Modellansatz - English episodes only

In the last two semesters Gudrun has taught the courses Advanced Mathematics I and II for Mechanical Engineers. This is a mandatory lecture for the International mechanical engineering students at KIT in their first year of the Bachelor program. This program is organized by the Carl Benz School of Engineering. Beside the study courses, the school also provides common housing for students coming to Karlsruhe from all over the world.

The general structure and topics of the first year in Advanced Mathematics were already discussed in our episode 146 Advanced Mathematics with Jonathan Rollin.

This time Gudrun invited two students from her course to have the student's perspective, talking about mathematics, life, and everything.
Yueyang Cai grew up mostly in China. In 2015, the work of her mother led Yueyang to Stuttgart. While looking for opportunities to study a technical subject in Germany the English speaking program in Karlsruhe somehow suggested itself. After one year she is sure to have made the right decision.

The second student in the conversation is Siddhant Dhanrajani. His family is Indian but lives in Dubai. For that he got his education in Dubai in an Indian community follwowing the Indian educational system (CBSE). He had never heard of the Engineering program in Karlsruhe but found it through thourough research. He is really amazed at how such an excellent study program and such an excellent university as the KIT are not better known for their value in the world.

In the conversation both students talk about their education in their respective countries, their hopes and plans for the study course mechanical engineering and their experiences in the first year here in Karlsruhe. It is very interesting to see how the different ways to teach mathematics, namely, either as a toolbox full of recipes (which the students get well-trained in) or secondly as a way to approach problems in a context of a mathematical education contribute to an experience to be well-equipped to work creative and with a lot of potential as an engineer.

Though the students finished only the first year in a three years course they already work towards applications and necessary certificates for their possible master program after finishing the course in Karlsruhe.

• G. Thäter, J. Rollin: Advanced Mathematics, Conversation in the Modellansatz Podcast, Episode 146, Department of Mathematics, Karlsruhe Institute for Technology (KIT), 2017.
• F. Hettlich, G. Thäter: Höhere Mathematik, Gespräch im Modellansatz Podcast, Folge 34, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2014.

Gudrun met the USA-based mathematician Roza Aceska from Macedonia in Turin at the Conference MicroLocal and Time-Frequency Analysis 2018.

The topic of the recorded conversation is dynamical sampling. The situation which Roza and other mathematician study is: There is a process which develops over time which in principle is well understood. In mathematical terms this means we know the equation which governs our model of the process or in other words we know the family of evolution operators. Often this is a partial differential equation which accounts for changes in time and in 1, 2 or 3 spatial variables. This means, if we know the initial situation (i.e. the initial conditions in mathematical terms), we can numerically calculate good approximations for the instances the process will have at all places and at all times in the future.

But in general when observing a process life is not that well sorted. Instead we might know the principal equation but only through (maybe only a few) measurements we can find information about the initial condition or material constants for the process. This leads to two questions: How many measurements are necessary in order to obtain the full information (i.e. to have exact knowledge)? Are there possibilities to choose the time and the spatial situation of a measurement so clever as to gain as much as possible new information from any measurement? These are mathematical questions which are answered through studying the equations.

The science of sampling started in the 1940s with Claude Shannon who found fundamental limits of signal processing. He developed a precise framework - the so-called information theory. Sampling and reconstruction theory is important because it serves as a bridge between the modern digital world and the analog world of continuous functions. It is surprising to see how many applications rely on taking samples in order to understand processes. A few examples in our everyday life are: Audio signal processing (electrical signals representing sound of speech or music), image processing, and wireless communication. But also seismology or genomics can only develop models by taking very intelligent sample measurements, or, in other words, by making the most scientific sense out of available measurements.

The new development in dynamical sampling is, that in following a process over time it might by possible to find good options to gain valuable information about the process at different time instances, as well as different spatial locations. In practice, increasing the number of spatially used sensors is more expensive (or even impossible) than increasing the temporal sampling density. These issues are overcome by a spatio-temporal sampling framework in evolution processes. The idea is to use a reduced number of sensors with each being activated more frequently. Roza refers to a paper by Enrique Zuazua in which he and his co-author study the heat equation and construct a series of later-time measurements at a single location throughout the underlying process. The heat equation is prototypical and one can use similar ideas in a more general setting. This is one topic on which Roza and her co-workers succeeded and want to proceed further.

