List of Projects

  1. Finding multiscale structures in high-dimensional data

  2. Predicting and analyzing turbulent flows

  3. Multiscale sparsity model

  4. Deep Learning of kinetics from large-scale molecular dynamics data

  5. Active Learning for better co-adaptation of user and BCIs

  6. Enhancing estimations of indiviodual head models

  7. Sparse identification of cellular kinetics models from data

  8. Regression for uncertainty quantification in PDEs with stochastic coefficients

  9. Image-based data assimilation in complex flows with low Reynolds number

  10. Detection and analysis of coherent structures in direct numerical simulations of complex flows

  11. Randomized techniques for model reduction

  12. Reduced order turbulence models in incompressible magnetohydrodynamics

  13. Reduced model based optimization for flows with schocks

  14. Numerical solution of the Hamilton-Jacobi-Bellman (HJB) equation

Project P1: Finding multiscale structures in high-dimensional data

Main Supervisor: G. Kutyniok

Additional Supervisor: K.-R. Müller and C. Schütte

Background: The area of data science provides various approaches to identifying structures in high-dimensional data. In numerical analysis of partial differential equations multiscale systems are today a common tool and often guarantee efficient algorithms by providing a decomposition into different resolution levels. Thus, it seems very natural to analyze high-dimensional data with respect to such multiscale structures as well. However, to date almost no approaches exist, and even the question of a precise definition of multiscale structures is far from being solved.

Objective: This project aims to develop a fundamental theory for multiscale structures in high-dimensional data and associated algorithmic approaches for detecting such structures. To achieve this goal, a carefully designed list of mathematical desiderata for multiscale structures will be specified, based on which learning algorithms will be developed and analyzed.

Project P2: Predicting and analyzing turbulent flows

Main Supervisor: K.-R. Müller

Additional Supervisors: W.-C. Müller and F. Noé

Background: The numerical cost of turbulence simulations and the lack of a comprehensive theory motivate the search for reduced models of this phenomenon. This project will explore the applicability of Machine Learning (ML) techniques to ultra-fast prediction and analysis of turbulence obtained from direct numerical simulations. This radically new approach treats a complex nonlinear differential equation (recently successful for the Schrödinger's (SE) equation) as a stochastic system and learns from data generated by it. Turbulent systems, e.g., Navier-Stokes and magnetohydrodynamic flow, exhibit due to their nonlinearity a complexity that is at least similar to the SE.

Objective: This project aims to transfer and adapt ML methods for studying multiscale properties of turbulent flow. In addition, the potential of the predictive approach to serve as a new kind of subgrid modeling technique will be investigated. Further study will focus on the usage of ML analysis for tracking structural turbulent flow properties.

Project P3: Multiscale sparsity models

Main Supervisor: G. Kutyniok

Additional Supervisors: J. Eisert, F. Noé, and C. Schütte

Background: In the past years, the classical sparsity model has been extensively explored and utilized. However, depending on the application, the sparsity of data typically exhibits certain structures, giving lately rise to different structured sparsity models. Despite models such as block, fusion frame, and model-based sparsity, research is still in its infancy. In fact, real-world applications require sparsity models with a much more sophisticated inner structure such as multiscale structures with dependencies.

Objective: The objective of this project is to first introduce a suitable model situation for such sparsity structures based on which a flexible theoretical framework including associated dictionary learning algorithms should be developed. This framework shall then be applied to derive model reduction strategies in molecular dynamics where one has to face an entire cascade of temporal and spatial scales bridging many orders of magnitude.

Project P4: Deep learning of kinetics from large-scale molecular dynamics data

Main Supervisor: F. Noé

Additional Supervisors: G. Kutyniok and K.-R. Müller

Background: High-throughput molecular dynamics (MD) simulations can now sample complex biologically relevant processes such as protein-protein binding. Due to the extensive amount and high dimension of such datasets, the task of analyzing such data and extracting the essential kinetic and thermodynamic features has become a major challenge. The currently most successful techniques to achieve this include Markov state models and blind source separation methods, as they both exploit the essential spectral properties of the Markov operator underlying the MD time series. However, these methods are shallow learning structures and are thus subject to the definition of suitable features, basis functions or kernels, which often leads to unsatisfactory and difficult-to-interpret results in complex MD processes that are often of intrinsically hierarchical nature.

Objective: In this project, we aim to design efficient deep learning structures that learn optimal, nonlinear feature transformations, provide interpretable models of the molecular kinetics, and permit simulating long-time molecular dynamics.

