# Research Projects

## Selected Research Projects at the Department of Mathematics

- 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. - 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. - PPP Frankreich 2019 Phase I

(Third Party Funds Single)**Term:**01-01-2019 - 31-12-2020**Funding source:**Deutscher Akademischer Austauschdienst (DAAD) - Theoretische Grenzen und algorithmische Verfahren verteilter komprimierender Abtastung

(Third Party Funds Single)**Term:**01-01-2019 - 31-12-2021**Funding source:**DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)The theoretical limits of distributed compressive sensing are studied by

tools from both information theory and statistical physics. The investigations

cover both noise-free and noisy distributed compressive sensing. The theoretical insights

are 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. - 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 - Holistische Optimierung von Trajektorien und Runway Scheduling

(Third Party Funds Group – Sub project)**Overall project:**Holistische Optimierung von Trajektorien und Runway Scheduling**Term:**01-09-2018 - 31-08-2021**Funding source:**Bundesministerium für Wirtschaft und Technologie (BMWi)**URL:**https://www.mso.math.fau.de/edom/projects/hotrun/Efficient runway utilization is a major issue in airport operation, as capacities are (nearly) reached in many aiports. But planing is highly affected by uncertainties arising from weather changes or disruptions in the operative business. Furthermore, the planing of flight trajectories in the terminal region is by now often neglected in runway scheduling, as time efficient solution methods are mathematically challenging. The overall goal of this project is to combine trajectory and runway schedule computation including resilience against uncertainties in order to obtain stable optimal solutions. - 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. - 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 Lisp

Implementierung SBCL verfügbar zu machen. SBCL ist der

populärste und am weitesten Entwickelte freie Compiler für Common

Lisp. Die Verbesserungen aus diesem Projekt machen es möglich

Common Lisp Programme zu schreiben, deren Ausführungsgeschwindigkeit

mit C++ und Fortran Programmen auf Augenhöhe liegt. Dadurch

ergeben sich interessante Möglichkeiten der Metaprogrammierung im

wissenschaftlichen Rechnen. - 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. - 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. - 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. - 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. - 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) - 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. - Optimierte Prozesse für Trajektorie, Instandhaltung, Management von Ressourcen und Abläufen in der Luftfahrt

(Third Party Funds Single)**Term:**01-01-2018 - 31-12-2021**Funding source:**Bundesministerium für Wirtschaft und Technologie (BMWi)**URL:**https://www.mso.math.fau.de/edom/projects/ops-timal/ - Mixed-Integer Non-Linear Optimisation: Algorithms and Applications

(Third Party Funds Group – Overall project)Building upon the achievements of the Marie-Curie ITN Mixed-Integer Non-Linear Optimization (MINO) (2012 - 2016), the goal of the Mixed-Integer Non-Linear Optimisation Applications (MINOA) proposal is to train the next generation of highly qualified researchers and managers in applied mathematics, operations research and computer science that are able to face the modern imperative challenges of European and international relevance in areas such as energy, logistics, engineering, natural sciences, and data analytics. Twelve Early-Stage Researchers (ESRs) will be trained through an innovative training programme based on individual research projects motivated by these applications that due to their high complexity will stimulate new developments in the field. The mathematical challenges can neither be met by using a single optimisation method alone, nor isolated by single academic partners. Instead, MINOA aims at building bridges between different mathematical methodologies and at creating novel and effective algorithmic enhancements. As special challenges, the ESRs will work on dynamic aspects and optimisation in real time, optimisation under uncertainty, multilevel optimization and non-commutativity in quantum computing. The ESRs will devise new effective algorithms and computer implementations. They will validate their methods for the applications with respect to metrics that they will define. All ESRs will derive recommendations, both for optimised MINO applications and for the effectiveness of the novel methodologies. These ESRs belong to a new generation of highly-skilled researchers that will strengthen Europe'e human capital base in R&I in the fast growing field of mathematical optimisation. The ESR projects will be pursued in joint supervision between experienced practitioners from leading European industries and leading optimisation experts, covering a wide range of scientific fields (from mathematics to quantum computing and real-world applications). - 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) - Flächenbezogene Modellierung, Simulation und Optimierung von Solar-Einspeisung, Lastfluss und Steuerung für Stromverteilnetze, unter Berücksichtigung von Einspeisungsunsicherheiten. Teilprojekt 5: Optimierung der Netzeingriffe.