After Roza graduated with a Ph.D. in Mathematics at the University of Vienna she worked as Assistant Professor at the University Ss Cyril and Methodius in Skopje (Macedonia), and after that at the Vanderbilt University in Nashville (Tennessee). Nowadays she is a faculty member of Ball State University in Muncie (Indiana).

• Overview on sampling theory and applications: M. Unser: Sampling-50 years after Shannon Proceedings of the IEEE 88 (4) 569 - 587, 2000.
• Dynamical sampling in shift-invariant spaces: R. Aceska e.a.: Dynamical Sampling in Shift-Invariant Spaces 2014 (Version at Archive}
• Dynamical sampling: R. Aceska, A. Petrosyan, S. Tang: Dynamical sampling of two-dimensional temporally-varying signals International Conference on Sampling Theory and Applications (SampTA), DOI:10.1109/SAMPTA.2015.7148929, 2015.
• DeVore, Ronald, and Enrique Zuazua: Recovery of an initial temperature from discrete sampling, Mathematical Models and Methods in Applied Sciences 24.12 (2014): 2487-2501, 2014.
• Evolution operators involved in dynamical sampling: S. Tang: System identification in dynamical sampling, Advances in Computational Mathematics 43 (3) 555–580, 2017.
• On Bessel systems, bases and frames in the dynamical sampling setup: A.Aldroubi e.a.:lterative actions of normal operators Journal of Functional Analysis 272 (3), 1121-1146, 2017.

• G. Thäter, E. Zuazua: Waves, Conversation in Modellansatz Podcast Episode 054, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2015.

Gudrun spent an afternoon at the Max Planck Institute for Mathematics in the Sciences (MPI MSI) in Leipzig. There she met the Colombian mathematician Eliana Maria Duarte Gelvez. Eliana is a PostDoc at the MPI MSI in the Research group in Nonlinear Algebra. Its head is Bernd Sturmfels.

They started the conversation with the question: What is algebraic geometry? It is a generalisation of what one learns in linear algebra insofar as it studies properties of polynomials such as its roots. But it considers systems of polynomial equations in several variables so-called multivariate polynomials. There are diverse applications in engineering, biology, statistics and topological data analysis. Among them Eliana is mostly interested in questions from computer graphics and statistics.

In any animated movie or computer game all objects have to be represented by the computer. Often the surface of the geometric objects is parametrized by polynomials. The image of the parametrization can as well be defined by an equation. For calculating interactions it can be necessary to know what is the corresponding equation in the three usual space variables. One example, which comes up in school and in the introductory courses at university is the circle. Its representation in different coordinate systems or as a parametrized curve lends itself to interesting problems to solve for the students.

Even more interesting and often difficult to answer is the simple question after the curve of the intersection of surfaces in the computer representation if these are parametrized objects. Moreover real time graphics for computer games need fast and reliable algorithms for that question. Specialists in computer graphics experience that not all curves and surfaces can be parametrized. It was a puzzling question until they talked to people working in algebraic geometry. They knew that the genus of the curve tells you about the possible vs. impossible parametrization.

For the practical work symbolic algebra packages help. They are based on the concept of the Gröbner basis. Gröbner basis help to translate between representations of surfaces and curves as parametrized objects and graphs of functions. Nevertheless, often very long polynomials with many terms (like 500) are the result and not so straightforward to analyse.

A second research topic of Eliana is algebraic statistics. It is a very recent field and evolved only in the last 20-30 years. In the typical problems one studies discrete or polynomial equations using symbolic computations with combinatorics on top. Often numerical algebraic tools are necessary. It is algebraic in the sense that many popular statistical models are parametrized by polynomials. The points in the image of the parameterization are the probability distributions in the statistical model. The interest of the research is to study properties of statistical models using algebraic geometry, for instance describe the implicit equations of the model.

Eliana already liked mathematics at school but was not always very good in it. When she decided to take a Bachelor course in mathematics she liked the very friendly environment at her faculty in the Universidad de los Andes, Bogotá. She was introduced to her research field through a course in Combinatorial commutative algebra there. She was encouraged to apply for a Master's program in the US and to work on elliptic curves at Binghamton University (State University of New York) After her Master in 2011 she stayed in the US to better understand syzygies within her work on a PhD at the University of Illinois at Urbana-Champaign. Since 2018 she has been a postdoc at the MPI MSI in Leipzig and likes the very applied focus especially on algebraic statistics.

