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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.