Snapshots of Modern Mathematics from Oberwolfach
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Browsing Snapshots of Modern Mathematics from Oberwolfach by Subject "Numerics and Scientific Computing"
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- ItemThe adaptive finite element method(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Gallistl, DietmarComputer simulations of many physical phenomena rely on approximations by models with a finite number of unknowns. The number of these parameters determines the computational effort needed for the simulation. On the other hand, a larger number of unknowns can improve the precision of the simulation. The adaptive finite element method (AFEM) is an algorithm for optimizing the choice of parameters so accurate simulation results can be obtained with as little computational effort as possible.
- ItemAnalogue mathematical instruments: Examples from the “theoretical dynamics” group (France, 1948–1964)(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Petitgirard, LoïcThroughout the history of dynamical systems, instruments have been used to calculate and visualize (approximate) solutions of differential equations. Here we describe the approach of a group of physicists and engineers in the period 1948–1964, and we give examples of the specific (analogue) mathematical instruments they conceived and used. These examples also illustrate how their analogue culture and practices faced the advent of the digital computer, which appeared at that time as a new instrument, full of promises.
- ItemChaos and chaotic fluid mixing(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Solomon, TomVery simple mathematical equations can give rise to surprisingly complicated, chaotic dynamics, with behavior that is sensitive to small deviations in the initial conditions. We illustrate this with a single recurrence equation that can be easily simulated, and with mixing in simple fluid flows.
- ItemComputational Optimal Transport(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Solomon, JustinOptimal transport is the mathematical discipline of matching supply to demand while minimizing shipping costs. This matching problem becomes extremely challenging as the quantity of supply and demand points increases; modern applications must cope with thousands or millions of these at a time. Here, we introduce the computational optimal transport problem and summarize recent ideas for achieving new heights in efficiency and scalability.
- ItemComputing the long term evolution of the solar system with geometric numerical integrators(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Fiorelli Vilmart, Shaula; Vilmart, GillesSimulating the dynamics of the Sun–Earth–Moon system with a standard algorithm yields a dramatically wrong solution, predicting that the Moon is ejected from its orbit. In contrast, a well chosen algorithm with the same initial data yields the correct behavior. We explain the main ideas of how the evolution of the solar system can be computed over long times by taking advantage of so-called geometric numerical methods. Short sample codes are provided for the Sun–Earth–Moon system.
- ItemDarcy's law and groundwater flow modelling(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Schweizer, BenFormulations of natural phenomena are derived, sometimes, from experimentation and observation. Mathematical methods can be applied to expand on these formulations, and develop them into better models. In the year 1856, the French hydraulic engineer Henry Darcy performed experiments, measuring water flow through a column of sand. He discovered and described a fundamental law: the linear relation between pressure difference and flow rate – known today as Darcy’s law. We describe the law and the evolution of its modern formulation. We furthermore sketch some current mathematical research related to Darcy’s law.
- ItemData assimilation: mathematics for merging models and data(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Morzfeld, Matthias; Reich, SebastianWhen you describe a physical process, for example, the weather on Earth, or an engineered system, such as a self-driving car, you typically have two sources of information. The first is a mathematical model, and the second is information obtained by collecting data. To make the best predictions for the weather, or most effectively operate the self-driving car, you want to use both sources of information. Data assimilation describes the mathematical, numerical and computational framework for doing just that.
- ItemDeep Learning and Inverse Problems(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Arridge, Simon; de Hoop, Maarten; Maass, Peter; Öktem, Ozan; Schönlieb, Carola; Unser, MichaelBig data and deep learning are modern buzz words which presently infiltrate all fields of science and technology. These new concepts are impressive in terms of the stunning results they achieve for a large variety of applications. However, the theoretical justification for their success is still very limited. In this snapshot, we highlight some of the very recent mathematical results that are the beginnings of a solid theoretical foundation for the subject.
- ItemDrugs, herbicides, and numerical simulation(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Benner, Peter; Mena, Hermann; Schneider, RenéThe Colombian government sprays coca fields with herbicides in an effort to reduce drug production. Spray drifts at the Ecuador-Colombia border became an international issue. We developed a mathematical model for the herbicide aerial spray drift, enabling simulations of the phenomenon.
- ItemFast Solvers for Highly Oscillatory Problems(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Barnett, AlexWaves of diverse types surround us. Sound, light and other waves, such as microwaves, are crucial for speech, mobile phones, and other communication technologies. Elastic waves propagating through the Earth bounce through the Earth’s crust and enable us to “see” thousands of kilometres in depth. These propagating waves are highly oscillatory in time and space, and may scatter off obstacles or get “trapped” in cavities. Simulating these phenomena on computers is extremely important. However, the achievable speeds for accurate numerical modelling are low even on large modern computers. Our snapshot will introduce the reader to recent progress in designing algorithms that allow for much more rapid solutions.