In her experience Mathematics is a good topic to work on in different places and it is important to have role models in your field.

• E. Duarte, Ch. Görgen: Equations defining probability tree models
• E. Duarte: Implicitization of tensor product surface in the presence of a generic set of basepoints. 2016. Journal of Algebra and Applications(to appear).
• Rigidity of Quasicrystal Frameworks - webpage
• E. M. Duarte, G. K. Francis: Stability of Quasicrystal Frameworks in 2D and 3D Proceedings of the First Conference Transformables 2013.In the Honor of Emilio Perez Piñero 18th-20th September 2013, Seville, Spain
• Portraits of people working in Nonlinear Algebra

• P. Schwer: Metrische Geometrie, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 102, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016.

Gudrun talks with Asher Zarth. He finished his Master thesis in the Lattice Boltzmann Research group at the Karlsruhe Institute for Technology (KIT) in April 2018.

Lattice Boltzmann methods (LBM) are an established method of computational fluid dynamics. Also, the solution of temperature-dependent problems - modeled by the Boussinesq approximation - with LBM has been done for some time. Moreover, LBM have been used to solve optimization problems, including parameter identification, shape optimization and topology optimization. Usual optimization approaches for partial differential equations are strongly based on using the corresponding adjoint problem. Especially since this method provides the sensitivities of quantities in the optimization process as well. This is very helpful. But it is also very hard to find the adjoint problem for each new problem. This needs a lot of experience and deep mathematical understanding.

For that, Asher uses automatic differentiation (AD) instead, which is very flexible and user friendly. His algorithm combines an extension of LBM to porous media models as part of the shape optimization framework. The main idea of that framework is to use the permeability as a geometric design parameter instead of a rigid object which changes its shape in the iterative process. The optimization itself is carried out with line search methods, whereby the sensitivities are calculated by AD instead of using the adjoint problem.

The method benefits from a straighforward and extensible implementation as the use of AD provides a way to obtain accurate derivatives with little knowledge of the mathematical formulation of the problem. Furthermore, the simplicity of the AD system allows optimization to be easily integrated into existing simulations - for example in the software package OpenLB which Asher used in his thesis.

One example to test the algorithm is the shape of an object under Stokes flow such that the drag becomes minimal. It is known that it looks like an american football ball. The new algorithm converges fast to that shape.

• F. Klemens e.a.: CFD- MRI: A Coupled Measurement and Simulation Approach for Accurate Fluid Flow Characterisation and Domain Identification. Computers & Fluids 166, 218-224, 2018.
• T. Dbouk: A review about the engineering design of optimal heat transfer systems using topology optimization. Applied Thermal Engineering 112, pp. 841-854, 2017.
• C. Geiger and C. Kanzow. Numerische Verfahren zur Lösung unrestringierter Optimierungsaufgaben. Springer-Verlag, 2013.
• M. J. Krause and V. Heuveline: Parallel fluid flow control and optimisation with lattice Boltzmann methods and automatic differentiation. Computers & Fluids 80, pp. 28-36, 2013.
• A. Kamikawa and M. Kawahara: Optimal control of thermal fluid flow using automatic differentiation. Computational Mechanics 43.6, pp. 839-846, 2009.
• A. Griewank and A. Walther. Evaluating derivatives: principles and techniques of algorithmic differentiation. Vol. 105. SIAM, 2008.

Gudrun had two podcast conversations at the FEniCS18 workshop in Oxford (21.-23. March 2018). FEniCS is an open source computing platform for solving partial differential equations with Finite Element methods.

This is the first of the two episodes from Oxford in 2018.

Roisin Hill works at the National University of Ireland in Galway on the west coast of Ireland. The university has 19.000 students and 2.000 staff. Roisin is a PhD student in Numerical Analysis at the School of Mathematics, Statistics & Applied Mathematics. Gudrun met her at her poster about Balanced norms and mesh generation for singularly perturbed reaction-diffusion problems. This is a collaboration with Niall Madden who is her supervisor in Galway.

