Drittmittelprojekte
DFG - Priority Program (SPP 2265) Random Geometric systems
Project: Scaling limits of evolving spanning trees and of random walks on evolving spanning trees
Anita Winter
In this project we study scaling limits of evolving uniform spanning trees (UST) on further classes of networks (including Erdös-Renyi graphs, sequences of densely connected expander graphs and low-dimensional tori). The main motivation comes from modeling large and sparsely connected networks. Trees are the extreme cases of sparsely connected networks. In real world networks, the structure of the network might change over time. One emphasis of the project concerns a particular network dynamics. This is the Aldous-Broder algorithm which is a tree-valued stochastic process that generates the UST. A random walk is a simple stochastic process on a network which allows to explore the structure of the network. In the context of communication networks (e.g.\ internet, wifi) it can be understood as a message sent from device to device. In the current research random walks on dynamic network models are compared with random walks on static networks. In this project we determine the space and time scales on which the random walk has a diffusive scaling limit.
Project: Spatial growth and information exchange in evolving environments and on evolving networks
Anton Klimovsky
What is the effect of the space-time varying environment on the long time behavior of spatially structured populations of interacting particles/individuals/ agents? This question is of high relevance, e.g., in life and social sciences/economics, computer science, artificial intelligence. Mathematically, the project focuses on two phenomenological models of interacting particle systems: (1) branching Brownian motion, which models population growth/ spatial spreading; and (2) voter model, which models information exchange in a population of agents. The novel features that this project introduces into these classical models are: (1) space-time-correlated environments and (2) evolving networks. These play the role of the geographic spaces and substantially change the underlying spatial geometry. The project aims at investigating (1) growth vs. extinction, population size, spread; and (2) clustering vs. consensus of agents, space-time scaling limits of stochastic processes on evolving networks and evolving graph limits.
DFG - Research grant Deep neural networks overcome the curse of dimensionality in the numerical approximation of stochastic control problems and of semilinear Poisson equations
Martin Hutzenthaler (with Thomas Kruse, Universität Gießen)
Partial differential equations (PDEs) are a key tool in the modeling of many real world phenomena. Several PDEs that arise in financial engineering, economics, quantum mechanics or statistical physics are nonlinear, high-dimensional, and cannot be solved explicitly. It is a highly challenging task to provably solve such high-dimensional nonlinear PDEs approximately without suffering from the so-called curse of dimensionality. Deep neural networks (DNNs) and other deep learning-based methods have recently been applied very successfully to a number of computational problems. In particular, simulations indicate that algorithms based on DNNs overcome the curse of dimensionality in the numerical approximation of solutions of certain nonlinear PDEs. For certain linear and nonlinear PDEs this has also been proven mathematically. The key goal of this project is to rigorously prove for the first time that DNNs overcome the curse of dimensionality for a class of nonlinear PDEs arising from stochastic control problems and for a class of semilinear Poisson equations with Dirichlet boundary conditions.
DFG - project Recursive and sparse approximation in reinforcement learning with applications
Denis Belomestny with Ch. Bayer and V. Spokoiny (WIAS)
Reinforcement learning (RL) is an integral part of Machine Learning concerned with controlling a system to maximize a performance measure that expresses a long-term objective. Reinforcement learning is of great interest because of the many practical applications, ranging from problems in artificial intelligence to operations research or financial mathematics. With recent breakthroughs in deep learning, deep reinforcement learning (DRL) demonstrates notable success in solving highly challenging problems. The DRL algorithms are compelling, but it remains an open issue how to relate the architecture of the networks involved to the structure of the underlying dynamic programming algorithm. Moreover, in DRL, the approximate dynamic programming algorithm involves solving a highly nonconvex statistical optimization problem. As an alternative to conventional deep neural network approximations in each backward step, one can construct a more problem-oriented nonlinear type approximation using information from the previous stages of the approximate dynamic programming algorithm. In this project, we aim at developing new types of recursive, sparse, and interpretable approximation methods for RL with provable theoretical guarantees. In particular, our objective is to present a new fitted Q-iteration algorithm with adaptively chosen approximation classes depending on previously constructed approximations. We shall compare this new approach to more conventional DRL algorithms regarding their theoretical guarantees and interpretability. Furthermore, we will extend our methods to the Mean-Field systems by combining our expertise on RL and McKean-Vlasov type processes. As a practical application we will provide an interpretation of the problem of consistent re-calibration of financial models as a RL problem, and study it using the methods developed in this project.