(Third Party Funds Single)**Term:**01-01-2018 - 31-12-2020**Funding source:**Bundesministerium für Bildung und Forschung (BMBF)**URL:**https://www.mso.math.fau.de/edom/projects/verteilnetze/ - Reduced Order Modelling, Simulation and Optimization of Coupled systems

(Third Party Funds Group – Sub project)**Overall project:**Reduced Order Modelling, Simulation and Optimization of Coupled systems**Term:**01-09-2017 - 31-08-2021**Funding source:**EU - 8. Rahmenprogramm - Horizon 2020**URL:**https://www.romsoc.eu/ - Innovationsfonds 2017: Urkunden und Buchgutscheine für gute Leistungen in Anfängervorlesungen

(FAU Funds)**Term:**01-07-2017 - 30-09-2020Um den Vorlesungs- und Prüfungsbetrieb persönlicher zu gestalten, wird bei sehr guten Leistungen in meinen Anfängervorlesungen "Mathematik für Ingenieure" ein wenig symbolisches Lob in der Form von Urkunden und auch ein wenig finanzielles Lob in der Form von Buchgutscheinen ausgeteilt. - Besov Regularität von parabolischen partiellen Differentialgleichungen auf Lipschitz Gebieten

(Third Party Funds Single)**Term:**01-04-2017 - 31-03-2019**Funding source:**DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)This project is concerned with partial differential equations (PDEs) of parabolic type on Lipschitz domains. We want to study the regularity of the solutions of such equations and are particularly interested in smoothness estimates in specific scales of Besov spaces, which determine the approximation order of adaptive and other nonlinear numerical approximation schemes. We wish to show that the Besov regularity is high enough to justify the use of adaptive schemes compared to non-adaptive (uniform) schemes. We shall mainly be concerned with approximation schemes based on wavelets.

Our starting point is to improve existing results for the heat equation and extend these results to linear parabolic PDEs with variable coefficients and also to nonlinear parabolic PDEs. We also want to show that when restricting ourselves to polygonal and polyhedral domains (compared to general Lipschitz domains) we get even higher Besov regularity.

Since the approximation rates obtained this way still depend on the spatial dimension, we secondly wish to study regularity of parabolic PDEs in generalized dominating mixed smoothness spaces. By using tensor-wavelets we want to beat the so-called 'curse of dimension' and obtain convergence rates independent of the underlying dimension of the domain.

- Rechenleistungsoptimierte Software-Strategien für auf unstrukturierten Gittern basierende Anwendungen in der Ozeanmodellierung

(Third Party Funds Single)**Term:**01-01-2017 - 30-09-2020**Funding source:**DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)Um akkurate Ozean, Atmosphären oder Klima Simulationen durchzuführen werden sehr effiziente numerische Verfahren und große Rechenkapazitäten benötigt, die in vielen Teilen der Welt und bei vielen Forschungsgruppen in diesen Anwendungsfeldern nicht verfügbar sind. Solche Beschränkungen führen auch dazu, dass Modelle und Softwarepakete basierend auf strukturierten Gittern derzeit in der Ozeanwissenschaft immer noch vorherrschend sind.In diesem Projekt soll zum einen die Rechenzeit für Modelle, die auf unstrukturierten Gittern und einer diskontinuierlichen Galerkin finite Elemente Methode basieren, deutlich reduziert werden, und zum anderen die Produktivität bei der Softwareentwicklung gesteigert werden. Das erste Ziel soll durch einen neuen Ansatz zur parallelen Gebietszerlegung und durch adaptive numerische Verfahren erreicht werden.Für das zweite Ziel kommen moderne Software Design Strategien zum Einsatz, vor allem Codegenerierung und automatische Optimierung von rechenintensiven Programmteilen. Die Fortschritte bei der Rechenzeit und dem Software Design, die aus dem Projekt resultieren, können einen wichtigen Beitrag leisten, um unstrukturierte Gitter für alle Forscher aus den Ozeanwissenschaften nutzbar zu machen, auch wenn sie nur Zugang zu moderat parallelen Systemen und nicht zu Höchstleistungsrechnern haben. - Energiemarktdesign

(Third Party Funds Group – Sub project)**Overall project:**Energie Campus Nürnberg (EnCN2)**Term:**01-01-2017 - 31-12-2021**Funding source:**andere Förderorganisation, Bayerische Staatsministerien**URL:**http://www.encn.de/en/forschungsthemen/energiemarktdesign/In the project “Energy Markt Design” within EnCN2 a team of researchers from economics, mathematics, and law analyses the economic and regulatory environment for the transformation of the energy system. The main objectives are to enhance the methods in energy market modeling and to contribute with well-grounded analyses to the policy discourse in Germany and Europe. For the electricity market, the focus is on the steering effect of market designs on regulated transmission expansion and private investments, as well as on the identification of frameworks at the distribution level that provide regional stakeholders with business models for the provision of flexibility measures. In order to address these complex issues, mathematical techniques are developed within the project that allow for solving the respective models. Another key research topic results from the advancing sector coupling in energy markets. Within EMD, gas market models, that are developed within DFG Transregio 154 (Simulation and Optimization of Gas Networks) in cooperation with project partners, are applied to evaluate the European gas market design. The long-term objective of the research group is an integrated assessment of the electricity and gas market design and their combined effects on investment decisions. - Verbundvorhaben proMT Teilprojekt 2: Modellreduktion zur prognostischen online MR-Thermometrie