The name of the poster refers to three topics which are interlinked in their research. Firstly, water flow is modelled as a singularly perturbed equation in a one-dimensional channel. Due to the fact that at the fluid does not move at the boundary there has to be a boundary layer in which the flow properties change. This might occur very rapidly. So, the second topic is that depending on the boundary layer the problem is singularly perturbed and in the limit it is even ill-posed. When solving this equation numerically, it would be best, to have a fine mesh at places where the error is large. Roisin uses a posteriori information to see where the largest errors occur and changes the mesh accordingly. To choose the best norm for errors is the third topic in the mix and strongly depends on the type of singularity.

More precisely as their prototypical test case they look for u(x) as the numerical solution of the problem

−ε2u″(x)+b(x)u(x) = f(x)onΩ=(0,1)

u(0) = u(1) = 0

for given functions b(x) and f(x). It is singularly perturbed in the sense that the positive real parameter ε may be arbitrarily small. If we formally set ε = 0, then it is ill-posed. The numercial schemes of choice are finite element methods - implemented in FEniCS with linear and quadratic elements. The numerical solution and its generalisations to higher-dimensional problems, and to the closely related convection-diffusion problem, presents numerous mathematical and computational challenges, particularly as ε → 0. The development of algorithms for robust solution is the subject of intense mathematical investigation. Here “robust” means two things:

1. The algorithm should yield a “reasonable” solution for all ranges of ε, including resolving any layers present;
2. The mathematical analysis of the method should be valid for all ranges of ε.

In order to measure the error, the energy norm sounds like a good basis - but as ε^2 → 0 the norm → 0 with order ε . They were looking for an alternative which they found in the literature as the so-called balanced norm. That remains O(1) as ε → 0. Therefore, it turns out that the balanced norm is indeed a better basis for error measurement.

After she finished school Roisin became an accountant. She believed what she was told: if you are good at mathematics, accountancy is the right career. Later her daughter became ill and had to be partially schooled at home. This was the moment when Roisin first encountered applied mathematics and fell in love with the topic. Inspired by her daughter - who did a degree in general science specialising in applied mathematics - Roisin studied mathematics and is a PhD student now (since Sept. 2017). Her enthusiasm has created impressive results: She won a prestigious Postgraduate Scholarship from the Irish Research Council for her four year PhD program.

• R. Lin, M. Stynes: A balanced finite element method for singularly perturbed reaction diffusion problems. SIAM J. Numer. Anal., 50(5):2729–2743, 2012.
• T. Linß: Layer-adapted meshes for reaction-convection-diffusion problems, volume 1985 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010.
• H.-G. Roos, M. Stynes, L. Tobiska: Robust Numerical Methods for Singularly Perturbed Differential Equations, volume 24 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd edition, 2008.

• M. E. Rognes: Cerebral Fluid Flow, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 134, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2017.

In March 2018 Gudrun visited University College London and recorded three conversations with mathematicians working there.

Her first partner was Karen Page. She works in Mathematical Biology and is interested in mathematical models for pattern formation. An example would be the question why (and how) a human embryo develops five fingers on each hand. The basic information for that is coded into the DNA but how the pattern develops over time is a very complicated process which we understand only partly. Another example is the patterning of neurons within the vertebrate nervous system. The neurons are specified by levels of proteins. Binding of other proteins at the enhancer region of DNA decides whether a gene produces protein or not. This type of work needs a strong collaboration with biologists who observe certain behaviours and do experiments. Ideally they are interested in the mathematical tools as well.

One focus of Karen's work is the development of the nervous system in its embryonic form as the neural tube. She models it with the help of dynamical systems. At the moment they contain three ordinary differential equations for the temporal changes in levels of three proteins. Since they influence each other the system is coupled. Moreover a fourth protein enters the system as an external parameter. It is called sonic hedgehog (Shh). It plays a key role in regulating the growth of digits on limbs and organization of the brain. It has different effects on the cells of the developing embryo depending on its concentration.

Concerning the mathematical theory the Poincaré Bendixson theorem completely characterizes the long-time behaviour of two-dimensional dynamical systems. Working with three equations there is room for more interesting long-term scenarios. For example it is possible to observe chaotic behaviour.

Karen was introduced to questions of Mathematical Biology when starting to work on her DPhil. Her topic was Turing patterns. These are possible solutions to systems of Partial differential equations that are thermodynamically non-equilibrium. They develop from random perturbations about a homogeneous state, with the help of an input of energy.

