Research Projects
Selected Research Projects at the Department of Mathematics
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Optimal decision making during pandemics
(Third Party Funds Single)
Term: 22-03-2022 - 01-06-2025
Funding source: Deutsche Forschungsgemeinschaft (DFG)Our project deals with various operations research problems for optimal decision making during pandemics. We formulate, solve, and analyze our problems with respect to sensitivity and stability. The common feature of our problems is the stochasticity of inputs --- we deal with one or multi-stage stochastic programming problems. Moreover, the probability distribution of the random inputs very often depends on the decisions, hence stochastic problems with endogenous randomness will be of interest. We are motivated by the applications of such problems towards optimal decision making, under a variety of settings, for pandemic response planning.
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Greedy algorithms for fair allocations and efficient assignments within facility location optimization problems
(Third Party Funds Single)
Term: since 03-01-2022
Funding source: Bayerisches Staatsministerium für Bildung und Kultus, Wissenschaft und Kunst (ab 10/2013) -
DAAD Projektbezogener Personenaustausch mit Finnland
DG Methoden und Parameterschätzer für Mikrostrukturmodelle in porösen Medien
(Third Party Funds Single)
Term: 01-01-2022 - 31-12-2023
Funding source: Deutscher Akademischer Austauschdienst (DAAD) -
Heisenberg Grants - Quantum Fields and Operator Algebras
(Third Party Funds Single)
Term: 01-09-2021 - 31-08-2026
Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)Das Ziel des Heisenberg-Programms ist es, herausragenden Wissenschaftlerinnen und Wissenschaftlern, die alle Voraussetzungen für die Berufung auf eine Langzeit-Professur erfüllen, zu ermöglichen, sich auf eine wissenschaftliche Leitungsfunktion vorzubereiten und in dieser Zeit weiterführende Forschungsthemen zu bearbeiten. In der Verfolgung dieses Ziels müssen nicht immer projektförmige Vorgehensweisen gewählt und realisiert werden. Aus diesem Grunde wird bei der Antragstellung und auch später bei der Abfassung von Abschlussberichten - anders als bei anderen Förderinstrumenten - keine "Zusammenfassung" von Projektbeschreibungen und Projektergebnissen verlangt. Somit werden solche Informationen auch in GEPRIS nicht zur Verfügung gestellt. -
Fairly allocating vaccines for COVID-19
(Third Party Funds Single)
Term: 15-03-2021 - 16-07-2021
Funding source: andere Förderorganisation -
Forschungskostenzuschuss zum Forschungsstipendium für erfahrene Wissenschaftler (Herr Dr. Vincenzo Morinelli)
(Third Party Funds Single)
Term: 01-03-2021 - 31-08-2022
Funding source: Alexander von Humboldt-Stiftung -
Process strategies for the production of thin-walled components during selective laser beam melting of plastics
(Third Party Funds Group – Sub project)
Overall project: CRC 814 - Additive Manufacturing
Term: 01-01-2021 - 31-12-2023
Funding source: DFG / Sonderforschungsbereich (SFB)
URL: https://www.crc814.research.fau.eu/projekte/t-transferprojekte/transferproject-t3/The aim of theproject is the systematic investigation of the process-geometry-interaction ofthin-walled components for the production of locally adapted properties as wellas the modeling of this effect in finite element simulations and structuraloptimization. In experimental tests, the main influencing factors areidentified and mapped in relation to the building position in the process. Newexposure technologies and strategies are used to manipulate the melting pooland homogenize component properties. The findings are incorporated into a wallthickness dependent material model for structural optimization, which isinvestigated in the project. The participating industrial partners willvalidate the results over the course of the project. The experimental findingsand the wall thickness dependent material model will be used to develop amethodology for the product development of thin-walled structures. In thefuture, the product development process can be accelerated, and the economicefficiency increased. Based on these findings, new application areas for theselective laser beam melting of plastics can be opened up in the future.