(Third Party Funds Group – Sub project)**Overall project:**Verbundvorhaben proMT Teilprojekt 2: Modellreduktion zur prognostischen online MR-Thermometrie**Term:**01-12-2016 - 30-11-2019**Funding source:**Bundesministerium für Bildung und Forschung (BMBF) - Optimization of medical care in rural environments

(Third Party Funds Group – Overall project)**Term:**01-12-2016 - 30-11-2019**Funding source:**BMBF / Verbundprojekt**URL:**https://www.mso.math.fau.de/edom/projects/healthfact/ - "Verbundprojekt MED4D: Dynamische Medizinische Bildgebung: Modellierung und Analyse medizinischer Daten für verbesserte Diagnose, Überwachung und Arzneimittelentwicklung"

(Non-FAU Project)**Term:**01-12-2016 - 30-11-2019**Funding source:**Bundesministerium für Bildung und Forschung (BMBF) - Welfare optimal nominations in gas networks and associated equilibria

(Third Party Funds Group – Sub project)**Overall project:**SFB TRR 154 “Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzen”**Term:**01-10-2016 - 30-06-2018**Funding source:**DFG / Sonderforschungsbereich / Transregio (SFB / TRR)**URL:**https://trr154.fau.de/index.php/de/teilprojekte/teilprojekte-phase1/b08-phase1The goal of this project is the analysis of the relation between (i) the equilibria of simple models of competitive natural gas markets, using complementarity problems for modeling the behavior of different players, and (ii) the solution of corresponding single-level welfare maximization problems. The understanding of this fundamental relation is a prerequisite for an analysis of the current entry-exit gas market design in Europe. Similar questions have been studied in detail in the context of electricity market modeling in the past. For natural gas markets, however, the addressed questions are much more complex and not yet well understood for adequate models of gas physics. The reasons for the high level of complexity is twofold: First, gas flow through pipeline systems is inherently nonconvex due to gas physics. This renders classical first-order optimality conditions possibly insufficient. Second, the operation of gas transport networks comprises the control of active network devices like (control) valves or compressors. These devices introduce binary aspects and thus a further type of non-convexity to the models of the underlying equilibrium problems. As a result of the project we will obtain a first reference model that combines gas physics and a market analysis in a well-understood way. This will lay the ground for multilevel models of entry-exit natural gas markets that account for network characteristics. Beyond that, our results will enhance the understanding of binary equilibrium problems. - Mathematische Schlüsselqualifikation für Energienetze im Wandel – Teilprojekt FAU

(Third Party Funds Group – Sub project)**Overall project:**Mathematische Schlüsselqualifikation für Energienetze im Wandel – Teilprojekt FAU**Term:**01-10-2016 - 30-04-2020**Funding source:**Bundesministerium für Wirtschaft und Technologie (BMWi) - Analyse und Anwendung reduzierter Modelle

(Third Party Funds Group – Sub project)**Overall project:**MathEnergy — Mathematische Schlüsseltechniken für Energienetze im Wandel**Term:**01-04-2016 - 30-04-2020**Funding source:**Bundesministerium für Wirtschaft und Technologie (BMWi)Das Kernziel dieses mathematisch orientierten Teilvorhabens ist die

Methodenentwicklung und Analyse reduzierter Modelle bzw.