Finished projects
Stochastic processes - DFG - Scientific network
Anton Klimovsky
In the last 20 years, complex networks became a key tool to model real-world complex systems in the sciences. Yet, the majority of networks evolve over time and this can have a substantial effect on the processes unfolding on them. Moreover, the influence can also go the other way around: processes happening on a network can affect the evolution of the network itself. This leads to what is called coevolution in adaptive networks or more generally complex adaptive systems. Examples include epidemiological and ecological networks, neural networks, systems biology networks, social networks, financial markets, etc. In all these contexts, there is a great deal of uncertainty/volatility in the structure and dynamics of the complex system.
What are the emerging global patterns in complex systems? How do they come about from the behavior of the elements? These are typical questions in the sciences, economy and policy making. These questions immediately lead to severe mathematical problems about the models of complex systems but also about their relationships with the real data.
It is the purpose of this scientific network to advance the rigorous mathematical theory of stochastic processes on (co)evolving networks.
The scientific network focuses on:
- The probabilistic underpinnings of stochastic processes on (co)evolving networks.
- Analysis and synthesis of key examples coming from the areas of information/opinion exchange processes, population dynamics, infection processes, and dynamics of artificial neural networks.
- Issues of statistical inference, estimation and uncertainty quantification for (evolving) complex networks and processes on them.
Probabilistic structures in evolution - DFG - Priority Program (SPP 1590)
Project: The effect of natural selection on genealogies
Martin Hutzenthaler
with Peter Pfaffelhuber (Freiburg)
Natural selection shapes genealogies within a population in various ways. In our proposal, we suggest both, a general qualitative study of some aspects of genealogies under selection, and a quantitative treatment of specific relevant models. More precisely, we study models with unbounded selection (so there are arbitraryly beneficial and/or deleterious fitness classes) using Girsanov transforms and approximate dualities, and selection in fluctuating environment (where an allele can be beneficial or deleterious, depending on the environment) using a general result on stochastic averaging in the limit of fast environmental changes. For both models, we use the previously developed technique of treating genealogical trees as metric measure spaces, leading to tree-valued stochastic Markov processes. The qualitative work is dealing with a comparison of genealogical distances under neutrality and under selection. We conjecture that many ssituations including selection lead to shorter genealogical distances.
Project: Evolution of altruistic defense traits in structured population
Martin Hutzenthaler
with Dirk Metzler (München)
In the first project phase we investigated under which conditions an inheritable behavioral trait of defense against parasites can spread in a structured population even if it is costly in the sense that individuals having a defense gene tend to have less offspring. In this proposed continuation project we study in a many-demes limit the time until the first fixation of a defense allele arising from rare mutations. We are going to show that this time to first fixation is logarithmic in the inverse mutation rate. So even for small mutation rates defense traits can appear on an evolutionary relevant time scale. Mathematically our central contribution is to prove and generalize the results of Dawson and Greven (2011) without using dual processes for a large class of processes.
Project: Evolving pathogen phylogenies: a two-level branching approach
Anita Winter
For many RNA viruses the lack of a proofreading mechanism in the virus' RNA polymerase results in frequent mutation. The high viral mutation rates, the large virus population size, and the short replication periods produce abundance of viral variability which is responsible for immune escape or drug resistance. Understanding in detail the forces which maintain this diversity can assist in the struggle against viral infections.
Pathogen patterns - and in particular the shapes of the phylogenies - are affected by the strength of selective pressure due to various levels of cross-immunity. We focus on the temporal structure of phylogenies associated with a persistent virus. We propose a two-level (host-pathogen) branching model with mutation and competition on both levels in dierent scaling regimes, where hosts can be either the infected patients or the infected cells within a single patient. We thereby extend our recent work on a panmitic virus population.
We will further rely on techniques developed for measure-valued (neutral) multilevel branching dynamics and two-level multi-type branching dynamics with mutation and competition.
Bayesian inference for generalised tempered stable Levy processes - Research grant
Denis Belomestny
mit Shota Gugushvili (Leiden) und Peter Spreij (Amsterdam)
The goal of the project is the development of new efficient methods of Bayesian inference for Levy processes based on their discrete-time observations and theoretical investigation of these methods. In particular, for the class of generalized tempered stable processes, we plan to estimate the tempering function using a nonparametric Bayesian approach. An important task is development of efficient MCMC algorithms and proof of the corresponding contraction rates. The implementation of proposed methods and their application to financial and insurance data is also foreseen.