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Parallel mesh loading and partitioning for large-scale simulation
(Third Party Funds Single)
Term: since 01-01-2021
Funding source: Bayerisches Staatsministerium für Wissenschaft und Kunst (StMWK) (seit 2018)
URL: https://www.konwihr.de/ -
Optimal Decision Making for COVID-19
(Third Party Funds Single)
Term: 13-07-2020 - 29-11-2020
Funding source: andere Förderorganisation -
Stability for doubly nonlinear parabolic equations (C07 intern)
(Third Party Funds Group – Sub project)
Overall project: TRR 154: Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken
Term: 01-05-2020 - 30-06-2022
Funding source: DFG / Sonderforschungsbereich / Transregio (SFB / TRR)
URL: https://trr154.fau.de/ -
EOSCsecretariat.eu: Optimal Spatiotemporal Antiviral Release under Uncertainty
(Third Party Funds Single)
Term: 01-05-2020 - 30-04-2022
Funding source: Research infrastructures, including e-infrastructures, EU - 8. Rahmenprogramm - Horizon 2020 -
Maschinelles Lernen bei korrelativer MR und Hochdurchsatz-NanoCT
(Third Party Funds Single)
Term: 01-04-2020 - 31-03-2023
Funding source: Bundesministerium für Bildung und Forschung (BMBF) -
Besov regularity of parabolic partial differential equations on Lipschitz domains (continuation)
(Third Party Funds Single)
Term: 01-04-2020 - 30-09-2021
Funding source: Deutsche Forschungsgemeinschaft (DFG)In this project we study parabolic partial differential equations (=PDEs) on bounded Lipschitz domains. We aim at justifying the use of adaptive numerical methods when treating such equations. In an adaptive strategy the choice of the underlying degrees of freedom is not a priori fixed but depends on the shape of the unknown solutions. Additional degrees of freedom are only spent in regions where the numerical approximation is still far away from the exact solution. The best one can expect from an adaptive algorithm is an optimal performance in the sense that it realizes the convergence rate of best N-term approximation (i.e., best approximation of the solution by linear combinations with at most N basis functions). However, this convergence order depends on the regularity of the solution in specific scales of Besov spaces. It is therefore our aim to investigate the Besov regularity of the solutions of parabolic PDEs in order to see whether adaptivity pays off in this context. In the first funding period of the project we were able to show that adaptivity is indeed justified for quite general classes of linear and nonlinear parabolic PDEs. Even better regularity results could be achieved for polyhedral cones (instead of general Lipschitz domains). In the second funding period of the project we wish to improve and develop these results further. Our achievements for cones have to be generalized to polyhedral domains. Moreover, the nonlinear results on the Besov regularity so far are only established on convex domains. Since from a numerical point of view non-convex domains are of particular interest, we want to prove similar results here. Furthermore, we plan to study the regularity in fractional Sobolev spaces for stochastic parabolic PDEs, which determines the convergence order of non-adaptive methods. Also an investigation of the Besov regularity of PDEs on more general manifolds (e.g. soap films) is intended. As another aspect we study the approximation classes of parabolic PDEs. Our goal here is a convergence analysis of the horizontal mothod of lines (Rothe's method), when we use a Galerkin-method for our discretization in time and adaptive discretizations in space. Moreover, instead of a time-marching algorithm (as described above) we could use a full space-time adaptive algorithm based on tensor wavelets. Numerical studies indicate that this is more efficient. In particular, the approximation order that can be achieved this way turns out to be independent of the spatial dimansion and depends on the regulairity of the exact solution in a specific scale of tensor products of Besov spaces. Therefore, we will systematically investigate the regularity of the solutions in these scales of dominating mixed smoothness spaces. -
Mechanistic, Integrative Multiscale Modelling of the Turnover of Soil Microaggregates
(Third Party Funds Group – Sub project)
Overall project: MAD Soil - Microaggregates: Formation and turnover of the structural building blocks of soils
Term: 01-04-2020 - 31-08-2024
Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)The grand goal of this group is the development, analytical, and numerical investigation of a mechanistic model describing processes of formation, stability, and turnover of soil micro-aggregates. In contrast to existing conceptual aggregation models and compartment models for carbon turnover and aggregation, we focus on specific, experimentally identified transformation processes of soil micro-aggregates. Since we are interested in giving an improved mechanistic, qualitative and even quantitative description of aggregation, we transfer the gained insights to a mechanistic model in terms of ordinary differential equations (ODEs), partial differential equations (PDEs), and perhaps algebraic equations (AEs). To that end, we aim to take into account information/identified processes on different spatial scales as well as spatial heterogeneity and variability. All our modeling is done in a rigorous, deterministic way and our modeling concepts are based on continuum mechanics. We start our investigations at the pore scale and apply multiscale techniques to obtain a comprehensive mathematical model at the macroscale (bottom up). In particular, the interplay of geochemistry and microbiology is considered, and also their link to soil functions. The coupled ODE/PDE systems and complex micro-macro problems cannot be treated numerically with standard software packages. The number of species, the nonlinearity of the processes and the heterogeneity of the medium results in a high computational effort that requires accurate and efficient discretization methods and solution algorithms. Moreover sophisticated numerical multiscale methods have to be applied. In our simulations, we do not aim to recreate reality in every detail. Instead, we aim to illustrate, compare, and reveal influencing factors and mechanisms by abstracting relevant processes. -
Applied index theory for quantum and classical systems
(Third Party Funds Single)
Term: 01-04-2020 - 31-03-2023
Funding source: Deutsche Forschungsgemeinschaft (DFG)The first goal of index theory is to relate topological invariants to indices of Fredholm operators. The most famous result in this direction is the Atiyah-Singer index theorem, but there exist far reaching non-commutative generalizations. While there is a general theory, such index theorems have to be established case by case in applications. The second goal of index theory is to connect invariants and indices of problems related via exact sequences. For example, this allows to read off the topology of boundary states or point defects from bulk invariants. The proposal aims to implement this program in situations which have not been tackled before like interacting spin systems, photonic crystals and lattices of classical springs, and also to further develop the index approach to scattering systems and topological materials. -
Process optimization for hospital logistics
(Third Party Funds Single)
Term: since 01-01-2020
Funding source: Industrie
URL: https://en.www.math.fau.de/edom/projects-edom/logistics-and-production/process-optimization-for-hospital-logistics/ -
Information management and computational science support
(Third Party Funds Group – Sub project)
Overall project: SFB 1411: Design of particulate products
Term: 01-01-2020 - 31-12-2023
Funding source: DFG / Sonderforschungsbereich (SFB) -
Topology, material and shape optimisation for particle ensembles
(Third Party Funds Group – Sub project)
Overall project: SFB 1411: Design of particulate products
Term: 01-01-2020 - 31-12-2023
Funding source: DFG / Sonderforschungsbereich (SFB)The objective is the development of a mathematical framework which allows to conclude from desired optical properties to a corresponding optimised configuration of single particles as well as particle assemblies. A structural optimisation approach based on discrete dipole approximations is explored to allow for a design space with sufficiently high resolution and enabling the prediction of structure-property relations of individual particles. For particle assemblies a structural optimisation method based on a generalised hybrid finite element approach is established. Finally, dispersity and angle independency are taken into account by a new stochastic optimisation method. -
Mechanistische, integrative Mehrskalenmodellierung der Umwandlung von Bodenmikroaggregaten
(Third Party Funds Single)
Term: 01-11-2019 - 31-10-2022
Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH) -
Geordnete Dilationsräume und Geometrie von Standard-Unterräumen
(Third Party Funds Single)
Term: 01-10-2019 - 30-09-2022
Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH) -
Multiscale modeling with evolving microstructure: An approach to
emergence in the rhizosphere via effective soil functions(Third Party Funds Group – Sub project)
Overall project: DFG Priority Programme 2089 “Rhizosphere Spatiotemporal Organisation – a Key to Rhizosphere Functions”
Term: 01-02-2019 - 31-01-2022
Funding source: DFG / Schwerpunktprogramm (SPP)