Prof. Page studied mathematics and physics in Cambridge and did her DPhil in Oxford in 1999. After that she spent two years at the Institute for Advanced Study in Princeton and has been working at UCL since 2001.

• A. Turing: The Chemical Basis of Morphogenesis Philosophical Transactions of the Royal Society of London B. 237 (641): 37–72.1952
• J.D. Murray: Mathematical Biology. Springer Science & Business Media, 2013. ISBN 978-3-662-08539-4
• M. Cohen, K.M. Page e.a: A theoretical framework for the regulation of Shh morphogen-controlled gene expression. Development, 141(20), 3868-3878, 2014.
• N. Balaskas e.a.: Gene regulatory logic for reading the Sonic Hedgehog signaling gradient in the vertebrate neural tube. Cell. 148, 273-284, 2012.
• J. Panovska-Griffiths e.a.: A gene regulatory motif that generates oscillatory or multiway switch outputs. J. Roy. Soc. Interface. 10.79, 2013.

• L. Adlung: Systembiologie, Gespräch mit G. Thäter und S. Ritterbusch im Modellansatz Podcast, Folge 39, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016.
• Omega Tau-Podcast 069: Grundlagen der Zellbiologie
• Omega Tau-Podcast 072: Forschung in der Zellbiologie
• Konscience-Podcast 024, Kapitel 5: Das Hochlandgen aus "Wie kam das bloß durch die Ethikkommission?"

Martina Hofmanová has been working as a professor at the University of Bielefeld since October 2017. Previously, she was a Junior Professor at TU Berlin from February 2016 onwards, and before that an Assistant Lecturer, there.

She studied at the Charles University in Prague, and got her PhD in 2013 at the École normale supérieure de Cachan in Rennes. Her time in Germany started in 2013 when she moved to the Max Planck Institute for Mathematics in the Sciences in Leipzig as a postdoc.

Gudrun and Martina talk about randomness in the modeling of fluid motion. This topic is connected to the study of turbulent flow. Of course, we observe turbulence all around us, i.e. chaotic behaviour of the pressure and the velocity field in fluid flow. One example is the smoke pattern of a freshly extinguished candle. Its first part is laminar, then we observe transitional turbulent flow and fully turbulent one the further away the smoke travels. A second example is Rayleigh Bénard convection. Under the influence of a temperature gradient and gravity, one observes convection rolls when the temperature difference between bottom and top becomes large enough. If we look more closely, one can prescribe the motion as a mean flow plus random fluctuations. These fluctuations are difficult to measure but their statistical properties are reproduced more easily.

A general procedure in physics and science is to replace expensive time averages by ensemble averages, which can be calculated together on a parallel computer. The concept why this often works is the so-called ergodic hypothesis. To justify this from the mathematical side, the main problem is to find the right measure in the ensemble average.

In the model problem y′=b(y) one can see that the solution is continuously dependent on the initial condition and the solution operator has a semigroup property. For random initial conditions, one can construct the solution operator correspondingly.

Already with this toy problem one sees that the justification of using ensemble averages is connected to the well-posedness of the problem. In general, this is not apriori known. The focus of Martina's work is to find the existence of steady solutions for the compressible flow system, including stochastic forces with periodic boundary conditions (i.e. on the torus).

At the moment, we know that there are global weak solutions but only local (in time) strong solutions.

It turned out that the right setting to study the problem are so-called dissipative martingale solutions: Unfortunately, in this setting, the velocity is not smooth enough to be a stochastic process. But the energy inequality can be proved. The proof rests on introducing artificial dissipation in the mass conservation, and a small term with higher order regularity for the density. Then, the velocity is approximated through a Faedo-Galerkin approximation and a lot of independent limiting processes can be carried out successfully.

The project is a collaboration with Dominic Breit and Eduard Feireisl.

• M. Hofmanová: Stochastic partial differential equations, Lecture notes, Technical University of Berlin, 2016.
• D. Breit, E. Feireisl, M. Hofmanová, B. Maslowski: Stationary solutions to the compressible Navier-Stokes system driven by stochastic forces, preprint, 2016.
• D. Breit, E. Feireisl, M. Hofmanová: Local strong solutions to the stochastic compressible Navier-Stokes system, preprint, 2016.
• D. Breit, E. Feireisl, M. Hofmanová: Compressible fluids driven by stochastic forcing: The relative energy inequality and applications, Comm. Math. Physics 350, 443-473, 2017.
• D. Breit, M. Hofmanová: Stochastic Navier-Stokes equations for compressible fluids, Indiana Univ. Math. J. 65 (4), 1183-1250, 2016.

