojalvo J, Neiman A, Schimansky-Geier L (2004), Effects of noise in excitable systems, Physics Reports 392, 321-424.
  • [6] Shardlow T (2005) Numerical simulation of stochastic PDEs for excitable media, J. Comput. Appl. Math. 175, 429-446.
  • [7] Shardlow T (2004) Nucleation of waves in excitable media, Multiscale Model. Simul. 3, 151-167.
  • [8] Tass P (1999) Phase Resetting in Medicine and Biology - Stochastic Modelling and Data Analys, Springer, Berlin.

  
  
  https://www.bccn-berlin.de/Research/Projects_II/Branch_A/A12/

• CH-AP21

  pH-dependent opioids

  PD Dr. Marcus Weber

  Projektleiter: PD Dr. Marcus Weber
  Projekt Mitglieder: -
  Laufzeit: 01.10.2012 - 30.06.2016
  Status: beendet
  Standort: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Beschreibung

  The goal of this project is design of pain relief drugs (opioids), which should be active only in inflamed tissue and therefore have reduced side effects compared to conventional opioids. We managed to develop a candidate, which was synthesized by the ASCA GmbH in Berlin. The opioid is currently undergoing in-vivo and in-vitro experiments at the Charité Berlin.
  
  http://www.zib.de/projects/ph-dependent-opioids

• CH-AP22

  Transformation products of trace pollutants

  PD Dr. Marcus Weber

  Projektleiter: PD Dr. Marcus Weber
  Projekt Mitglieder: -
  Laufzeit: 01.11.2011 - 31.10.2014
  Status: beendet
  Standort: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Beschreibung

  Das am 01. November 2011 gestartete Projekt TransRisk richtet den Blick besonders auf Transformationsprodukte, die durch oxidativen Abbau aus Spurenstoffen hervorgehen. Das daraus entstehende Risiko wird genauer analysiert und in ein handlungsorientiertes Risikomanagementkonzept integriert. Um einen weitergehenden Abbau von Spurenstoffen und eine Minimierung der Bildung von Transformationsprodukten zu erreichen, werden in TransRisk verschiedene Verfahrenskombinationen aus konventionellen Aufreinigungsverfahren wie z.B. Nitrifikation mit erweiterten Behandlungstechniken wie beispielsweise Ozonung und Aktivkohlefiltration kombiniert. Darüber hinaus werden aber auch neue Verfahren wie die Verwendung von Eisenbakterien in der biologischen Abwasserreinigung detailliert untersucht. Weitere Schwerpunkte von TransRisk sind neu aufkommende Krankheitserreger und die antibiotikaresistenten Keime. Hierbei werden neue Nachweismethoden entwickelt, um die Verbreitung dieser Bakterien besser zu verstehen und geeignete Maßnahmen einleiten zu können. Die erzielten Projektergebnisse werden in der Modellregion Donauried mit den Betroffenen vor Ort diskutiert und – soweit möglich – auch umgesetzt. TransRisk ist ein Verbundprojekt, welches sich aus insgesamt 15 Teilprojekten von 14 Institutionen wie Universitäten, Wasserversorgern, Verbänden, Industrie und Forschungseinrichtungen zusammensetzt. TransRisk wird durch das Bundesministerium für Bildung und Forschung (BMBF) im Förderschwerpunkt „NaWaM - Nachhaltiges Wassermanagement“ im Rahmen der Fördermaßnahme „RiSKWa - Risikomanagement von neuen Schadstoffen und Krankheitserregern im Wasserkreislauf“ gefördert. Der Förderschwerpunkt NaWaM bündelt die Aktivitäten des BMBF im Bereich der Wasserforschung innerhalb des BMBF-Rahmenprogramms „FONA - Forschung für nachhaltige Entwicklungen“.
  