URL: https://www.ufz.de/spp-rhizosphere/index.php?en=46495The self-organization of the aggregates in the rhizosphere by various
attracting forces influenced by geochemistry, and microbiology shall
be studied by a novel, comprehensive model. This model should
account for processes on the microscale (single roots, pore scale),
and then be upscaled to the root system scale (macroscale) by
mathematical homogenization. This goal exceeds the functional range
of existing models for aggregation and needs the introduction of an
explicit phase of mucilage, and attachment properties of root hairs in
the rhizosheath. The project aims at the development of a mechanistic modeling approach that allows for dynamic structural reorganization of the rhizosphere at the single root scale and couples this evolving microscale model to the root system scale including the inference of soil functions. This means that we do not assume a static rhizosphere but develop a tool that is capable to dynamically track this zone on the basis of the underlying spatiotemporal aggregegate formation and geochemical patterns. The collaboration with experimental groups – analyzing CT images in various moisture and growth conditions - the Central Experiment will allow to derive the properties of the mucilage phase, the pore structure and thus the
influence of root hairs on aggregation mechanisms. -
Teilprojekt P11 - Fracture Control by Material Optimization
(Third Party Funds Group – Sub project)
Overall project: Skalenübergreifende Bruchvorgänge: Integration von Mechanik, Materialwissenschaften, Mathematik, Chemie und Physik (FRASCAL)
Term: 02-01-2019 - 30-06-2023
Funding source: DFG / Graduiertenkolleg (GRK)
URL: https://www.frascal.research.fau.eu/home/research/p-11-fracture-control-by-material-optimization/In previous works, the dependence of failure mechanisms in composite materials like debonding of the matrix-fibre interface or fibre breakage have been discussed. The underlying model was based on specific cohesive zone elements, whose macroscopic properties could be derived from DFT. It has been shown that the dissipated energy could be increased by appropriate choices of cohesive parameters of the interface as well as aspects of the fibre. However due to the numerical complexity of applied simulation methods the crack path had to be fixed a priori. Only recently models allow computing the full crack properties at macroscopic scale in a quasi-static scenario by the solution of a single nonlinear variational inequality for a given set of material parameters and thus model based optimization of the fracture properties can be approached.
The goal of the project is to develop an optimization method, in the framework of which crack properties (e.g. the crack path) can be optimized in a mathematically rigorous way. Thereby material properties of matrix, fibre and interfaces should serve as optimization variables.
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Teilprojekt P10 - Configurational Fracture/Surface Mechanics
(Third Party Funds Group – Sub project)
Overall project: Fracture across Scales: Integrating Mechanics, Materials Science, Mathematics, Chemistry, and Physics (FRASCAL)
Term: 02-01-2019 - 30-06-2023
Funding source: DFG / Graduiertenkolleg (GRK)
URL: https://www.frascal.research.fau.eu/home/research/p-10-configurational-fracture-surface-mechanics/In a continuum the tendency of pre-existing cracks to propagate through the ambient material is assessed based on the established concept of configurational forces. In practise crack propagation is however prominently affected by the presence and properties of either surfaces and/or interfaces in the material. Here materials exposed to various surface treatments are mentioned, whereby effects of surface tension and crack extension can compete. Likewise, surface tension in inclusion-matrix interfaces can often not be neglected. In a continuum setting the energetics of surfaces/interfaces is captured by separate thermodynamic potentials. Surface potentials in general result in noticeable additions to configurational mechanics. This is particularly true in the realm of fracture mechanics, however its comprehensive theoretical/computational analysis is still lacking.
The project aims in a systematic account of the pertinent surface/interface thermodynamics within the framework of geometrically nonlinear configurational fracture mechanics. The focus is especially on a finite element treatment, i.e. the Material Force Method [6]. The computational consideration of thermodynamic potentials, such as the free energy, that are distributed within surfaces/interfaces is at the same time scientifically challenging and technologically relevant when cracks and their kinetics are studied.