• N. Vercauteren: Lokale Turbulenzen, Gespräch mit S. Ritterbusch im Modellansatz Podcast, Folge 144, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2017. http://modellansatz.de/lokale-turbulenzen
• B.Valsler, D. Ansell: The Science of Turbulence, The Naked Scientists Podcast, 2010.

Gudrun is speaking with the portuguese engineer Bruno Pousinho. He has been a student of the Energy Technologies (ENTECH) Master program. This is an international and interdisciplinary program under the label of the European Institute of Innovation and Technology (EIT) inbetween a number of European technical universities.

Bruno spent his second master year at the Karlsruhe Institute of Technology (KIT). Gudrun had the role of his supervisor at KIT while he worked on his Master's thesis at the Chair of Renewable and Sustainable Energy Systems (ENS) at TUM in Garching. His direct contact person there was Franz Christange from the group of Prof. Thomas Hamacher.

Renewable energy systems are a growing part of the energy mix. In Germany between 1990 and 2016 it grew from 4168 GW to 104024 GW. This corresponds to an annual power consumption share of 3.4% and 31.7%, respectively. But in the related research this means a crucial shift. The conventional centralized synchronous machine dominated models have to be exchanged for decentralized power electronic dominated networks - so-called microgrids. This needs collaboration of mechanical and electrical engineers. The interdisciplinary group at TUM has the goal to work on modeling future microgrids in order to easily configure and simulate them.

One additional factor is that for most renewable energy systems it is necessary to have the right weather conditions. Moreover, there is always the problem of reliability. Especially for Photovoltaics (PV) and wind turbines Weather phenomena as solar irradiation, air temperature and wind speed have to be known in advance in order to plan for these types of systems.

There are two fundamentally different approaches to model weather data. Firstly the numerical weather and climate models, which provide the weather forecast for the next days and years. Secondly, so-called weather generators. The numerical models are very complex and have to run on the largest computer systems available. For that in order to have a simple enough model for planning the Renewable energy resources (RER) at a certain place weather generators are used. They produce synthetic weather data on the basis of the weather conditions in the past. They do not predict/forecast the values of a specific weather phenomenon for a specific time but provides random simulations whose outputs show the same or very similar distributional properties as the measured weather data in the past.

The group in Garching wanted to have a time dynamic analytical model. The model is time continuous which grant it the ability of having any time sampling interval. This means it wanted to have a system of equations for the generation of synthetic weather data with as few as possible parameters. When Bruno started his work, there existed a model for Garching (developped by Franz Christange) with about 60 parameters. The aim of Bruno's work was to reduce the number of parameters and to show that the general concept can be used worldwide, i.e. it can adapt to different weather data in different climate zones. In the thesis the tested points range from 33º South to 40º North.

In the synthesis of the weather generator the crucial tool is to use stochastic relations. Mostly the standard normal distribution is applied and shaped for the rate of change and corelation between RER. In particular this means that it describes the fundamental behavior of weather (mean, standard deviation, time- and cross-correlation) and introduces them into white noise in an analytical way. This idea was first introduced for crop estimation by Richardson in 1985. Time-dependence works on different time scales - through days and through seasons, e.g..

In the Analysis it is then necessary to parametrize the measured weather data and to provide a parameter set to the weather model.

Bruno started his Master course in Lisbon at Instituto Superior tecnico (IST). In his second year he changed to KIT in Karlsruhe and put his focus on Energy systems. In his thesis he uses a lot of mathematics which he learned during his Bachelor education and had to recall and refresh.

The results of the project are published in the open source model 'solfons' in Github, which uses Python and was developed in MATLAB.

• F. Christange & T. Hamacher: Analytical Modeling Concept for Weather Phenomena as Renewable Energy Resources, in IEEE International Conference on Renewable Energy Research and Applications (ICRERA), 2016. doi: 10.1109/ICRERA.2016.7884551
• P. Ailliot, D. Allard, P. Naveau, C. D. Beaulieu, R. Cedex: Stochastic weather generators, an overview of weather type models, Journal de la Société Française de Statistique, Vol. 156, No 1, pp. 1-14, 2015.
• C.L. Wiegand, A.J. Richardson: Leaf area, light interception, and yield estimates from spectral components analysis, Agron. J., 76, 543, 1984.
• solfons: Artificial wheater data for energy system modeling, Software at GitHub.

