Dr. Jan Heiland

My research focus is the robust control and optimization of complex systems at the interface of theory for the ∞-dimensional PDEs and the numerical implementation in the high-dimensional state space. With regard to applications, data-driven methods for modeling, simulation and control have become an important topic in my research. 

Systems, Control, and Optimization. For the incompressible Navier-Stokes equations (see e.g. [A4], [B7], [B3]), so-called H∞-robust controller can be theoretically defined using projected Riccati equations and implemented numerically efficiently with a guaranteed degree of robustness; see [B8]. In addition, we have shown in the infinitedimensional model that the required degree of robustness continuously depends on a possible linearization error (cf. [B5], [A13]) and that the error due to Galerkin spatial discretizations can be compensated in theory by such controllers; see [B6].

For more general cases we have been exploring a class of Riccati equations, which are not symmetrical but can be formulated without projections. The results enable a direct generalization of the H∞ theory for ODEs (cf. [B12]) and also prove to be useful for qualitative studies of the linear quadratic control problem; see [A16].

In my dissertation [T2], I have investigated the interplay between optimization, discretization, and transformations in the decoupling of (P)DAEs. We adapted the results obtained there with regard to the optimization of flow equations for multi-body systems [A5] and also implemented them for additional couplings [A7]. Building on earlier work on the discrete approximation of I/O maps (cf. [A1] [B1]), we have generalized the POD approach [B4], adapted it for efficient optimization of PDEs on finite time horizons [A10] and currently for PDEs with multivariate uncertainties [A21]. 

Applications. Generally, the model equations that I have considered describe coupled and possibly high- or ∞-dimensional systems. Direct applications are given in the simulation of flows and multi-body systems; see [A12], [A7], [A5]. Also, my recent inclusions of data-driven approaches enable the consideration of arbitrarily complex systems if only simulation or measurement data is available. On the concrete side, we implemented³ the control and regulation of an inverted 3-way pendulum with all the difficulties of a real test setup.

During my time at Bombardier Transportations, I experienced model– and simulationbased development in industry. There, I implemented an automated environment for optimizing the shape of regional and high-speed trains [P1] with validation in wind tunnel tests and relevant contributions to the design of the Zefiro train series⁴. Further application-oriented experience with modeling and optimization results from an interdisciplinary project on dispersion processes [P2] and, most recently, from the simulation of a methanation reactor [B13], which is a promising modern tool for energy conversion.

³www.mpi-magdeburg.mpg.de/3952045/showcase
⁴See also the relevant publications engineering.esteco.com/resources/success-story/bombardier-usesmodefrontier-to-optimize-high-speed-trains and www.zevrail.de/artikel/die-zefiro-familie-eine-neuedimension-der-hochgeschwindigkeit.

Data-informed Galerkin-approximations of Multivariate Uncertainty Quantification and Optimization Problems.
Especially with regard to optimization and control in applications, I will continue to work on the generalization of the data-informed Galerkin method (cf. [A10], [A21]) and further establish this approach to the numerical analysis and simulation of optimization problems under uncertainties. This approach also allows the complexity reduction of optimization problems without having to rely on model equations (cf. [B13]), creates direct links to various applications for example in bioengineering and is also the link in my existing cooperation with the department for process engineering at the MPI Magdeburg.

Analysis of Convolutional Neural Networks for Parametrizations of PDE State Spaces.
When advancing our works ([A8], [B7], [A23]) on extended linearizations towards efficient nonlinear feedback controller design by so-called gain scheduling⁵ or other LPV⁶ techniques, it appeared that very low-dimensional parametrizations of state spaces are incredibly useful for these approaches. Standard linear model reduction approaches have their natural limits that are best overcome by nonlinear reduction techniques. In a recent proof of concept for the use of neural networks for model reduction of flow equations (see [A19] and [B14]), we showed that convolutional neural networks can provide the right compromise between a drastic complexity reduction and accuracy. In a current PhD project, using the links to well understood wavelet approximations, we are developing the analysis of convolutional neural networks for the quantitative and qualitative approximation in Sobolev spaces. In parallel, we will combine the numerical treatment of PDEs with the optimization of network parameters which brings both needs and potentials for the development of efficient algorithms for modern hardware.
 

Data-driven Model Synthesis and Interdisciplinary Application Oriented Projects.
For situations where a model is not available at all or simply too complex for, say, long time simulations, I want to further develop on completely data-driven methods like DMD and operator inference (cp. e.g. [A14] and [A17]) that we have been exploring in Master–and Bachelor projects for an inverted triple pendulum and a flow setup from medical engineering. These methods can be applied to the design, inferrence, and analysis of models in all natural and life sciences in particular if combined with relevant knowledge of measurements or simulation data from high-fidelity models. Generally, my relevant experience with data-based model synthesis and model reduction provides a direct link to, say, applications and modelling in engineering science and an asset in possible interdisciplinary research projects and grant proposals.
 

⁵Gain scheduling – an approach to interpolate locally valid, linear controllers to a globally valid, nonlinear controller.
⁶here, we will make use of the embedding of nonlinear systems into the class of LPV (linear parameter varying) systems

since 2018

tba

¹ mardi4nfdi.de/consortium/organization

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