  http://www.transrisk-projekt.de/TRANSRISK/DE/01_Home/home_node.html

• CH-AP23

  Regularity, complexity, and approximability of electronic wavefunctions

  Prof. Dr. Harry Yserentant

  Projektleiter: Prof. Dr. Harry Yserentant
  Projekt Mitglieder: -
  Laufzeit: 01.10.2013 - 30.09.2016
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  The project considers the electronic Schrödinger equation of quantum chemistry that describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. Solutions of this equation depend on 3N variables, three spatial dimensions for each electron. Approximating the solutions is thus inordinately challenging. It is conventionally believed that the accuracy cannot be systematically improved without the effort truly exploding for larger numbers of electrons and that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach for the approximation of the solutions. Results of the applicant indicate that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that of a system of two electrons or even only one electron. Goal of the project is to extend and refine these results and to identify structural properties of the wavefunctions that could ideally enable breaking the curse of dimensionality and to develop the present approximation methods further to true discretications of the Schrödinger equation.
  
  http://www.dfg-spp1324.de/abstracts.php?lang=de#20

• CH-AP26

  Branching random walks in random environment with a special focus on the intermittent behavior of the particle flow

  Prof. Dr. Wolfgang König

  Projektleiter: Prof. Dr. Wolfgang König
  Projekt Mitglieder: -
  Laufzeit: 01.04.2013 - 31.08.2016
  Status: beendet
  Standort: Weierstraß-Institut
  

  Beschreibung

  We study the long-time behaviour of branching random walk in random environment (BRWRE) on the d-dimensional lattice. We consider one of the basic models, which includes migration and branching/killing of the particles, given a random potential of spatially dependent branching/killing rates. Based on the observation that the expectation of the population size over the migration and the branching and killing is equal to the solution to the well-known and much-studied parabolic Anderson model (PAM), we will use our understanding of the long-time behaviour of the PAM to develop a detailed picture of the BRWRE. Furthermore, we will exploit methods that were successful in the treatment of the PAM to prove at least part of this picture. Particular attention is payed to the study of the concentration of the population in sites that determine the long-time behaviour of the PAM, which shows a kind of intermittency. One fundamental thesis that we want to make precise and rigorous is that the overwhelming contribution to the total population size of the BRWRE comes from small islands where most of the particles travel to and have a extremely high reproduction activity. We aim at a detailed analysis for the case of the random potential being Pareto-distributed, in which case the rigorous study of the PAM has achieved a particularly clear picture. This project has the following four main goals. I. For a variety of random potentials, we derive large-time asymptotics for the n-th moments of the local and total population size, based on techniques from the study of the PAM. II. We want to understand and identify the limiting distributions of the global population size by a finer analysis for Pareto-distributed potentials. III. We want to investigate, for Pareto-distributed potentials, the long-time (de)correlation properties of the evolution of the particles such as aging, in particular, slow/fast evolution phenomena and what type of aging functions will appear. IV. We want to study, for Paretodistributed potentials, almost surely with respect to the potential, the particle flow of the BRWRE in a geometric sense by finding trajectories along which most of the particles travel and branch, in particular the sites and the time intervals where, respectively when, most of the particles show an extremely high reproduction activity.
  
  http://www.dfg-spp1590.de/abstracts.php#34

• CH-AP27

  Application of rough path theory for filtering and numerical integration methods

  Prof. Dr. Peter Karl Friz / Prof. Dr. Wilhelm Stannat

  Projektleiter: Prof. Dr. Peter Karl Friz / Prof. Dr. Wilhelm Stannat
  Projekt Mitglieder: -
  Laufzeit: 01.11.2011 - 31.10.2014
  Status: beendet
  Standort: Technische Universität Berlin
  

  Beschreibung

  In 1998 T. Lyons (Oxford) suggested a new approach for the robust pathwise solution of stochastic differential equations which is nowadays known as the rough path analysis. Based on this approach a new class of numerical algorithms for the solution of stochastic differential equations have been developed. Recently, the rough path approach has been successfully extended also to stochastic partial differential equations. In stochastic filtering, the (unnormalized) conditional distribution of a Markovian signal observed with additive noise is called the optimal filter and it can be described as the solution of a stochastic partial differential equation which is called the Zakai equation. In the proposed project we want to apply the rough path analysis to a robust pathwise solution of the Zakai equation in order to construct robust versions of the optimal filter. Subsequently, we want to apply known algorithms based on the rough path approach to the numerical approximation of these robust estimators and further investigate their properties.
  
  http://www.dfg-spp1324.de/abstracts.php?lang=de#8