• S. Seier, T. Alexandrin: Blindstrom - Der Energie Podcast, 2016-2017.
• M. Völter, V. Hagenmeyer: Stromnetze, ein Überblick, omega tau Podcast, Episode 246, 2017.
• K. A. Zach, L. Bodingbauer: Energiespeicher, PHS186 in der Physikalischen Soiree, 2013.
• F. Trieb, T. Pritlive: Energie der Zukunft, RZ033 im Raumzeit Podcast, Metaebene Personal Media, 2012.

Gudrun Thäter and Jonathan Rollin talk about their plans for the course Advanced Mathematics (taught in English) for mechanical engineers at the Karlsruhe Institute of Technology (KIT). The topics of their conversation are relevant in the mathematical education for engineers in general (though the structure of courses differs between universities). They discuss

• how to embrace university mathematics,
• how to study,
• what is the structure of the educational program and
• what topics will be covered in the first semester in Karlsruhe.

For students starting an engineering study course it is clear, that a mathematical education will be an important part. Nevertheless, most students are not aware that their experiences with mathematics at school will not match well with the mathematics at university. This is true in many ways. Mathematics is much more than calculations. As the mathematical models become more involved, more theoretical knowledge is needed in order to learn how and why the calculations work. In particular the connections among basic ideas become more and more important to see why certain rules are valid. Very often this knowledge also is essential since the rules need to be adapted for different settings.

In their everyday work, engineers combine the use of well-established procedures with the ability to come up with solutions to yet unsolved problems. In our mathematics education, we try to support that skills insofar as we train certain calculations with the aim that they become routine for the future engineers. But we also show the ideas and ways how mathematicians came up with these ideas and how they are applied again and again at different levels of abstraction. This shall help the students to become creative in their engineering career.

Moreover seeing how the calculation procedures are derived often helps to remember them. So it makes a lot of sense to learn about proofs behind calculations, even if we usually do not ask to repeat proofs during the written exam at the end of the semester.

The course is structured as 2 lectures, 1 problem class and 1 tutorial per week. Moreover there is a homework sheet every week. All of them play their own role in helping students to make progress in mathematics.

The lecture is the place to see new material and to learn about examples, connections and motivations. In this course there are lecture notes which cover most topics of the lecture (and on top of that there are a lot of books out there!). So the lecture is the place where students follow the main ideas and take these ideas to work with the written notes of the lecture later on.

The theory taught in the lecture becomes more alive in the problem classes and tutorials. In the problem classes students see how the theory is applied to solve problems and exercises. But most importantly, students must solve problems on their own, with the help of the material from the lecture. Only in this way they learn how to use the theory. Very often the problems seem quite hard in the sense that it is not clear how to start or proceed. This is due to the fact that students are still learning to translate the information from the lecture to a net of knowledge they build for themselves. In the tutorial the tutor and the fellow students work together to find first steps onto a ladder to solving problems on the homework.

Gudrun and Jonathan love mathematics. But from their own experience they can understand why some of the students fear mathematics and expect it to be too difficult to master. They have the following tips:

• just take one step after the other, and do not give up too early
• discuss problems, questions and topics of the lecture with fellow students - talking about mathematics helps to understand it
• teach fellow students about things you understand - you will be more confident with your arguments, or find some gaps to fix
• take time to think about mathematics and the homework problems
• sit down after the lecture, and repeat the arguments and ideas in your own words in order to make them your own
• use the problem classes and tutorials to ask questions

In the lecture course, students see the basic concepts of different mathematical fields. Namely, it covers calculus, linear algebra, numerics and stochastics. Results from all these fields will help them as engineers to calculate as well as to invent. There is no standard or best way to organize the topics since there is a network of connections inbetween results and a lot of different ways to end up with models and calculation procedures. In the course in Karlsruhe in the first semester we mainly focus on calculus and touch the following subjects:

• Numbers
• Methods of proof
• Complex numbers
• Sequences and convergence
• Functions and continuity
• Series
• Differential calculus of one real variable
• Integral calculus
• Numerical integration
• Elementary differential equations

All of these topics have applications and typical problems which will be trained in the problem class. But moreover they are stepping stones in order to master more and more complex problems. This already becomes clear during the first semester but will become more clear at the end of the course.