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PPP Frankreich 2019 Phase I
(Third Party Funds Single)
Term: 01-01-2019 - 31-12-2020
Funding source: Deutscher Akademischer Austauschdienst (DAAD) -
Integriertes und an Raum-Zeit-Messungsskalen angepasstes Global Random Walk - Modell für reaktiven Transport im Grundwasser
(Third Party Funds Single)
Term: 01-10-2018 - 30-09-2021
Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH) -
Nonlocal Methods for Arbitrary Data Sources
(Third Party Funds Group – Sub project)
Overall project: Nonlocal Methods for Arbitrary Data Sources
Term: 01-10-2018 - 28-02-2022
Funding source: EU - 8. Rahmenprogramm - Horizon 2020 -
Parity Sheaves on Kashiwara's flag manifold
(Third Party Funds Group – Overall project)
Term: 01-09-2018 - 31-12-2019
Funding source: Deutscher Akademischer Austauschdienst (DAAD)The project is located in pure mathematics and deals with a problem in geometric representation theory. Parity sheaves and moment graph techniques have proven to be extremely effective in answering questions in modular representation theory. In the finite-dimensional case a hypercohomology functor establishes a connection between parity sheaves and sheaves on moment graphs. However the geometry controlling representation theoretic phenomena in this case is often infinite-dimensional. We plan to study the category of parity sheaves on Kashiwara's infinite-dimensional thick-flag variety Y, to define a hypercohomology functor, to interpret its image as a category of moment graph sheaves and to establish an equivalence between parity sheaves and canonical sheaves on the moment graph. In a second phase, we intend to study base change and torsion phenomena in the category of parity sheaves on the thick flag manifold, in order to establish an equivalence between the category of projective objects in the category O of an affine Kac-Moody algebra at negative level and parity sheaves on Y.
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Implementation of vector operations for SBCL
(Third Party Funds Single)
Term: 10-07-2018 - 31-03-2019
Funding source: Bayerisches Staatsministerium für Bildung und Kultus, Wissenschaft und Kunst (ab 10/2013)Ziel des Projekts ist es, AVX2 Vektoroperationen für die Common LispImplementierung SBCL verfügbar zu machen. SBCL ist derpopulärste und am weitesten Entwickelte freie Compiler für CommonLisp. Die Verbesserungen aus diesem Projekt machen es möglichCommon Lisp Programme zu schreiben, deren Ausführungsgeschwindigkeitmit C++ und Fortran Programmen auf Augenhöhe liegt. Dadurchergeben sich interessante Möglichkeiten der Metaprogrammierung imwissenschaftlichen Rechnen. -
Theoretische Grenzen und algorithmische Verfahren verteilter komprimierender Abtastung
(Third Party Funds Single)
Term: 01-07-2018 - 31-12-2021
Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)The theoretical limits of distributed compressive sensing are studied bytools from both information theory and statistical physics. The investigationscover both noise-free and noisy distributed compressive sensing. The theoretical insightsare utilized to design approximate message passing algorithms for joint recovery of large distributed compressive sensing networks with feasible computational complexity. These algo-rithms enable us to verify the non-rigorous results obtained by the replica method from statistical mechanics, and also, to propose theoretically optimal approaches for sampling and low complexity. The proposed research will lead to improved performance of reconstruction algorithms for distributed compressive sensing, e.g. higher compression rates and/or higher fidelity of reconstruction. -
Decomposition methods for mixed-integer optimal control (A05) (2018 - 2022)
(Third Party Funds Group – Sub project)
Overall project: TRR 154: Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken
Term: 01-07-2018 - 30-06-2022
Funding source: DFG / Sonderforschungsbereich / Transregio (SFB / TRR)The focus lies on the development of mathematical decomposition methods for mixed-integer nonlinear optimal control problems on networks. On the top level (master) mixed-integer linear problems are in place, whereas in the sub-problem only continuous variables are considered. The exchange between the levels is performed not only via cutting planes, but also via the modelling of disjunctions to deal with non-convex optimal control problems as well. The overall emphasis is the mathematical analysis of structured mixed nonlinear optimization problems based on hierarchical models.