- ItemHigh performance computing on smartphones(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Patera, Anthony T.; Urban, KarstenNowadays there is a strong demand to simulate even real-world engineering problems on small computing devices with very limited capacity, such as a smartphone. We explain, using a concrete example, how we can obtain a reduction in complexity – to enable such computations – using mathematical methods.
- ItemIs it possible to predict the far future before the near future is known accurately?(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Gander, Martin J.It has always been the dream of mankind to predict the future. If the future is governed by laws of physics, like in the case of the weather, one can try to make a model, solve the associated equations, and thus predict the future. However, to make accurate predictions can require extremely large amounts of computation. If we need seven days to compute a prediction for the weather tomorrow and the day after tomorrow, the prediction arrives too late and is thus not a prediction any more. Although it may seem improbable, with the advent of powerful computers with many parallel processors, it is possible to compute a prediction for tomorrow and the day after tomorrow simultaneously. We describe a mathematical algorithm which is designed to achieve this.
- ItemMathematics plays a key role in scientific computing(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Shu, Chi-WangI attended a very interesting workshop at the research center MFO in Oberwolfach on “Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws”. The title sounds a bit technical, but in plain language we could say: The theme is to survey recent research concerning how mathematics is used to study numerical algorithms involving a special class of equations. These equations arise from computer simulations to solve application problems including those in aerospace engineering, automobile design, and electromagnetic waves in communications as examples. This topic belongs to the general research area called “scientific computing”.
- ItemMathematische Modellierung von Krebswachstum(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Engwer, Christian; Knappitsch, MarkusKrebs ist eine der größten Herausforderungen der modernen Medizin. Der WHO zufolge starben 2012 weltweit 8,2 Millionen Menschen an Krebs. Bis heute sind dessen molekulare Mechanismen nur in Teilen verstanden, was eine erfolgreiche Behandlung erschwert. Mathematische Modellierung und Computersimulationen können helfen, die Mechanismen des Tumorwachstums besser zu verstehen. Sie eröffnen somit neue Chancen für zukünftige Behandlungsmethoden. In diesem Schnappschuss steht die mathematische Modellierung von Glioblastomen im Fokus, einer Klasse sehr agressiver Tumore im menschlichen Gehirn.
- ItemMixed-dimensional models for real-world applications(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Nordbotten, Jan MartinWe explore mathematical models for physical problems in which it is necessary to simultaneously consider equations in different dimensions; these are called mixed-dimensional models. We first give several examples, and then an overview of recent progress made towards finding a general method of solution of such problems.
- ItemModeling communication and movement: from cells to animals and humans(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Eftimie, RalucaCommunication forms the basis of biological interactions. While the use of a single communication mechanism (for example visual communication) by a species is quite well understood, in nature the majority of species communicate via multiple mechanisms. Here, I review some mathematical results on the unexpected behaviors that can be observed in biological aggregations where individuals interact with each other via multiple communication mechanisms.
- ItemModelling the spread of brain tumours(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Swan, Amanda; Murtha, AlbertThe study of mathematical biology attempts to use mathematical models to draw useful conclusions about biological systems. Here, we consider the modelling of brain tumour spread with the ultimate goal of improving treatment outcomes.
- ItemMolecular Quantum Dynamics(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Hagedorn, George A.; Lasser, CarolineWe provide a brief introduction to some basic ideas of Molecular Quantum Dynamics. We discuss the scope, strengths and main applications of this field of science. Finally, we also mention open problems of current interest in this exciting subject.
- ItemThe mystery of sleeping sickness – why does it keep waking up?(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Funk, SebastianSleeping sickness is a neglected tropical disease that affects rural populations in Africa. Deadly when untreated, it is being targeted for elimination through case finding and treatment. Yet, fundamental questions about its transmission cycle remain unanswered. One of them is whether transmission is limited to humans, or whether other species play a role in maintaining circulation of the disease. In this snapshot, we introduce a mathematical model for the spread of Trypanosoma brucei, the parasite responsible for causing sleeping sickness, and present some results based on data collected in Cameroon. Understanding how important animals are in harbouring Trypanosoma brucei that can infect humans is important for assessing whether the disease could be reintroduced in human populations even after all infected people have been successfully treated.
- ItemOn radial basis functions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Buhmann, Martin; Jäger, JaninMany sciences and other areas of research and applications from engineering to economics require the approximation of functions that depend on many variables. This can be for a variety of reasons. Sometimes we have a discrete set of data points and we want to find an approximating function that completes this data; another possibility is that precise functions are either not known or it would take too long to compute them explicitly. In this snapshot we want to introduce a particular method of approximation which uses functions called radial basis functions. This method is particularly useful when approximating functions that depend on very many variables. We describe the basic approach to approximation with radial basis functions, including their computation, give several examples of such functions and show some applications.