• K. F. Riley, M. P. Hobson, S. J. Bence: Mathematical Methods for Physics and Engineering. Cambridge University Press.
• K. F. Riley, M. P. Hobson: Foundation Mathematics for the Physical Sciences. Cambridge University Press.
• T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K. Lichtenegger, H. Stachel: Mathematik.Spektrum Akademischer Verlag, Heidelberg (in German).
• J. Stewart: Calculus, Early Transcendentals. Brooks/Cole Publishing Company.
• K. Burg, H. Haf, F. Wille: Höhere Mathematik für Ingenieure. Volumes I-III. Teubner Verlag, Stuttgart (in German).
• E. Kreyszig: Advanced Engineering Mathematics. John Wiley & Sons.
• E.W. Swokowski, M. Olinick, D. Pence, J.A. Cole: Calculus. PWS Publishing Company. Boston.

• F. Hettlich: Höhere Mathematik, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 34, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2014.
• J. Eilinghoff: Analysis, Gespräch mit S. Ritterbusch im Modellansatz Podcast, Folge 36, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2014.

This is one of two conversations which Gudrun Thäter recorded alongside the conference Women in PDEs which took place at our faculty in Karlsruhe on 27-28 April 2017. Marie Elisabeth Rognes was one of the seven invited speakers.

Marie is Chief Research Scientist at the Norwegian research laboratory Simula near Oslo. She is Head of department for Biomedical Computing there. Marie got her university education with a focus on Applied Mathematics, Mechanics and Numerical Physics as well as her PhD in Applied mathematics at the Centre for Mathematics for Applications in the Department of Mathematics at the University of Oslo.

Her work is devoted to providing robust methods to solve Partial Differential Equations (PDEs) for diverse applications. On the one hand this means that from the mathematical side she works on numerical analysis, optimal control, robust Finite Element software as well as Uncertainty quantification while on the other hand she is very much interested in the modeling with the help of PDEs and in particular Mathematical models of physiological processes. These models are useful to answer What if type-questions much more easily than with the help of laboratory experiments.

In our conversation we discussed one of the many applications - Cerebral fluid flow, i.e. fluid flow in the context of the human brain.

Medical doctors and biologists know that the soft matter cells of the human brain are filled with fluid. Also the space between the cells contains the water-like cerebrospinal fluid. It provides a bath for human brain. The brain expands and contracts with each heartbeat and appoximately 1 ml of fluid is interchanged between brain and spinal area. What the specialists do not know is: Is there a circulation of fluid? This is especially interesting since there is no traditional lymphatic system to transport away the biological waste of the brain (this process is at work everywhere else in our body). So how does the brain get rid of its litter? There are several hyotheses:

• Diffusion processes,
• Fast flow (and transport) along the space near blood vessel,
• Convection.

The aim of Marie's work is to numerically test these (and other) hypotheses. Basic testing starts on very idalised geometries. For the overall picture one useful simplified geometry is the annulus i.e. a region bounded by two concentric circles. For the microlevel-look a small cube can be the chosen geometry.

As material law the flow in a porous medium which is based on Darcy flow is the starting point - maybe taking into account the coupling with an elastic behaviour on the boundary.

The difficult non-mathematical questions which have to be answered are:

• How to use clinical data for estabilishing and testing models
• How to prescribe the forces

In the near future she hopes to better understand the multiscale character of the processes. Here especially for embedding 1d- into 3d-geometry there is almost no theory available.

For the project Marie has been awarded a FRIPRO Young Research Talents Grant of the Research Council of Norway (3 years - starting April 2016) and the very prestegious ERC Starting Grant (5 years starting - 2017).

• M.E. Rognes: Mathematics that cures us.TEDxOslo 3 May 2017
• Young academy of Norway
• ERC Starting Grant: Mathematical and computational foundations for modeling cerebral fluid flow 5 years
• P.E. Farrell e.a.: Dolfin adjoint (Open source software project)
• FEniCS computing platform for PDEs (Open source software project)
• Wikipedia on FEniCS
• Collection of relevant literature implemented in FEniCS