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MIP techniques for equilibrium models with integer constraints (B07) (2018 - 2022)
(Third Party Funds Group – Sub project)
Overall project: TRR 154: Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken
Term: 01-07-2018 - 30-06-2022
Funding source: DFG / Sonderforschungsbereich / Transregio (SFB / TRR)In this subproject we will develop techniques to solve equilibrium problems with integer constraints using MIP techniques. To this end, we will consider first mixed-integer linear, then mixed-integer nonlinear problems as subproblems. To solve the resulting problems we will study both complete descriptions as also generalized KKT theorems for mixed-integer nonlinear optimization problems. -
Multilevel mixed-integer nonlinear optimization for gas markets (B08) (2018 - 2022)
(Third Party Funds Group – Sub project)
Overall project: TRR 154: Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken
Term: 01-07-2018 - 30-06-2022
Funding source: DFG / Sonderforschungsbereich / Transregio (SFB / TRR)
URL: https://trr154.fau.de/index.php/de/teilprojekte/b08The main goal of this project is the development of mathematical methods for the solution of multilevel, mixed-integer, and nonlinear optimization models for gas markets. To this end, the focus is on a genuine four-level model of the entry-exit system that can be reformulated as a Bilevel model. The mathematical and algorithmic insights shall then be used to characterize market solutions in the entry-exit system and to compare them to system optima. Particular attention is paid to booking prices for entry or exit capacity.
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Robustification of Physics Parameters in Gas Networks (B06) (2018 - 2022)
(Third Party Funds Group – Sub project)
Overall project: TRR 154: Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken
Term: 01-07-2018 - 30-06-2022
Funding source: DFG / Sonderforschungsbereich / Transregio (SFB / TRR)The goal of this research project is to study uncertain optimization problems using robust optimization methods. Focusing on transport networks, we aim at the development of tractable robust counterparts for uncertain optimization problems and an analysis of the problem structure. For the arising adjustable robust optimization tasks, good relaxations as well as effective branch-and-bound implementations shall be developed.
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Free boundary propagation and noise: analysis and numerics of stochastic degenerate parabolic equations
(Third Party Funds Single)
Term: 01-04-2018 - 31-03-2020
Funding source: Deutsche Forschungsgemeinschaft (DFG), DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
URL: https://www1.am.uni-erlangen.de/~gruen/The porous-medium equation and the thin-film equation are prominent examples of nonnegativity preserving degenerate parabolic equations which give rise to free boundary problems with the free boundary at time t > 0 defined as the boundary of the solution’s support at that time.
As they are supposed to describe the spreading of gas in a porous-medium or the spreading of a viscous droplet on a horizontal surface, respectively, mathematical results on the propagation of free boundaries become relevant in applications. In contrast to, e.g., the heat equation, where solutions to initial value problems with compactly supported nonnegative initial data
instantaneously become globally positive, finite propagation and waiting time phenomena are characteristic features of degenerate parabolic equations.
In this project, stochastic partial differential equations shall be studied which arise from the aforementioned degenerate parabolic equations by adding multiplicative noise in form of source terms or of convective terms. The scope is to investigate the impact of noise on the propagation of free boundaries, including in particular necessary and sufficient conditions for the occurrence
of waiting time phenomena and results on the size of waiting times. Technically, the project relies both on rigorous mathematical analysis and on numerical simulation. -
Interfaces, complex structures, and singular limits in continuum mechanics
(Third Party Funds Group – Overall project)
Term: 01-04-2018 - 30-09-2022
Funding source: DFG / Graduiertenkolleg (GRK) -
Optimierung der Netzeingriffe
(Third Party Funds Group – Sub project)
Overall project: Flächenbezogene Modellierung, Simulation und Optimierung von Solar-Einspeisung, Lastfluss und Steuerung für Stromverteilnetze, unter Berücksichtigung von Einspeisungsunsicherheiten
Term: 01-01-2018 - 31-12-2021
Funding source: Bundesministerium für Bildung und Forschung (BMBF)
URL: https://en.www.math.fau.de/edom/projects-edom/analytics/optimal-control-of-electrical-distribution-networks-with-uncertain-solar -
Adaptive Verfahren zur Optimierung gekoppelter pH-Systeme
(Third Party Funds Group – Sub project)
Overall project: EiFer: Energieeffizienz durch intelligente Fernwärmenetze
Term: 01-01-2018 - 31-12-2020
Funding source: Bundesministerium für Bildung und Forschung (BMBF)