Further information
Statistical modelling of nonlinear dynamic processes - SFB 823
Project B 3: Statistical Modelling of Highly-Resolved Spectro-Temporal Audio Data in Hearing Aids
Denis Belomestny
mit Rainer Martin (Bochum)
This project develops and optimizes signal processing methods to better reproduce music signals in cochlear implants and hearing aids. It proposes new algorithms for change point detection in high-resolution spectro-temporal audio data and for the dimensionality reduction of spectral audio data using a priori information.
Project C 5: Statistical inference for complex dynamical models in Empirical finance
Denis Belomestny
mit Jeannette Woerner (Dortmund)
This project develops novel procedures for generalized moving average processes and generalized diffusions of McKean-Vlasov- and Dunkl type. It focuses on instationary and non-linear processes and allows for a complex dependence structure in space and time including long range dependence. Among its goals are parametric and non-parametic estimating procedures for both low and high frequency data which combine methods from stochastic and time series analysis and generalized Fourier techniques.
On numerical approximations of high-dimensional nonlinear parabolic partial differential equations and of backward stochastic differential equations - Research grant
Martin Hutzenthaler
Parabolic partial differential equations (PDEs) are a fundamental tool in the state-of-the-art pricing and hedging of financial derivatives. The PDEs appearing in such financial engineering applications are often high- dimensional and nonlinear. Since explicit solutions of such PDEs are typically not available, it is a very active topic of research to solve such PDEs approximately. However existing approximation methods are either computationally expensive or are only applicable to a (in our view) small class of PDEs. The goal of this project is to establish a class of generally applicable numerical approximations for semilinear parabolic partial differential equations and, in particular, to prove that the computational effort grows at most linearly in the dimension and cubic in the reciprocal of the prescribed accuracy. In particular, in the notation of information-based complexity, our goal is to show that the numerical problem of approximating the solution of a semilinear parabolic PDE at single space-time points is polynomially tractable.
Further information
High-dimensional Phenomena in Probability - Fluctuations and Discontinuity - RTG 2131
Martin Hutzenthaler and Anita Winter
The Research Training Group (RTG) High-dimensional Phenomena in Probability - Fluctuations and Discontinuity offers excellent national and international graduates in the mathematical sciences the opportunity to conduct internationally visible doctoral research in probability theory. The goal of the RTG is to bring together the joint expertise on aspects of high dimension in probability. In the study of random structures in high dimensions, one frequently observes universality in limit theorems (fluctuations) as well as phase transitions (discontinuities). These aspects form the common focus of a large number of currently active research projects in stochastic processes. The cooperation of several research groups will offer the Ph.D. students the unique opportunity to gain experience beyond their own research topic, thus giving a broad scientific education. The RTG is supported by top level research groups in probability theory and its applications, stochastic analysis, stochastic geometry and mathematical physics. The research groups involved in the RTG have recently successfully carried out externally funded research projects in probability and statistics. As a rule, each doctoral student in the RTG will be supervised by two PIs.
Further information
Markov-Ketten-Approximation von eindimensionalen singulären Diffusionen - Research grant
Mikhail Urusov
Ziel dieses Projektes ist es, ein schwaches Approximationsschema zu konstruieren, das für große Klassen von eindimensionalen stetigen starken Markov-Prozessen mit möglicherweise singulärem Verhalten anwendbar ist. Z. B. sollten solche Phänomene wie stickiness im Inneren des Zustandsraumes oder (instantaneous oder slow) Reflexion an der Grenze eingeschlossen sein. Approximationen sollten Markov-Ketten sein, die auf einem Computer implementiert werden können. Wir sind an Approximationen von Pfadfunktionalen interessiert. Daher sollte ein für unseren Zweck geeigneter funktionaler Grenzwertsatz abgeleitet werden. Da sich in der Praxis oft die Frage nach der Approximation bestimmter unstetiger Pfadfunktionale (z. B. Austrittszeiten aus einem Bereich) stellt, wollen wir das Schema, das auch dazu in der Lage ist, konstruieren. Die Haupteigenschaften des Schemas, einschließlich die Konvergenzrate, und die Leistung bei wichtigen Beispielen sollten ebenfalls untersucht werden.