Project P5: Active learning for better co-adaptation of user and BCIs

Main Supervisor: A. Carpentier

Additional Supervisors: B. Blankertz and H. Siebert

Background: Brain-Computer Interfaces (BCIs) mostly target medical applications. More recently, BCI research aims at improving human-computer interaction (HCI) in general. For example, in information seeking, users quickly scan through a larger amount of information and provide an explicit response only for few items, while the BCI may concurrently collect and integrate implicit information about all scanned items. Still, a major hurdle in such cases is the relatively long procedure that is required to calibrate the system.

Objective: Recent developments of machine learning in general, and sequential/active learning in particular, are excellent tools to accelerate the calibration in a co-adaptive process between the user and the BCI. Applying those techniques in the context of EEG data with its inherent complexity (spatial/temporal dependency, high dimensionality, etc.) poses considerable challenges that will be tackled in this project.

Project P6: Enhancing estimations of individual head models

Main Supervisor: B. Blankertz

Additional Supervisors: A. Carpentier

Background: Localization of brain activity is of great importance for clinical applications and for basic neuroscience. However, this is difficult to achieve from surface EEG data as it requires the solution of an ill-posed inverse problem. The most accurate way of (noninvasive) localization is the estimation of individual head models from anatomical MRI scans, which are very costly and time consuming to acquire.

Objective: This project will provide a method to estimate a 4-shell model with the boundary-element-method (BEM) which describes the anatomical and electrical properties of the head of an individual test person. Our proposed methods will use as input data from electrical impedance tomography (EIT) to avoid costly MRI scans. In order to fit the 4-shell BEM to the EIT data, we will exploit a large scale database and develop as well as evaluate different methods to restrict the fitting to a low dimensional search space, based on non-negative matrix factorization and on tensor approximations.

Project P7: Sparse identification of cellular kinetics models from data

Main Supervisor: C. Schütte

Additional Supervisors: K.-R. Müller

Background: Modeling cellular dynamics is a very active field of research with a huge variety of possible models. Selecting the appropriate model for a certain system is a challenging art, in particular, if a mixture of deterministic and stochastic effects have to be considered. Simultaneously the amount of data that is produced by advanced experimental techniques like single cell analysis is growing exponentially which raises the challenges of identifying appropriate models algorithmically. Recently, new algorithms for finding appropriate differential equation (DE) models based on time-resolved experimental data have been proposed and developed into a family of methods under the name data-driven sparse identification of nonlinear dynamics (SINDy). Presently these algorithms cannot account for the multiscale structure with deterministic and stochastic reactions exhibited by real-life cellular dynamics, and also require the data to be given in time slices with rather small lag times with it almost never valid for real-world data.

Objective: This project aims to develop a new SINDy framework that allows for application to data-based identification of deterministic-stochastic hybrid models.

Project P8: Regression for uncertainty quantification in PDE's with stochastic coefficients

Main Supervisor: A. Carpentier

Additional Supervisors: J. Eisert and R. Schneider

Background: In the context of modeling and quantifying the uncertainty in porous flow models, extremely high dimensional regression problems appear. A very promising novel approach is to cast such problems in a tensor regression framework, which is far from being fully understood. Beyond classical estimation, an almost unexplored question in this context despite its importance for applications is uncertainty quantification.

Objective: This project aims to perform complete inference in the regression model, i.e., not only estimating the parameter in the model but also estimating the precision of the estimators from data. We highlight that the possibility of optimally estimating the solution or the model by minimizing a constrained/penalised risk functional does not imply the possibility of quantifying optimally the risk of the estimator. Our approach will be based on acceleration phenomena for risk estimation combined with machine learning methods, which we expect to also ensure extremely fast and efficient methods in practice.

Project P9: Image-based data assimilation in complex flows with low Reynolds number

Main Supervisor: J. Sesterhenn

Additional Supervisors: F. Noé

Background: Fluid dynamic experiments usually suffer from measurement noise and influences of the measurement technique on the flow field, e.g., by probes. Furthermore, there are strong restrictions on measurable quantities, as in general only a subset of quantities is experimentally accessible. However, image-based data assimilation techniques using data from optical methods like PIV or Schlieren enable a reliable analysis of experimental flow configurations by means of image-based data assimilation.

Objective: The main objective of this project is to develop a theoretical foundation for image-based data assimilation, also taking into account compressed sensing ideas on the optimal measurement data to make quantities of interest measurable with minimal effort. Preliminary works on the topic, in particular, the adjoint-based pressure determination from PIV data, already give evidence of the extensive possibilities of image-based data assimilation techniques for research and industrial use.