Modellhierarchien und ihrer Anwendung zur dynamischen Zustandsschätzung

in einer modellprädiktiven Regelung. - Analysis and Implementation of Highly Compressive Sensing

(Third Party Funds Single)**Term:**01-03-2016 - 28-02-2019**Funding source:**DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)The project will investigate the performance limits of compressive sensing and propose practical algorithms that approach these performance limits closely.The project will focus on the regime of high compression ratios where L1-norm regularization is suboptimal.Compressive sensing will be investigated from the viewpoint of statistical physics and considered as a particular instance of a spin glass system.Both average distortion and minimax distortion will be addressed as objective functions (Hamiltonians).The analysis will rely on the replica method. Particular emphasis will be put on the implications of replica symmetry breaking.The main objectives of the project are:1) to find a system of saddle point equations that describe the replica symmetry breaking solution to the compressive sensing problem,2) to prove rigorous one-sided bounds for these solutions by adapting Guerra's arguments for the Sherrington-Kirkpatrick spin glass model to spin glass model of compressive sensing and, hereby, justifying the correctness of the replica method in compressive sensing,3) to numerically solve the system of saddle point equations, and 4) to determine which practical algorithms work well for compressive sensing at high compression ratios. - Invariant convexity in infinite dimensional Lie algebras

(Third Party Funds Single)**Term:**01-01-2016 - 01-01-2019**Funding source:**DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)Infinite dimensional Lie groups show up in all areas of mathematics and other sciences, wherever symmetries depending on infinitely many parameters arise. The goal of this project is to develop a systematic understanding of convexity properties of infinite dimensional Lie algebras. More precisely, we are aiming at a classification of open convex cones in an infinite dimensional Lie algebra that are invariant under the adjoint action. In the dual of the Lie algebra we would like to determine those invariant convex subsets which are semi-equicontinuous, which means that their support functional is bounded in the neighborhood of some point. A key point of this project is to understand closed convex hulls of projections of adjoint and coadjoint orbits to subalgebras; results of this type are called convexity theorems. Classically convexity theorems mostly concern orbit projections onto abelian subalgebras, where they are often convex hulls of Weyl group orbits. The convexity theorems of Schur-Horn, Kostant, Atiyah-Pressley and Kac-Peterson are of this type. We are aiming at a systematic extension of these results to larger classes of Lie algebras and to projections onto more general subalgebras. This project is motivated to a large extent by its applications to unitary representations, where knowledge on open invariant cones is crucial to determine spectral bounds of operators from the derived representation. The set of all elements represented by operators bounded from below is an invariant convex cone. That it has interior points means that the representation is semibounded. Semiboundedness is a stable version of the positive energy condition which characterizes many representations arising in quantum mechanics. Typical Lie algebras we plan to study in this context are direct limits of finite dimensional Lie algebras and their completions, hermitian Lie algebras (corresponding to automorphism groups of symmetric Hilbert domains) and so-called double extensions of Hilbert-Lie algebras (close infinite dimensional relatives of compact Lie algebras) and of twisted loop algebras with infinite dimensional target groups. The latter lead to infinite rank generalizations of affine Kac-Moody Lie algebras. The focus of the present project lies on combining structural properties on infinite dimensional Lie algebras with functional analytic and geometric methods to obtain a concrete description of invariant convex cones and semi-equicontinuous coadjoint orbits. - Applied index theory for quantum and classical systems

(Third Party Funds Single)**Term:**01-01-2016 - 31-12-2018**Funding source:**DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)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. - Mechanistic modelling of the formation and consolidation of soil microaggregates

(Third Party Funds Group – Sub project)**Overall project:**DFG RU 2179 “MAD Soil - Microaggregates: Formation and turnover of the structural building blocks of soils”**Term:**01-01-2016 - 31-12-2019**Funding source:**Deutsche Forschungsgemeinschaft (DFG) - Modellreduktion zur prognostischen online MR-Thermometrie

(Third Party Funds Group – Sub project)**Overall project:**proMT — Prognostische modellbasierte online MR-Thermometrie bei minimalinvasiver Thermoablation zur Behandlung von Lebertumoren**Term:**01-01-2016 - 31-12-2019**Funding source:**Bundesministerium für Bildung und Forschung (BMBF)In Hinblick auf die angestrebte prognostische Online-Simulationsfähigkeit

zielt dieses Teilprojekt auf die Entwicklung und Analyse reduzierter

Modelle mittels Techniken der Modellordnungsreduktion (MOR). Die

Kombination von MOR und Space-Mapping lässt eine weitere Steigerung der

Performance erhoffen, die für die konkrete Anwendung der MR-Thermometrie

qualitativ und quantitativ bewertet wird. - German-Norwegian collaborative research support scheme

(Third Party Funds Single)**Term:**01-01-2016 - 31-12-2017**Funding source:**Deutscher Akademischer Austauschdienst (DAAD)Homogenisierung reaktiven Transports in variablen Mikrostrukturen - Spatial continuum limit of tree-valued state-dependent spatial branching processes