Project P10: Detection and analysis of coherent structures in direct numerical simulations of complex flows

Main Supervisor: W.-C. Müller

Additional Supervisors: V. Mehrmann, J. Reiss, and R. Schneider

Background: Temporal or spatial structures are readily extracted from complex data by, e.g., modal decompositions like proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) or Lagrangian coherent structure (LCS) analysis of Lyapunov exponent manifolds. Subspaces obtained by these techniques can serve as reduced order models and define spatial structures in time or temporal structures in space.

Objective: This project aims at combining the mode decomposition and the LCS approach, in particular the underlying statistical sampling of passive tracers, to increase the efficiency of complexity reduction and the robustness of detection of spatio-temporally coherent structures in turbulent flows. We will combine sub-ensembles of tracers associated with an LCS with the mode decomposition representations, since the geometry of these tracer groups should be amenable to complexity reduction by representation of topologically simpler geometrical objects such as ellipsoids or tetrahedra.

Project P11: Randomized techniques for model reduction

Main Supervisor: V. Mehrmann

Additional Supervisors: G. Kutyniok and M. Voigt

Background: In simulation, control, and optimization for large scale complex systems, model reduction is an essential requirement. Typically model reduction techniques used for the solution of PDEs or control problems are based on SVD-techniques in suitable finite dimensional representation systems. Although this is very successful in many applications, sparsity of the solution or the solution operator is not promoted well.

Objective: The objective of this project is to combine the construction of appropriate representation systems with randomized techniques for the construction of reduced order models and sparse solutions. To achieve this goal, we will exploit adequate multi-level representations of operators and solutions that promote sparsity and combine them with randomized techniques to compute these sparse representations. As model class we will discuss transport equations in reactive flows.

Project P12: Reduced order turbulence models in incompressible magnetohydrodynamics

Main Supervisor: W.-C. Müller

Additional Supervisors: V. Mehrmann

Background: The nonlinear dynamics of turbulence represents a longstanding physical and mathematical problem. Incompressible turbulence can serve as a simplified model, since it corresponds to an ensemble of discrete Fourier mode excitations in wavenumber space. The presence of a magnetic field reduces each nonlinear three-mode interaction to a pair exchange mediated by the third mode. Thus, the mathematically difficult quadratic advection nonlinearity becomes amenable to a detailed study in the case of incompressible magnetohydrodynamics (MHD).

Objective: We aim at a detailed understanding of the spectral turbulent transport of ideal invariants such as energy or magnetic helicity in numerical simulations of incompressible MHD turbulence and the development of reduced turbulence models that go beyond the classical shell-model ansatz. Insights gained from the above-mentioned nonlinear dynamics will help to construct a new reduced triad-interaction model of turbulence.

Project P13: Reduced model based optimization for flows with shocks

Main Supervisor: V. Mehrmann

Additional Supervisors: J. Reiss and J. Sesterhenn

Background: Flows with shocks appear in many technical situations, since a pressure-drop by a factor of two is sufficient to create supersonic flows and thereby nearly inevitable shocks. These shocks are crucial for the performance yet depend sensitively and non-linearly on parameters. Classical methods of model reduction in industrial use, like the POD often fail at the very positions of the shocks.

Objective: The goal of this PhD project is to study the new shifted POD method for shocks in transonic flows, and to produce parameterized reduced order models. This new method offers a way to circumvent structural problems and allows an efficient optimization of a configuration. Optimal configurations predicted in the low dimensional model will be validated via simulation and then be improved by expansion of the model. This would lead to an important methodological progress in this field, which will be tested in the reduction of total pressure loss in the optimization of internal flows.

Project P14: Numerical solution of the Hamilton-Jacobi-Bellman (HJB) equation

Main Supervisor: R. Schneider

Additional Supervisors: W.-C. Müller

Background: The minimizer of an optimal control problem for a Bolza type functional constrained by a system of stochastic differential equations can be found by solving the Hamilton-Jacobi-Bellman (HJB) equations. It was recently shown that if the optimal control problem has a unique solution, then the desired value function is the adjoint state of a PDE constrained optimization problem, which can be obtained by solving a backward Kolmogorov equation. In direct numerical simulations of incompressible turbulence, driven by a small set of spatial Fourier modes, driver efficiency, turbulent energy content, and dissipation rate will be controlled; uncertainties are accounted for by a noise term.

Objective: We will treat this problem directly, using sparse representations in the spectral ansatz functions combined with via appropriate finite dimensional basis of Hermite polynomials as spectral ansatz functions, where for the high-dimensional problems we will apply low rank tensor product approximation. Particular challenges are the preconditioning of the high-dimensional backward Kolmogorov equation and the treatment of the dominating non-symmetric drift term.