(Third Party Funds Single)**Term:**01-01-2016 - 28-02-2019**Funding source:**DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)The project studies the spatial continuum limit of one- and multitype branching processes with state-dependent branching rates and analyses the characteristics of limit processes. Particular emphasis is placed on genealogies in the corresponding populations, i.e. tree-valued processes are studied. Typical examples are branching random walks, catalytic branching random walks, mutually branching random walks, self-catalytic random walks and logistic branching random walks and the relevant continuous mass limits (interacting branching diffusions). The goal is to show the existence of the limits, the characteristics of their longtime behaviour and the structure on small space and time scales. The question of their stochasticity shall be clarified and in the case of a deterministic limit the asymptotic of ''hotspots'' shall be studied by means of volatility or size biasing and subsequent rescaling. As basis the calculus of infinitely divisible genealogical processes and the description of genealogies of matri- and patrilinear ancestor lines are to be developed. - Scientific Computing for Improved Detection and Therapy of Sepsis

(Non-FAU Project) - EXIST-Gründerstipendium: FoodOptimizer ist ein wissenschaftliches Ernährungs- und Symptomtagebuch

(Third Party Funds Single)**Term:**01-10-2015 - 30-09-2016**Funding source:**Bundesministerium für Wirtschaft und Technologie (BMWi)FoodOptimizer ist ein wissenschaftliches Ernährungs- und Symptomtagebuch, das eine systematische Analyse und Diagnoseunterstützung zur Aufdeckung von Nahrungsmittelallergien und -unverträglichkeiten anbietet. - Verteiltes Höchstleistungsrechnen in Common Lisp

(Third Party Funds Single)**Term:**01-10-2015 - 31-03-2016**Funding source:**Bayerisches Staatsministerium für Wissenschaft, Forschung und Kunst (StMWFK) (bis 09/2013)The Message Passing Interface\cite{mpi-standard} (MPI) is the de facto

standard for distributed programming on all modern compute clusters and

supercomputers. It features a large number of communication patterns with

virtually no overhead. Our work on bringing MPI functionality to Common

Lisp resulted in vast improvements to the message passing library CL-MPI

and the development of severaly new approaches to distributed computing.

- DAAD Exchange Service: PPP Finnland 2017: Bayesian Inverse Problems in Banach Space

(Non-FAU Project)**Term:**25-01-2015 - 31-12-2017**Funding source:**Deutscher Akademischer Austauschdienst (DAAD) - Modeling, simulation and optimization of process chains

(Third Party Funds Group – Sub project)**Overall project:**SPP 1679: Dynamic Simulation of Interconnected Solids Processes**Term:**01-01-2015 - 01-01-2018**Funding source:**DFG / Schwerpunktprogramm (SPP)The projected research aims at the simulation-based optimization of predictive models of coupled process chains. This makes it possible to optimize both the conditions of the single processes and the configuration of the entire process chain integrated into an MRT with respect to the desired particle properties. The methods and algorithms to be developed during the funding period build up a general toolbox for simulation and optimization of dynamic processes relevant to the SPP, thereby generalizing from the exemplary applications provided by batch processes, as nucleation and ripening, to process chains. This is made possible by integrating mathematical models including processes of more general but structurally equivalent type. - Existence and regularity for parabolic quasi minimizers on metric measure spaces

(Third Party Funds Single)**Term:**01-01-2015 - 01-01-2018**Funding source:**DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)The aim of this project is to make a substantial contrubution to existence and regularity theory for solutions of nonlinear parabolic minimization problems on general metric measure spaces. It is intended to establish a systematic approach to the generalization of a result by A. Grigor'yan and L. Saloff-Coste on the relation between solutions of the heat equation on Riemannian manifolds and the validity of Harnack estimates. The interest is to generalize this result in two ways: Firstly, one intends to consider metric measure spaces, supporting a doubling property of the measure and a Poincaré inequality, instead of a Riemannian manifold. Secondly, instead of the (linear) heat equation, nonlinear problems should be investigated.Main difficulties of the project consist on one hand in the very general concept of metric measure spaces, on the other hand in the nonlinearity of the partial differential equations and integral functionals under consideration. On general metric measure spaces it is not possible to speak of "direction" or "integration by parts" and consequently there's a lack of a suitable form of derivative. This makes it necessary to work with so-called "upper gradients" which are defined according to a characterization of Sobolev functions in the Euklidean space by to the power p integrable vector fields. This concept does not allow to introduce a reasonable definition of a partial differential equation, however it helps to generalize minimization problems to the context of metric measure spaces. The nonlinearity of the problem causes a number of further severe difficulties, which are basically already known from the theory of nonlinear parabolic differential equations in the n dimensional Euklidean space. These difficulties have to be overcome on the level of pure minimization problems -- without any associated partial differential equation behind.